\(\int x^4 (c+d x^4)^2 (a+b x^8)^p \, dx\) [18]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int x^4 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {d^2 x^5 \left (a+b x^8\right )^{1+p}}{b (13+8 p)}-\frac {\left (5 a d^2-b c^2 (13+8 p)\right ) x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )}{5 b (13+8 p)}+\frac {2}{9} c d x^9 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right ) \] Output:

d^2*x^5*(b*x^8+a)^(p+1)/b/(13+8*p)-1/5*(5*a*d^2-b*c^2*(13+8*p))*x^5*(b*x^8 
+a)^p*hypergeom([5/8, -p],[13/8],-b*x^8/a)/b/(13+8*p)/((1+b*x^8/a)^p)+2/9* 
c*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 = 0.62 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.69 \[ \int x^4 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {1}{585} x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (117 c^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+5 d x^4 \left (26 c \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )+9 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {13}{8},-p,\frac {21}{8},-\frac {b x^8}{a}\right )\right )\right ) \] Input:

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

Output:

(x^5*(a + b*x^8)^p*(117*c^2*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)] 
 + 5*d*x^4*(26*c*Hypergeometric2F1[9/8, -p, 17/8, -((b*x^8)/a)] + 9*d*x^4* 
Hypergeometric2F1[13/8, -p, 21/8, -((b*x^8)/a)])))/(585*(1 + (b*x^8)/a)^p)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 1865

\(\displaystyle \int \left (c^2 x^4 \left (a+b x^8\right )^p+2 c d x^8 \left (a+b x^8\right )^p+d^2 x^{12} \left (a+b x^8\right )^p\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} c^2 x^5 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+\frac {2}{9} c d x^9 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )+\frac {1}{13} d^2 x^{13} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {13}{8},-p,\frac {21}{8},-\frac {b x^8}{a}\right )\)

Input:

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

Output:

(c^2*x^5*(a + b*x^8)^p*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)])/(5* 
(1 + (b*x^8)/a)^p) + (2*c*d*x^9*(a + b*x^8)^p*Hypergeometric2F1[9/8, -p, 1 
7/8, -((b*x^8)/a)])/(9*(1 + (b*x^8)/a)^p) + (d^2*x^13*(a + b*x^8)^p*Hyperg 
eometric2F1[13/8, -p, 21/8, -((b*x^8)/a)])/(13*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d^2*x^12 + 2*c*d*x^8 + c^2*x^4)*(b*x^8 + a)^p, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(1024*(a + b*x**8)**p*a*c*d*p**3*x + 2304*(a + b*x**8)**p*a*c*d*p**2*x + 1 
040*(a + b*x**8)**p*a*c*d*p*x + 512*(a + b*x**8)**p*a*d**2*p**3*x**5 + 640 
*(a + b*x**8)**p*a*d**2*p**2*x**5 + 72*(a + b*x**8)**p*a*d**2*p*x**5 + 512 
*(a + b*x**8)**p*b*c**2*p**3*x**5 + 1472*(a + b*x**8)**p*b*c**2*p**2*x**5 
+ 1112*(a + b*x**8)**p*b*c**2*p*x**5 + 117*(a + b*x**8)**p*b*c**2*x**5 + 1 
024*(a + b*x**8)**p*b*c*d*p**3*x**9 + 2432*(a + b*x**8)**p*b*c*d*p**2*x**9 
 + 1328*(a + b*x**8)**p*b*c*d*p*x**9 + 130*(a + b*x**8)**p*b*c*d*x**9 + 51 
2*(a + b*x**8)**p*b*d**2*p**3*x**13 + 960*(a + b*x**8)**p*b*d**2*p**2*x**1 
3 + 472*(a + b*x**8)**p*b*d**2*p*x**13 + 45*(a + b*x**8)**p*b*d**2*x**13 - 
 4194304*int((a + b*x**8)**p/(4096*a*p**4 + 14336*a*p**3 + 16256*a*p**2 + 
6496*a*p + 585*a + 4096*b*p**4*x**8 + 14336*b*p**3*x**8 + 16256*b*p**2*x** 
8 + 6496*b*p*x**8 + 585*b*x**8),x)*a**2*c*d*p**7 - 24117248*int((a + b*x** 
8)**p/(4096*a*p**4 + 14336*a*p**3 + 16256*a*p**2 + 6496*a*p + 585*a + 4096 
*b*p**4*x**8 + 14336*b*p**3*x**8 + 16256*b*p**2*x**8 + 6496*b*p*x**8 + 585 
*b*x**8),x)*a**2*c*d*p**6 - 53936128*int((a + b*x**8)**p/(4096*a*p**4 + 14 
336*a*p**3 + 16256*a*p**2 + 6496*a*p + 585*a + 4096*b*p**4*x**8 + 14336*b* 
p**3*x**8 + 16256*b*p**2*x**8 + 6496*b*p*x**8 + 585*b*x**8),x)*a**2*c*d*p* 
*5 - 59015168*int((a + b*x**8)**p/(4096*a*p**4 + 14336*a*p**3 + 16256*a*p* 
*2 + 6496*a*p + 585*a + 4096*b*p**4*x**8 + 14336*b*p**3*x**8 + 16256*b*p** 
2*x**8 + 6496*b*p*x**8 + 585*b*x**8),x)*a**2*c*d*p**4 - 32472064*int((a...