\(\int (c+d x^4)^3 (a+b x^8)^p \, dx\) [302]

Optimal result

Integrand size = 19, antiderivative size = 209 \[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\frac {3 c d^2 x \left (a+b x^8\right )^{1+p}}{b (9+8 p)}+\frac {d^3 x^5 \left (a+b x^8\right )^{1+p}}{b (13+8 p)}-\frac {c \left (3 a d^2-b c^2 (9+8 p)\right ) x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )}{b (9+8 p)}-\frac {d \left (5 a d^2-3 b c^2 (13+8 p)\right ) x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )}{5 b (13+8 p)} \] Output:

3*c*d^2*x*(b*x^8+a)^(p+1)/b/(9+8*p)+d^3*x^5*(b*x^8+a)^(p+1)/b/(13+8*p)-c*( 
3*a*d^2-b*c^2*(9+8*p))*x*(b*x^8+a)^p*hypergeom([1/8, -p],[9/8],-b*x^8/a)/b 
/(9+8*p)/((1+b*x^8/a)^p)-1/5*d*(5*a*d^2-3*b*c^2*(13+8*p))*x^5*(b*x^8+a)^p* 
hypergeom([5/8, -p],[13/8],-b*x^8/a)/b/(13+8*p)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.66 \[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\frac {1}{195} x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (195 c^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+d x^4 \left (117 c^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+5 d x^4 \left (13 c \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )+3 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {13}{8},-p,\frac {21}{8},-\frac {b x^8}{a}\right )\right )\right )\right ) \] Input:

Integrate[(c + d*x^4)^3*(a + b*x^8)^p,x]
 

Output:

(x*(a + b*x^8)^p*(195*c^3*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)] + 
d*x^4*(117*c^2*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)] + 5*d*x^4*(1 
3*c*Hypergeometric2F1[9/8, -p, 17/8, -((b*x^8)/a)] + 3*d*x^4*Hypergeometri 
c2F1[13/8, -p, 21/8, -((b*x^8)/a)]))))/(195*(1 + (b*x^8)/a)^p)
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx\)

\(\Big \downarrow \) 1770

\(\displaystyle \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^pdx\)

Input:

Int[(c + d*x^4)^3*(a + b*x^8)^p,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1770

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] 
 :> Unintegrable[(d + e*x^n)^q*(a + c*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, 
n, p, q}, x] && EqQ[n2, 2*n]
 

Maple [F]

\[\int \left (x^{4} d +c \right )^{3} \left (b \,x^{8}+a \right )^{p}d x\]

Input:

int((d*x^4+c)^3*(b*x^8+a)^p,x)
 

Output:

int((d*x^4+c)^3*(b*x^8+a)^p,x)
 

Fricas [F]

\[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{3} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((d*x^4+c)^3*(b*x^8+a)^p,x, algorithm="fricas")
 

Output:

integral((d^3*x^12 + 3*c*d^2*x^8 + 3*c^2*d*x^4 + c^3)*(b*x^8 + a)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\text {Timed out} \] Input:

integrate((d*x**4+c)**3*(b*x**8+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{3} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((d*x^4+c)^3*(b*x^8+a)^p,x, algorithm="maxima")
 

Output:

integrate((d*x^4 + c)^3*(b*x^8 + a)^p, x)
 

Giac [F]

\[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{3} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((d*x^4+c)^3*(b*x^8+a)^p,x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)^3*(b*x^8 + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\int {\left (b\,x^8+a\right )}^p\,{\left (d\,x^4+c\right )}^3 \,d x \] Input:

int((a + b*x^8)^p*(c + d*x^4)^3,x)
 

Output:

int((a + b*x^8)^p*(c + d*x^4)^3, x)
 

Reduce [F]

\[ \int \left (c+d x^4\right )^3 \left (a+b x^8\right )^p \, dx=\text {too large to display} \] Input:

int((d*x^4+c)^3*(b*x^8+a)^p,x)
 

Output:

(1536*(a + b*x**8)**p*a*c*d**2*p**3*x + 3456*(a + b*x**8)**p*a*c*d**2*p**2 
*x + 1560*(a + b*x**8)**p*a*c*d**2*p*x + 512*(a + b*x**8)**p*a*d**3*p**3*x 
**5 + 640*(a + b*x**8)**p*a*d**3*p**2*x**5 + 72*(a + b*x**8)**p*a*d**3*p*x 
**5 + 512*(a + b*x**8)**p*b*c**3*p**3*x + 1728*(a + b*x**8)**p*b*c**3*p**2 
*x + 1816*(a + b*x**8)**p*b*c**3*p*x + 585*(a + b*x**8)**p*b*c**3*x + 1536 
*(a + b*x**8)**p*b*c**2*d*p**3*x**5 + 4416*(a + b*x**8)**p*b*c**2*d*p**2*x 
**5 + 3336*(a + b*x**8)**p*b*c**2*d*p*x**5 + 351*(a + b*x**8)**p*b*c**2*d* 
x**5 + 1536*(a + b*x**8)**p*b*c*d**2*p**3*x**9 + 3648*(a + b*x**8)**p*b*c* 
d**2*p**2*x**9 + 1992*(a + b*x**8)**p*b*c*d**2*p*x**9 + 195*(a + b*x**8)** 
p*b*c*d**2*x**9 + 512*(a + b*x**8)**p*b*d**3*p**3*x**13 + 960*(a + b*x**8) 
**p*b*d**3*p**2*x**13 + 472*(a + b*x**8)**p*b*d**3*p*x**13 + 45*(a + b*x** 
8)**p*b*d**3*x**13 - 6291456*int((a + b*x**8)**p/(4096*a*p**4 + 14336*a*p* 
*3 + 16256*a*p**2 + 6496*a*p + 585*a + 4096*b*p**4*x**8 + 14336*b*p**3*x** 
8 + 16256*b*p**2*x**8 + 6496*b*p*x**8 + 585*b*x**8),x)*a**2*c*d**2*p**7 - 
36175872*int((a + b*x**8)**p/(4096*a*p**4 + 14336*a*p**3 + 16256*a*p**2 + 
6496*a*p + 585*a + 4096*b*p**4*x**8 + 14336*b*p**3*x**8 + 16256*b*p**2*x** 
8 + 6496*b*p*x**8 + 585*b*x**8),x)*a**2*c*d**2*p**6 - 80904192*int((a + b* 
x**8)**p/(4096*a*p**4 + 14336*a*p**3 + 16256*a*p**2 + 6496*a*p + 585*a + 4 
096*b*p**4*x**8 + 14336*b*p**3*x**8 + 16256*b*p**2*x**8 + 6496*b*p*x**8 + 
585*b*x**8),x)*a**2*c*d**2*p**5 - 88522752*int((a + b*x**8)**p/(4096*a*...