Integrand size = 20, antiderivative size = 101 \[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{6} c x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )+\frac {1}{10} d x^{10} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^8}{a}\right ) \] Output:
1/6*c*x^6*(b*x^8+a)^p*hypergeom([3/4, -p],[7/4],-b*x^8/a)/((1+b*x^8/a)^p)+ 1/10*d*x^10*(b*x^8+a)^p*hypergeom([5/4, -p],[9/4],-b*x^8/a)/((1+b*x^8/a)^p )
Time = 0.56 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77 \[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{30} x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (5 c \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )+3 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^8}{a}\right )\right ) \] Input:
Integrate[x^5*(c + d*x^4)*(a + b*x^8)^p,x]
Output:
(x^6*(a + b*x^8)^p*(5*c*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^8)/a)] + 3* d*x^4*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^8)/a)]))/(30*(1 + (b*x^8)/a)^ p)
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.
| \(\displaystyle \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx\) |
| \(\Big \downarrow \) 1815 |
| \(\displaystyle \frac {1}{2} \int x^4 \left (d x^4+c\right ) \left (b x^8+a\right )^pdx^2\) |
| \(\Big \downarrow \) 1675 |
| \(\displaystyle \frac {1}{2} \int \left (d x^8 \left (b x^8+a\right )^p+c x^4 \left (b x^8+a\right )^p\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} c x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )+\frac {1}{5} d x^{10} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^8}{a}\right )\right )\) |
Input:
Int[x^5*(c + d*x^4)*(a + b*x^8)^p,x]
Output:
((c*x^6*(a + b*x^8)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^8)/a)])/(3*(1 + (b*x^8)/a)^p) + (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))/2
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])
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 ]
\[\int x^{5} \left (x^{4} d +c \right ) \left (b \,x^{8}+a \right )^{p}d x\]
Input:
int(x^5*(d*x^4+c)*(b*x^8+a)^p,x)
Output:
int(x^5*(d*x^4+c)*(b*x^8+a)^p,x)
\[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{8} + a\right )}^{p} x^{5} \,d x } \] Input:
integrate(x^5*(d*x^4+c)*(b*x^8+a)^p,x, algorithm="fricas")
Output:
integral((d*x^9 + c*x^5)*(b*x^8 + a)^p, x)
Result contains complex when optimal does not.
Time = 144.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {a^{p} c x^{6} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{p} d x^{10} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate(x**5*(d*x**4+c)*(b*x**8+a)**p,x)
Output:
a**p*c*x**6*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**8*exp_polar(I*pi)/a)/ (8*gamma(7/4)) + a**p*d*x**10*gamma(5/4)*hyper((5/4, -p), (9/4,), b*x**8*e xp_polar(I*pi)/a)/(8*gamma(9/4))
\[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{8} + a\right )}^{p} x^{5} \,d x } \] Input:
integrate(x^5*(d*x^4+c)*(b*x^8+a)^p,x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*(b*x^8 + a)^p*x^5, x)
\[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{8} + a\right )}^{p} x^{5} \,d x } \] Input:
integrate(x^5*(d*x^4+c)*(b*x^8+a)^p,x, algorithm="giac")
Output:
integrate((d*x^4 + c)*(b*x^8 + a)^p*x^5, x)
Timed out. \[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int x^5\,{\left (b\,x^8+a\right )}^p\,\left (d\,x^4+c\right ) \,d x \] Input:
int(x^5*(a + b*x^8)^p*(c + d*x^4),x)
Output:
int(x^5*(a + b*x^8)^p*(c + d*x^4), x)
\[ \int x^5 \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx =\text {Too large to display} \] Input:
int(x^5*(d*x^4+c)*(b*x^8+a)^p,x)
Output:
(16*(a + b*x**8)**p*a*d*p**2*x**2 + 12*(a + b*x**8)**p*a*d*p*x**2 + 16*(a + b*x**8)**p*b*c*p**2*x**6 + 24*(a + b*x**8)**p*b*c*p*x**6 + 5*(a + b*x**8 )**p*b*c*x**6 + 16*(a + b*x**8)**p*b*d*p**2*x**10 + 16*(a + b*x**8)**p*b*d *p*x**10 + 3*(a + b*x**8)**p*b*d*x**10 + 8192*int(((a + b*x**8)**p*x**5)/( 64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b*p**2*x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a*b*c*p**6 + 30720*int(((a + b*x**8)**p*x**5 )/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b*p**2*x* *8 + 92*b*p*x**8 + 15*b*x**8),x)*a*b*c*p**5 + 41984*int(((a + b*x**8)**p*x **5)/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b*p**2 *x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a*b*c*p**4 + 25344*int(((a + b*x**8)** p*x**5)/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b*p **2*x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a*b*c*p**3 + 6560*int(((a + b*x**8) **p*x**5)/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b *p**2*x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a*b*c*p**2 + 600*int(((a + b*x**8 )**p*x**5)/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144* b*p**2*x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a*b*c*p - 2048*int(((a + b*x**8) **p*x)/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b*p* *2*x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a**2*d*p**5 - 6144*int(((a + b*x**8) **p*x)/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**8 + 144*b*p* *2*x**8 + 92*b*p*x**8 + 15*b*x**8),x)*a**2*d*p**4 - 6400*int(((a + b*x*...