\(\int (c+d x^4)^2 (a+b x^8)^p \, dx\) [303]

Optimal result

Integrand size = 19, antiderivative size = 150 \[ \int \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {d^2 x \left (a+b x^8\right )^{1+p}}{b (9+8 p)}-\frac {\left (a d^2-b c^2 (9+8 p)\right ) x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )}{b (9+8 p)}+\frac {2}{5} c d x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right ) \] Output:

d^2*x*(b*x^8+a)^(p+1)/b/(9+8*p)-(a*d^2-b*c^2*(9+8*p))*x*(b*x^8+a)^p*hyperg 
eom([1/8, -p],[9/8],-b*x^8/a)/b/(9+8*p)/((1+b*x^8/a)^p)+2/5*c*d*x^5*(b*x^8 
+a)^p*hypergeom([5/8, -p],[13/8],-b*x^8/a)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.71 \[ \int \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {1}{45} x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (45 c^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+d x^4 \left (18 c \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )+5 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )\right )\right ) \] Input:

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

Output:

(x*(a + b*x^8)^p*(45*c^2*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)] + d 
*x^4*(18*c*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)] + 5*d*x^4*Hyperg 
eometric2F1[9/8, -p, 17/8, -((b*x^8)/a)])))/(45*(1 + (b*x^8)/a)^p)
 

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.

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

\(\Big \downarrow \) 1770

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1770

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((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)
 

Giac [F]

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

integrate((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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(64*(a + b*x**8)**p*a*d**2*p**2*x + 40*(a + b*x**8)**p*a*d**2*p*x + 64*(a 
+ b*x**8)**p*b*c**2*p**2*x + 112*(a + b*x**8)**p*b*c**2*p*x + 45*(a + b*x* 
*8)**p*b*c**2*x + 128*(a + b*x**8)**p*b*c*d*p**2*x**5 + 160*(a + b*x**8)** 
p*b*c*d*p*x**5 + 18*(a + b*x**8)**p*b*c*d*x**5 + 64*(a + b*x**8)**p*b*d**2 
*p**2*x**9 + 48*(a + b*x**8)**p*b*d**2*p*x**9 + 5*(a + b*x**8)**p*b*d**2*x 
**9 - 32768*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* 
*2*p**5 - 81920*int((a + b*x**8)**p/(512*a*p**3 + 960*a*p**2 + 472*a*p + 4 
5*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**2*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**2*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**2*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**2*p + 262144*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*b*c**2*p**6 + 950272*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*b*c**2*p**5 + 1286144*int((a + b*x**8)**p/(512*a*p**3 +...