[?] Use substitution (or other) method to get a specific form integral?

Questions
,
1

Hi, from a book need to solve an integral:

inte.png

, was handwrite solved, but Mathematica show the answer in different form, with code:

Integrate[x^3 (2 - x^2)^12, x]

and

Integrate[x^3 (2 - x^2)^m, x] /. m-> 12

get this:

mathematica-example-1.png
mathematica-example-1.png816×316 25 KB

but the question is, can get a form that looks like this?:

maple-1.png

which is the answer by handwrite and solution book. Thanks!

2

Yep. You can get that, but it is somewhat, say, artificial

jj = Integrate[x^3 (2 - x^2)^m, x] /. m -> 12


(jj /. (2 - x^2) -> p /. (2 + 13 x^2) -> q /. 
q -> ((2 - x^2) (PolynomialQuotient[2 + 13 x^2, 2 - x^2, x] + 
PolynomialRemainder[2 + 13 x^2, 2 - x^2, 
x]/(2 - x^2))) /. (2 - x^2) -> p // Expand) /. 
p -> 2 - x^2
3

This is sightly different from the method provided by Hans Dolhaine, and slightly different from the desired form, but it might suffice depending on requirements.

jj = Integrate[x^3 (2 - x^2)^12, x];
divisor = x^2 - 2;
pq = PolynomialQuotientRemainder[jj, divisor, x];
Factor[pq[[1]]]*divisor + pq[[2]]

(* Out[3011]= 4096/91 + 1/364 (-2 + x^2)^13 (2 + 13 x^2) *)