\(\int \frac {(a+b x^8)^p}{(c+d x)^2} \, dx\) [320]

Optimal result

Integrand size = 17, antiderivative size = 970 \[ \int \frac {\left (a+b x^8\right )^p}{(c+d x)^2} \, dx =\text {Too large to display} \] Output:

x*(b*x^8+a)^p*AppellF1(1/8,2,-p,9/8,d^8*x^8/c^8,-b*x^8/a)/c^2/((1+b*x^8/a) 
^p)-d*x^2*(b*x^8+a)^p*AppellF1(1/4,2,-p,5/4,d^8*x^8/c^8,-b*x^8/a)/c^3/((1+ 
b*x^8/a)^p)+d^2*x^3*(b*x^8+a)^p*AppellF1(3/8,2,-p,11/8,d^8*x^8/c^8,-b*x^8/ 
a)/c^4/((1+b*x^8/a)^p)-d^3*x^4*(b*x^8+a)^p*AppellF1(1/2,2,-p,3/2,d^8*x^8/c 
^8,-b*x^8/a)/c^5/((1+b*x^8/a)^p)+d^4*x^5*(b*x^8+a)^p*AppellF1(5/8,2,-p,13/ 
8,d^8*x^8/c^8,-b*x^8/a)/c^6/((1+b*x^8/a)^p)-d^5*x^6*(b*x^8+a)^p*AppellF1(3 
/4,2,-p,7/4,d^8*x^8/c^8,-b*x^8/a)/c^7/((1+b*x^8/a)^p)+d^6*x^7*(b*x^8+a)^p* 
AppellF1(7/8,2,-p,15/8,d^8*x^8/c^8,-b*x^8/a)/c^8/((1+b*x^8/a)^p)+7/9*d^8*x 
^9*(b*x^8+a)^p*AppellF1(9/8,2,-p,17/8,d^8*x^8/c^8,-b*x^8/a)/c^10/((1+b*x^8 
/a)^p)-3/5*d^9*x^10*(b*x^8+a)^p*AppellF1(5/4,2,-p,9/4,d^8*x^8/c^8,-b*x^8/a 
)/c^11/((1+b*x^8/a)^p)+5/11*d^10*x^11*(b*x^8+a)^p*AppellF1(11/8,2,-p,19/8, 
d^8*x^8/c^8,-b*x^8/a)/c^12/((1+b*x^8/a)^p)-1/3*d^11*x^12*(b*x^8+a)^p*Appel 
lF1(3/2,2,-p,5/2,d^8*x^8/c^8,-b*x^8/a)/c^13/((1+b*x^8/a)^p)+3/13*d^12*x^13 
*(b*x^8+a)^p*AppellF1(13/8,2,-p,21/8,d^8*x^8/c^8,-b*x^8/a)/c^14/((1+b*x^8/ 
a)^p)-1/7*d^13*x^14*(b*x^8+a)^p*AppellF1(7/4,2,-p,11/4,d^8*x^8/c^8,-b*x^8/ 
a)/c^15/((1+b*x^8/a)^p)+1/15*d^14*x^15*(b*x^8+a)^p*AppellF1(15/8,2,-p,23/8 
,d^8*x^8/c^8,-b*x^8/a)/c^16/((1+b*x^8/a)^p)-b*c^7*d^7*(b*x^8+a)^(p+1)*hype 
rgeom([2, p+1],[2+p],d^8*(b*x^8+a)/(a*d^8+b*c^8))/(a*d^8+b*c^8)^2/(p+1)
 

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

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 2457

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2457

Int[(u_.)*(Px_)*(Qx_)^(q_), x_Symbol] :> Module[{Rx = PolyGCD[Px, Qx, x]}, 
Int[u*Rx^(q + 1)*PolynomialQuotient[Px, Rx, x]*PolynomialQuotient[Qx, Rx, x 
]^q, x] /; NeQ[Rx, 1]] /; ILtQ[q, 0] && PolyQ[Px, x] && PolyQ[Qx, x]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

((a + b*x**8)**p + 64*int((a + b*x**8)**p/(8*a*c**2*p - a*c**2 + 16*a*c*d* 
p*x - 2*a*c*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c**2 
*x**8 + 16*b*c*d*p*x**9 - 2*b*c*d*x**9 + 8*b*d**2*p*x**10 - b*d**2*x**10), 
x)*a*c*d*p**2 - 8*int((a + b*x**8)**p/(8*a*c**2*p - a*c**2 + 16*a*c*d*p*x 
- 2*a*c*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c**2*x** 
8 + 16*b*c*d*p*x**9 - 2*b*c*d*x**9 + 8*b*d**2*p*x**10 - b*d**2*x**10),x)*a 
*c*d*p + 64*int((a + b*x**8)**p/(8*a*c**2*p - a*c**2 + 16*a*c*d*p*x - 2*a* 
c*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c**2*x**8 + 16 
*b*c*d*p*x**9 - 2*b*c*d*x**9 + 8*b*d**2*p*x**10 - b*d**2*x**10),x)*a*d**2* 
p**2*x - 8*int((a + b*x**8)**p/(8*a*c**2*p - a*c**2 + 16*a*c*d*p*x - 2*a*c 
*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c**2*x**8 + 16* 
b*c*d*p*x**9 - 2*b*c*d*x**9 + 8*b*d**2*p*x**10 - b*d**2*x**10),x)*a*d**2*p 
*x - 64*int(((a + b*x**8)**p*x**7)/(8*a*c**2*p - a*c**2 + 16*a*c*d*p*x - 2 
*a*c*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c**2*x**8 + 
 16*b*c*d*p*x**9 - 2*b*c*d*x**9 + 8*b*d**2*p*x**10 - b*d**2*x**10),x)*b*c* 
*2*p**2 + 8*int(((a + b*x**8)**p*x**7)/(8*a*c**2*p - a*c**2 + 16*a*c*d*p*x 
 - 2*a*c*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c**2*x* 
*8 + 16*b*c*d*p*x**9 - 2*b*c*d*x**9 + 8*b*d**2*p*x**10 - b*d**2*x**10),x)* 
b*c**2*p - 64*int(((a + b*x**8)**p*x**7)/(8*a*c**2*p - a*c**2 + 16*a*c*d*p 
*x - 2*a*c*d*x + 8*a*d**2*p*x**2 - a*d**2*x**2 + 8*b*c**2*p*x**8 - b*c*...