Integrand size = 22, antiderivative size = 157 \[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {d^2 x^3 \left (a+b x^8\right )^{1+p}}{b (11+8 p)}-\frac {\left (3 a d^2-b c^2 (11+8 p)\right ) x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},-p,\frac {11}{8},-\frac {b x^8}{a}\right )}{3 b (11+8 p)}+\frac {2}{7} c d x^7 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{8},-p,\frac {15}{8},-\frac {b x^8}{a}\right ) \] Output:
d^2*x^3*(b*x^8+a)^(p+1)/b/(11+8*p)-1/3*(3*a*d^2-b*c^2*(11+8*p))*x^3*(b*x^8 +a)^p*hypergeom([3/8, -p],[11/8],-b*x^8/a)/b/(11+8*p)/((1+b*x^8/a)^p)+2/7* c*d*x^7*(b*x^8+a)^p*hypergeom([7/8, -p],[15/8],-b*x^8/a)/((1+b*x^8/a)^p)
Time = 0.62 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.69 \[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\frac {1}{231} x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (77 c^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{8},-p,\frac {11}{8},-\frac {b x^8}{a}\right )+3 d x^4 \left (22 c \operatorname {Hypergeometric2F1}\left (\frac {7}{8},-p,\frac {15}{8},-\frac {b x^8}{a}\right )+7 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {11}{8},-p,\frac {19}{8},-\frac {b x^8}{a}\right )\right )\right ) \] Input:
Integrate[x^2*(c + d*x^4)^2*(a + b*x^8)^p,x]
Output:
(x^3*(a + b*x^8)^p*(77*c^2*Hypergeometric2F1[3/8, -p, 11/8, -((b*x^8)/a)] + 3*d*x^4*(22*c*Hypergeometric2F1[7/8, -p, 15/8, -((b*x^8)/a)] + 7*d*x^4*H ypergeometric2F1[11/8, -p, 19/8, -((b*x^8)/a)])))/(231*(1 + (b*x^8)/a)^p)
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.
| \(\displaystyle \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx\) |
| \(\Big \downarrow \) 1865 |
| \(\displaystyle \int \left (c^2 x^2 \left (a+b x^8\right )^p+2 c d x^6 \left (a+b x^8\right )^p+d^2 x^{10} \left (a+b x^8\right )^p\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} c^2 x^3 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},-p,\frac {11}{8},-\frac {b x^8}{a}\right )+\frac {2}{7} c d x^7 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{8},-p,\frac {15}{8},-\frac {b x^8}{a}\right )+\frac {1}{11} d^2 x^{11} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {11}{8},-p,\frac {19}{8},-\frac {b x^8}{a}\right )\) |
Input:
Int[x^2*(c + d*x^4)^2*(a + b*x^8)^p,x]
Output:
(c^2*x^3*(a + b*x^8)^p*Hypergeometric2F1[3/8, -p, 11/8, -((b*x^8)/a)])/(3* (1 + (b*x^8)/a)^p) + (2*c*d*x^7*(a + b*x^8)^p*Hypergeometric2F1[7/8, -p, 1 5/8, -((b*x^8)/a)])/(7*(1 + (b*x^8)/a)^p) + (d^2*x^11*(a + b*x^8)^p*Hyperg eometric2F1[11/8, -p, 19/8, -((b*x^8)/a)])/(11*(1 + (b*x^8)/a)^p)
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]
\[\int x^{2} \left (x^{4} d +c \right )^{2} \left (b \,x^{8}+a \right )^{p}d x\]
Input:
int(x^2*(d*x^4+c)^2*(b*x^8+a)^p,x)
Output:
int(x^2*(d*x^4+c)^2*(b*x^8+a)^p,x)
\[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(d*x^4+c)^2*(b*x^8+a)^p,x, algorithm="fricas")
Output:
integral((d^2*x^10 + 2*c*d*x^6 + c^2*x^2)*(b*x^8 + a)^p, x)
Timed out. \[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**2*(d*x**4+c)**2*(b*x**8+a)**p,x)
Output:
Timed out
\[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(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^2, x)
\[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(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^2, x)
Timed out. \[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\int x^2\,{\left (b\,x^8+a\right )}^p\,{\left (d\,x^4+c\right )}^2 \,d x \] Input:
int(x^2*(a + b*x^8)^p*(c + d*x^4)^2,x)
Output:
int(x^2*(a + b*x^8)^p*(c + d*x^4)^2, x)
\[ \int x^2 \left (c+d x^4\right )^2 \left (a+b x^8\right )^p \, dx=\text {too large to display} \] Input:
int(x^2*(d*x^4+c)^2*(b*x^8+a)^p,x)
Output:
(64*(a + b*x**8)**p*a*d**2*p**2*x**3 + 56*(a + b*x**8)**p*a*d**2*p*x**3 + 64*(a + b*x**8)**p*b*c**2*p**2*x**3 + 144*(a + b*x**8)**p*b*c**2*p*x**3 + 77*(a + b*x**8)**p*b*c**2*x**3 + 128*(a + b*x**8)**p*b*c*d*p**2*x**7 + 224 *(a + b*x**8)**p*b*c*d*p*x**7 + 66*(a + b*x**8)**p*b*c*d*x**7 + 64*(a + b* x**8)**p*b*d**2*p**2*x**11 + 80*(a + b*x**8)**p*b*d**2*p*x**11 + 21*(a + b *x**8)**p*b*d**2*x**11 + 524288*int(((a + b*x**8)**p*x**6)/(512*a*p**3 + 1 344*a*p**2 + 1048*a*p + 231*a + 512*b*p**3*x**8 + 1344*b*p**2*x**8 + 1048* b*p*x**8 + 231*b*x**8),x)*a*b*c*d*p**6 + 2293760*int(((a + b*x**8)**p*x**6 )/(512*a*p**3 + 1344*a*p**2 + 1048*a*p + 231*a + 512*b*p**3*x**8 + 1344*b* p**2*x**8 + 1048*b*p*x**8 + 231*b*x**8),x)*a*b*c*d*p**5 + 3751936*int(((a + b*x**8)**p*x**6)/(512*a*p**3 + 1344*a*p**2 + 1048*a*p + 231*a + 512*b*p* *3*x**8 + 1344*b*p**2*x**8 + 1048*b*p*x**8 + 231*b*x**8),x)*a*b*c*d*p**4 + 2824192*int(((a + b*x**8)**p*x**6)/(512*a*p**3 + 1344*a*p**2 + 1048*a*p + 231*a + 512*b*p**3*x**8 + 1344*b*p**2*x**8 + 1048*b*p*x**8 + 231*b*x**8), x)*a*b*c*d*p**3 + 967296*int(((a + b*x**8)**p*x**6)/(512*a*p**3 + 1344*a*p **2 + 1048*a*p + 231*a + 512*b*p**3*x**8 + 1344*b*p**2*x**8 + 1048*b*p*x** 8 + 231*b*x**8),x)*a*b*c*d*p**2 + 121968*int(((a + b*x**8)**p*x**6)/(512*a *p**3 + 1344*a*p**2 + 1048*a*p + 231*a + 512*b*p**3*x**8 + 1344*b*p**2*x** 8 + 1048*b*p*x**8 + 231*b*x**8),x)*a*b*c*d*p - 98304*int(((a + b*x**8)**p* x**2)/(512*a*p**3 + 1344*a*p**2 + 1048*a*p + 231*a + 512*b*p**3*x**8 + ...