\(\int x^5 (c+d x^4)^2 (a+b x^8)^p \, dx\) [12]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {d^2 x^6 \left (a+b x^8\right )^{1+p}}{2 b (7+4 p)}-\frac {\left (3 a d^2-b c^2 (7+4 p)\right ) x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )}{6 b (7+4 p)}+\frac {1}{5} c d x^{10} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^8}{a}\right ) \] Output:

1/2*d^2*x^6*(b*x^8+a)^(p+1)/b/(7+4*p)-1/6*(3*a*d^2-b*c^2*(7+4*p))*x^6*(b*x 
^8+a)^p*hypergeom([3/4, -p],[7/4],-b*x^8/a)/b/(7+4*p)/((1+b*x^8/a)^p)+1/5* 
c*d*x^10*(b*x^8+a)^p*hypergeom([5/4, -p],[9/4],-b*x^8/a)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.68 \[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {1}{210} x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (35 c^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )+3 d x^4 \left (14 c \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^8}{a}\right )+5 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^8}{a}\right )\right )\right ) \] Input:

Integrate[x^5*(c + d*x^4)^2*(a + b*x^8)^p,x]
 

Output:

(x^6*(a + b*x^8)^p*(35*c^2*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^8)/a)] + 
 3*d*x^4*(14*c*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^8)/a)] + 5*d*x^4*Hyp 
ergeometric2F1[7/4, -p, 11/4, -((b*x^8)/a)])))/(210*(1 + (b*x^8)/a)^p)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1815, 1675, 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.

\(\displaystyle \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx\)

\(\Big \downarrow \) 1815

\(\displaystyle \frac {1}{2} \int x^4 \left (d x^4+c\right )^2 \left (b x^8+a\right )^pdx^2\)

\(\Big \downarrow \) 1675

\(\displaystyle \frac {1}{2} \int \left (d^2 x^{12} \left (b x^8+a\right )^p+2 c d x^8 \left (b x^8+a\right )^p+c^2 x^4 \left (b x^8+a\right )^p\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} c^2 x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )+\frac {2}{5} c d x^{10} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^8}{a}\right )+\frac {1}{7} d^2 x^{14} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^8}{a}\right )\right )\)

Input:

Int[x^5*(c + d*x^4)^2*(a + b*x^8)^p,x]
 

Output:

((c^2*x^6*(a + b*x^8)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^8)/a)])/(3* 
(1 + (b*x^8)/a)^p) + (2*c*d*x^10*(a + b*x^8)^p*Hypergeometric2F1[5/4, -p, 
9/4, -((b*x^8)/a)])/(5*(1 + (b*x^8)/a)^p) + (d^2*x^14*(a + b*x^8)^p*Hyperg 
eometric2F1[7/4, -p, 11/4, -((b*x^8)/a)])/(7*(1 + (b*x^8)/a)^p))/2
 

Defintions of rubi rules used

rule 1675

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, 
x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q, 0] | 
| IntegersQ[m, q])
 

rule 1815

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_ 
.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/ 
k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k != 1] 
/; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m 
]
 

rule 2009

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

Maple [F]

\[\int x^{5} \left (x^{4} d +c \right )^{2} \left (b \,x^{8}+a \right )^{p}d x\]

Input:

int(x^5*(d*x^4+c)^2*(b*x^8+a)^p,x)
 

Output:

int(x^5*(d*x^4+c)^2*(b*x^8+a)^p,x)
 

Fricas [F]

\[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(d*x^4+c)^2*(b*x^8+a)^p,x, algorithm="fricas")
 

Output:

integral((d^2*x^13 + 2*c*d*x^9 + c^2*x^5)*(b*x^8 + a)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\text {Timed out} \] Input:

integrate(x**5*(d*x**4+c)**2*(b*x**8+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(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^5, x)
 

Giac [F]

\[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(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^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int x^5\,{\left (b\,x^8+a\right )}^p\,{\left (d\,x^4+c\right )}^2 \,d x \] Input:

int(x^5*(a + b*x^8)^p*(c + d*x^4)^2,x)
 

Output:

int(x^5*(a + b*x^8)^p*(c + d*x^4)^2, x)
 

Reduce [F]

\[ \int x^5 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\text {too large to display} \] Input:

int(x^5*(d*x^4+c)^2*(b*x^8+a)^p,x)
 

Output:

(128*(a + b*x**8)**p*a*c*d*p**3*x**2 + 320*(a + b*x**8)**p*a*c*d*p**2*x**2 
 + 168*(a + b*x**8)**p*a*c*d*p*x**2 + 64*(a + b*x**8)**p*a*d**2*p**3*x**6 
+ 96*(a + b*x**8)**p*a*d**2*p**2*x**6 + 20*(a + b*x**8)**p*a*d**2*p*x**6 + 
 64*(a + b*x**8)**p*b*c**2*p**3*x**6 + 208*(a + b*x**8)**p*b*c**2*p**2*x** 
6 + 188*(a + b*x**8)**p*b*c**2*p*x**6 + 35*(a + b*x**8)**p*b*c**2*x**6 + 1 
28*(a + b*x**8)**p*b*c*d*p**3*x**10 + 352*(a + b*x**8)**p*b*c*d*p**2*x**10 
 + 248*(a + b*x**8)**p*b*c*d*p*x**10 + 42*(a + b*x**8)**p*b*c*d*x**10 + 64 
*(a + b*x**8)**p*b*d**2*p**3*x**14 + 144*(a + b*x**8)**p*b*d**2*p**2*x**14 
 + 92*(a + b*x**8)**p*b*d**2*p*x**14 + 15*(a + b*x**8)**p*b*d**2*x**14 - 9 
8304*int(((a + b*x**8)**p*x**5)/(256*a*p**4 + 1024*a*p**3 + 1376*a*p**2 + 
704*a*p + 105*a + 256*b*p**4*x**8 + 1024*b*p**3*x**8 + 1376*b*p**2*x**8 + 
704*b*p*x**8 + 105*b*x**8),x)*a**2*d**2*p**7 - 540672*int(((a + b*x**8)**p 
*x**5)/(256*a*p**4 + 1024*a*p**3 + 1376*a*p**2 + 704*a*p + 105*a + 256*b*p 
**4*x**8 + 1024*b*p**3*x**8 + 1376*b*p**2*x**8 + 704*b*p*x**8 + 105*b*x**8 
),x)*a**2*d**2*p**6 - 1148928*int(((a + b*x**8)**p*x**5)/(256*a*p**4 + 102 
4*a*p**3 + 1376*a*p**2 + 704*a*p + 105*a + 256*b*p**4*x**8 + 1024*b*p**3*x 
**8 + 1376*b*p**2*x**8 + 704*b*p*x**8 + 105*b*x**8),x)*a**2*d**2*p**5 - 11 
85792*int(((a + b*x**8)**p*x**5)/(256*a*p**4 + 1024*a*p**3 + 1376*a*p**2 + 
 704*a*p + 105*a + 256*b*p**4*x**8 + 1024*b*p**3*x**8 + 1376*b*p**2*x**8 + 
 704*b*p*x**8 + 105*b*x**8),x)*a**2*d**2*p**4 - 610944*int(((a + b*x**8...