\(\int \frac {(a+b x^8)^p}{c+d x^2} \, dx\) [313]

Optimal result

Integrand size = 19, antiderivative size = 255 \[ \int \frac {\left (a+b x^8\right )^p}{c+d x^2} \, dx=\frac {x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{8},-p,1,\frac {9}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{c}-\frac {d x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{8},-p,1,\frac {11}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{3 c^2}+\frac {d^2 x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{8},-p,1,\frac {13}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{5 c^3}-\frac {d^3 x^7 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{8},-p,1,\frac {15}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{7 c^4} \] Output:

x*(b*x^8+a)^p*AppellF1(1/8,1,-p,9/8,d^4*x^8/c^4,-b*x^8/a)/c/((1+b*x^8/a)^p 
)-1/3*d*x^3*(b*x^8+a)^p*AppellF1(3/8,1,-p,11/8,d^4*x^8/c^4,-b*x^8/a)/c^2/( 
(1+b*x^8/a)^p)+1/5*d^2*x^5*(b*x^8+a)^p*AppellF1(5/8,1,-p,13/8,d^4*x^8/c^4, 
-b*x^8/a)/c^3/((1+b*x^8/a)^p)-1/7*d^3*x^7*(b*x^8+a)^p*AppellF1(7/8,1,-p,15 
/8,d^4*x^8/c^4,-b*x^8/a)/c^4/((1+b*x^8/a)^p)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1388

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 2584

Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbo 
l] :> Int[ExpandToSum[(c - d*x^n)^(-q), x]*((a + b*x^nn)^p/(c^2 - d^2*x^(2* 
n))^(-q)), x] /; FreeQ[{a, b, c, d, n, nn, p}, x] &&  !IntegerQ[p] && ILtQ[ 
q, 0] && IGtQ[Log[2, nn/n], 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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