Goku blasted off of the rock that he was perched on, intent on ending the fight before Vegeta powered up far enough to force him to use the Kaio-ken times four. First, he punched Vegeta in the face, then he kicked him into the air. The moment that Vegeta managed to get his bearings again to gather enough energy for any sort of attack, it was already too late. Goku's foot met his face, throwing Vegeta back down, two or three feet deep into one of the nearby rocks.
It looked pretty devastating from my angle, but Vegeta wasn't down yet. He reduced the rock that Goku had kicked him into to rubble in a burst of rage. Unfortunately for him, he simply wasn't as fast as Goku was currently. Unlike in the anime, Goku had full control over his Kaio-ken Times Three transformation, and he was putting it to good use. He kicked Vegeta again, and when Vegeta came back at him with a punch instead, he slammed his fist into Vegeta's stomach instead, causing the Prince to double over in pain.
Everything Vegeta tried to throw at Goku was given back to him with interest, and after taking an impressive beating, Vegeta realized the truth. Goku was stronger than him right now.
'His power... He's so strong... blocking my attacks like they're nothing!' Vegeta thought, doubled over in pain.
"You really don't give up, huh?" Goku said. He was winded, but unlike in the Anime, he'd probably be able to keep this level of power going for hours. Vegeta didn't stand a chance against him in a head-to-head, physical fight. He might have been able to win in some sort of war of attrition, but the Saiyan Prince was far too proud to attempt such a feat, even given how injured he was.
Groaning, Vegeta started muttering to himself, "This shouldn't be happening. I don't understand! I am a super elite! Now he's sealed his fate! This planet is history! I'LL DESTROY EVERYTHING! EVERYTHING! DO YOU HEAR ME KAKAROT!? YOU'RE FINISHED!" Vegeta was having a breakdown. He'd never been defeated in a battle with his life on the line, especially not by some low-class scum like Kakarot. By the end of his mutterings, they'd turned into hysterical screamings instead.
Vegeta's idol for his whole life had been his father. The undefeated Saiyan King. He knew that it was his duty to live up to the legacy that his father had left him, even if there wasn't a Saiyan race left to rule, he'd be the strongest one out there... now, this low-class scum was taking even that from him. Unbeknownst to Vegeta, Goku's own father had taken the title of 'Strongest Saiyan' from King Vegeta before the planet had exploded, and perhaps this was the universe's way of repeating the past, in hopes of a better future or something, but it didn't really matter.
It was Vegeta's destiny to be surpassed by my father here, just as his father was surpassed by Goku's long ago. That jealousy would inevitably lead the duo to the final confrontation with Frieza on Namek, where the past would finally be put to rest, just as Bardock's original surpassing of the King had begun this whole mess by attracting Frieza's attention.
The universe had a funny way of aligning fates...
'Who does this fool think he's dealing with?' Vegeta thought to himself, wiping the blood that he had vomited out during Goku's last attack from his mouth, 'he's beneath me! I'll show him! I'll show them all! I'LL REDUCE THIS PLACE TO ASHES!' He thought. Enraged, Vegeta screamed, his power spiking further with every passing moment.
"Kakarot! Dodge this next attack if you can, but know that if you do, your precious little planet will be gone!" He shouted.
"No! You can't!" Goku shouted. Vegeta was too far gone to listen, though. He flew up into the sky, purple ki gathering around him as he prepared to release his full-powered Galick Gun.
"The choice is yours, Kakarot! But either way, you are going to perish!" Vegeta screamed, cackling.
"He's actually serious!" Goku realized, "I guess this is it! Now or never! Kaio-Ken times three! KA-ME"
"RAAH!"
"HA-"
"Now, Kakarot, prepare yourself for oblivion!" Vegeta shouted.
"ME-"
"GALICK GUN! RAAAH!"
"HAAAA!"
"SHINING TUESDAY!" Raditz added his energy to Goku's blast. It may have gone against Saiyan etiquette, but Goku wasn't going to complain, since the entire planet was literally at stake. I chose not to add anything to it, since it wouldn't have mattered.
"YOU THINK THIS WILL STOP ME?!" Vegeta screamed, "I AM THE PRINCE OF ALL SAIYANS! AND I WILL NOT BE DEFEATED!" Honestly, I'd not expected Vegeta to be capable of putting up anything more than a token resistance against two Kaio-ken Times Threes, but he must have been running on hysterical strength now, because his Galick Gun seemed to be even with the two other beams. It was incredible. Unfortunately for Vegeta, Goku wasn't satisfied with just 'matching' his blast.
"KAIO-KEN ATTACK!!!!! TIMES FOUR!!!" Goku screamed. In an instant, the balance shifted. Now, Vegeta's Galick Gun was being gobbled up with earnest by the two other beams.
"NOO! AAGH! NOOOO!!!!" Vegeta screamed, a horrifying, gut wrenching noise as the two beams gobbled up his Galick Gun. But an instant before he was entirely swallowed, something about Vegeta's scream changed. It got even more heart wrenching, somehow. Like his entire existence was riding on this one blast.
'Shay. Don't let him die.' I thought to my trusty assistant as Vegeta's Galick Gun pushed back against the combined ki waves from Goku and Raditz for its final effort before being snuffed out.
...
In the upper atmosphere, a few moments later.
'So... this is how I die, huh?' Vegeta thought as his body was eaten up by the combined power of Goku and Raditz. 'As a weakling... Heh. I don't even have the energy to pull myself out of this beam... I suppose dying as a warrior is better than living on as a weakling, though. I'm sorry... father...'
"Not happening." Shay appeared, just as Vegeta was about to fly into space. At this point, he was only holding onto consciousness through sheer force of will. She magically plucked him out of the blast and fixed his body to the point where he'd be able to get back to the Frieza force's headquarters to get healed.
"W-why?" Vegeta croaked out, barely capable of forming the words. He needed to know why someone would save him, though.
"Because you're not dying yet. Mistress doesn't want you to. You have potential, still." Shay told him, as if it was the most obvious thing in the world. Honestly, to her, it was. She dug something out of his pocket before hitting a couple buttons, "Let's get Nappa, gone, too." She decided, exactly as her Mistress had told her to decide.
While technically in the anime, Vegeta came back after being hit by that blast, I had to change around a few things to account for the combination of Raditz being there, and Gohan not being there (due to the whole tail thing). Had Vegeta attempted to use his 'fake moon' again, it would've been a waste of his time, with Raditz there, so I had him pour all of his remaining strength into lasting a bit longer with his Galick Gun, instead.
A lot of people might be mad about how it turned out, but I'm actually pretty satisfied with how the Saiyan Invasion arc turned out. It all came together pretty smoothly.
Remember to give feedback and support!
The aftermath is next! Then, off to Namek!