\(\int \frac {(a+b x^8)^p}{(c+d x^2)^2} \, dx\) [314]

Optimal result

Integrand size = 19, antiderivative size = 453 \[ \int \frac {\left (a+b x^8\right )^p}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{8},-p,2,\frac {9}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{c^2}-\frac {2 d x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{8},-p,2,\frac {11}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{3 c^3}+\frac {3 d^2 x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{8},-p,2,\frac {13}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{5 c^4}-\frac {4 d^3 x^7 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{8},-p,2,\frac {15}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{7 c^5}+\frac {d^4 x^9 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {9}{8},-p,2,\frac {17}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{3 c^6}-\frac {2 d^5 x^{11} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {11}{8},-p,2,\frac {19}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{11 c^7}+\frac {d^6 x^{13} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {13}{8},-p,2,\frac {21}{8},-\frac {b x^8}{a},\frac {d^4 x^8}{c^4}\right )}{13 c^8} \] Output:

x*(b*x^8+a)^p*AppellF1(1/8,2,-p,9/8,d^4*x^8/c^4,-b*x^8/a)/c^2/((1+b*x^8/a) 
^p)-2/3*d*x^3*(b*x^8+a)^p*AppellF1(3/8,2,-p,11/8,d^4*x^8/c^4,-b*x^8/a)/c^3 
/((1+b*x^8/a)^p)+3/5*d^2*x^5*(b*x^8+a)^p*AppellF1(5/8,2,-p,13/8,d^4*x^8/c^ 
4,-b*x^8/a)/c^4/((1+b*x^8/a)^p)-4/7*d^3*x^7*(b*x^8+a)^p*AppellF1(7/8,2,-p, 
15/8,d^4*x^8/c^4,-b*x^8/a)/c^5/((1+b*x^8/a)^p)+1/3*d^4*x^9*(b*x^8+a)^p*App 
ellF1(9/8,2,-p,17/8,d^4*x^8/c^4,-b*x^8/a)/c^6/((1+b*x^8/a)^p)-2/11*d^5*x^1 
1*(b*x^8+a)^p*AppellF1(11/8,2,-p,19/8,d^4*x^8/c^4,-b*x^8/a)/c^7/((1+b*x^8/ 
a)^p)+1/13*d^6*x^13*(b*x^8+a)^p*AppellF1(13/8,2,-p,21/8,d^4*x^8/c^4,-b*x^8 
/a)/c^8/((1+b*x^8/a)^p)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 2584

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

\(\Big \downarrow \) 1380

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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