\(\int \frac {x^2 (a+b x^8)^p}{(c+d x^4)^2} \, dx\) [43]

Optimal result

Integrand size = 22, antiderivative size = 194 \[ \int \frac {x^2 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\frac {x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{8},-p,2,\frac {11}{8},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c^2}-\frac {2 d x^7 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{8},-p,2,\frac {15}{8},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{7 c^3}+\frac {d^2 x^{11} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {11}{8},-p,2,\frac {19}{8},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{11 c^4} \] Output:

1/3*x^3*(b*x^8+a)^p*AppellF1(3/8,2,-p,11/8,d^2*x^8/c^2,-b*x^8/a)/c^2/((1+b 
*x^8/a)^p)-2/7*d*x^7*(b*x^8+a)^p*AppellF1(7/8,2,-p,15/8,d^2*x^8/c^2,-b*x^8 
/a)/c^3/((1+b*x^8/a)^p)+1/11*d^2*x^11*(b*x^8+a)^p*AppellF1(11/8,2,-p,19/8, 
d^2*x^8/c^2,-b*x^8/a)/c^4/((1+b*x^8/a)^p)
 

Mathematica [F]

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

Integrate[(x^2*(a + b*x^8)^p)/(c + d*x^4)^2,x]
 

Output:

Integrate[(x^2*(a + b*x^8)^p)/(c + d*x^4)^2, 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.

\(\displaystyle \int \frac {x^2 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1888

\(\displaystyle \int \frac {x^2 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2}dx\)

Input:

Int[(x^2*(a + b*x^8)^p)/(c + d*x^4)^2,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]

\[\int \frac {x^{2} \left (b \,x^{8}+a \right )^{p}}{\left (x^{4} d +c \right )^{2}}d x\]

Input:

int(x^2*(b*x^8+a)^p/(d*x^4+c)^2,x)
 

Output:

int(x^2*(b*x^8+a)^p/(d*x^4+c)^2,x)
 

Fricas [F]

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

integrate(x^2*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**2*(b*x**8+a)**p/(d*x**4+c)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^8 + a)^p*x^2/(d*x^4 + c)^2, x)
 

Giac [F]

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

integrate(x^2*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="giac")
 

Output:

integrate((b*x^8 + a)^p*x^2/(d*x^4 + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int \frac {x^2\,{\left (b\,x^8+a\right )}^p}{{\left (d\,x^4+c\right )}^2} \,d x \] Input:

int((x^2*(a + b*x^8)^p)/(c + d*x^4)^2,x)
 

Output:

int((x^2*(a + b*x^8)^p)/(c + d*x^4)^2, x)
 

Reduce [F]

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

int(x^2*(b*x^8+a)^p/(d*x^4+c)^2,x)
 

Output:

int(((a + b*x**8)**p*x**2)/(c**2 + 2*c*d*x**4 + d**2*x**8),x)