\(\int \frac {x^4 (a+b x^8)^p}{c+d x^4} \, dx\) [30]

Optimal result

Integrand size = 22, antiderivative size = 128 \[ \int \frac {x^4 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\frac {x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{8},-p,1,\frac {13}{8},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{5 c}-\frac {d x^9 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {9}{8},-p,1,\frac {17}{8},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{9 c^2} \] Output:

1/5*x^5*(b*x^8+a)^p*AppellF1(5/8,1,-p,13/8,d^2*x^8/c^2,-b*x^8/a)/c/((1+b*x 
^8/a)^p)-1/9*d*x^9*(b*x^8+a)^p*AppellF1(9/8,1,-p,17/8,d^2*x^8/c^2,-b*x^8/a 
)/c^2/((1+b*x^8/a)^p)
 

Mathematica [F]

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

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

Output:

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

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.

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

\(\Big \downarrow \) 1888

\(\displaystyle \int \frac {x^4 \left (a+b x^8\right )^p}{c+d x^4}dx\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1888

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x^8 + a)^p*x^4/(d*x^4 + c), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\frac {\left (b \,x^{8}+a \right )^{p} x -8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) a c p -\left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) a c -64 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{8}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) b c \,p^{2}-16 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{8}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) b c p -\left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{8}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) b c +64 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{4}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) a d \,p^{2}+8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{4}}{8 b d p \,x^{12}+b d \,x^{12}+8 b c p \,x^{8}+b c \,x^{8}+8 a d p \,x^{4}+a d \,x^{4}+8 a c p +a c}d x \right ) a d p}{d \left (8 p +1\right )} \] Input:

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

Output:

((a + b*x**8)**p*x - 8*int((a + b*x**8)**p/(8*a*c*p + a*c + 8*a*d*p*x**4 + 
 a*d*x**4 + 8*b*c*p*x**8 + b*c*x**8 + 8*b*d*p*x**12 + b*d*x**12),x)*a*c*p 
- int((a + b*x**8)**p/(8*a*c*p + a*c + 8*a*d*p*x**4 + a*d*x**4 + 8*b*c*p*x 
**8 + b*c*x**8 + 8*b*d*p*x**12 + b*d*x**12),x)*a*c - 64*int(((a + b*x**8)* 
*p*x**8)/(8*a*c*p + a*c + 8*a*d*p*x**4 + a*d*x**4 + 8*b*c*p*x**8 + b*c*x** 
8 + 8*b*d*p*x**12 + b*d*x**12),x)*b*c*p**2 - 16*int(((a + b*x**8)**p*x**8) 
/(8*a*c*p + a*c + 8*a*d*p*x**4 + a*d*x**4 + 8*b*c*p*x**8 + b*c*x**8 + 8*b* 
d*p*x**12 + b*d*x**12),x)*b*c*p - int(((a + b*x**8)**p*x**8)/(8*a*c*p + a* 
c + 8*a*d*p*x**4 + a*d*x**4 + 8*b*c*p*x**8 + b*c*x**8 + 8*b*d*p*x**12 + b* 
d*x**12),x)*b*c + 64*int(((a + b*x**8)**p*x**4)/(8*a*c*p + a*c + 8*a*d*p*x 
**4 + a*d*x**4 + 8*b*c*p*x**8 + b*c*x**8 + 8*b*d*p*x**12 + b*d*x**12),x)*a 
*d*p**2 + 8*int(((a + b*x**8)**p*x**4)/(8*a*c*p + a*c + 8*a*d*p*x**4 + a*d 
*x**4 + 8*b*c*p*x**8 + b*c*x**8 + 8*b*d*p*x**12 + b*d*x**12),x)*a*d*p)/(d* 
(8*p + 1))