\(\int x^4 (c+d x^4) (a+b x^8)^p \, dx\) [7]

Optimal result

Integrand size = 20, antiderivative size = 101 \[ \int x^4 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{5} c x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+\frac {1}{9} d x^9 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77 \[ \int x^4 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{45} x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (9 c \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+5 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )\right ) \] Input:

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

Output:

(x^5*(a + b*x^8)^p*(9*c*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)] + 5 
*d*x^4*Hypergeometric2F1[9/8, -p, 17/8, -((b*x^8)/a)]))/(45*(1 + (b*x^8)/a 
)^p)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1865, 2009}

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.

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

\(\Big \downarrow \) 1865

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} c x^5 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+\frac {1}{9} d x^9 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )\)

Input:

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

Output:

(c*x^5*(a + b*x^8)^p*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)])/(5*(1 
 + (b*x^8)/a)^p) + (d*x^9*(a + b*x^8)^p*Hypergeometric2F1[9/8, -p, 17/8, - 
((b*x^8)/a)])/(9*(1 + (b*x^8)/a)^p)
 

Defintions of rubi rules used

rule 1865

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ 
(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2*n))^p, (f*x)^m*(d 
 + e*x^n)^q, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && I 
GtQ[n, 0] && IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

(64*(a + b*x**8)**p*a*d*p**2*x + 40*(a + b*x**8)**p*a*d*p*x + 64*(a + b*x* 
*8)**p*b*c*p**2*x**5 + 80*(a + b*x**8)**p*b*c*p*x**5 + 9*(a + b*x**8)**p*b 
*c*x**5 + 64*(a + b*x**8)**p*b*d*p**2*x**9 + 48*(a + b*x**8)**p*b*d*p*x**9 
 + 5*(a + b*x**8)**p*b*d*x**9 - 32768*int((a + b*x**8)**p/(512*a*p**3 + 96 
0*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2*x**8 + 472*b*p*x* 
*8 + 45*b*x**8),x)*a**2*d*p**5 - 81920*int((a + b*x**8)**p/(512*a*p**3 + 9 
60*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2*x**8 + 472*b*p*x 
**8 + 45*b*x**8),x)*a**2*d*p**4 - 68608*int((a + b*x**8)**p/(512*a*p**3 + 
960*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2*x**8 + 472*b*p* 
x**8 + 45*b*x**8),x)*a**2*d*p**3 - 21760*int((a + b*x**8)**p/(512*a*p**3 + 
 960*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2*x**8 + 472*b*p 
*x**8 + 45*b*x**8),x)*a**2*d*p**2 - 1800*int((a + b*x**8)**p/(512*a*p**3 + 
 960*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2*x**8 + 472*b*p 
*x**8 + 45*b*x**8),x)*a**2*d*p + 262144*int(((a + b*x**8)**p*x**4)/(512*a* 
p**3 + 960*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2*x**8 + 4 
72*b*p*x**8 + 45*b*x**8),x)*a*b*c*p**6 + 819200*int(((a + b*x**8)**p*x**4) 
/(512*a*p**3 + 960*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960*b*p**2* 
x**8 + 472*b*p*x**8 + 45*b*x**8),x)*a*b*c*p**5 + 892928*int(((a + b*x**8)* 
*p*x**4)/(512*a*p**3 + 960*a*p**2 + 472*a*p + 45*a + 512*b*p**3*x**8 + 960 
*b*p**2*x**8 + 472*b*p*x**8 + 45*b*x**8),x)*a*b*c*p**4 + 394240*int(((a...