Integrand size = 22, antiderivative size = 104 \[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {2 B x^3 \left (a+b x^n\right )^{7/2}}{b (6+7 n)}+\frac {a^2 \left (A-\frac {6 a B}{6 b+7 b n}\right ) x^3 \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{n},\frac {3+n}{n},-\frac {b x^n}{a}\right )}{3 \sqrt {1+\frac {b x^n}{a}}} \] Output:
2*B*x^3*(a+b*x^n)^(7/2)/b/(6+7*n)+1/3*a^2*(A-6*a*B/(7*b*n+6*b))*x^3*(a+b*x ^n)^(1/2)*hypergeom([-5/2, 3/n],[(3+n)/n],-b*x^n/a)/(1+b*x^n/a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {a^2 x^3 \sqrt {a+b x^n} \left (A (3+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{n},\frac {3+n}{n},-\frac {b x^n}{a}\right )+3 B x^n \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{n},2+\frac {3}{n},-\frac {b x^n}{a}\right )\right )}{3 (3+n) \sqrt {1+\frac {b x^n}{a}}} \] Input:
Integrate[x^2*(a + b*x^n)^(5/2)*(A + B*x^n),x]
Output:
(a^2*x^3*Sqrt[a + b*x^n]*(A*(3 + n)*Hypergeometric2F1[-5/2, 3/n, (3 + n)/n , -((b*x^n)/a)] + 3*B*x^n*Hypergeometric2F1[-5/2, (3 + n)/n, 2 + 3/n, -((b *x^n)/a)]))/(3*(3 + n)*Sqrt[1 + (b*x^n)/a])
Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 889, 888}
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.
| \(\displaystyle \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \left (A-\frac {6 a B}{7 b n+6 b}\right ) \int x^2 \left (b x^n+a\right )^{5/2}dx+\frac {2 B x^3 \left (a+b x^n\right )^{7/2}}{b (7 n+6)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \frac {a^2 \sqrt {a+b x^n} \left (A-\frac {6 a B}{7 b n+6 b}\right ) \int x^2 \left (\frac {b x^n}{a}+1\right )^{5/2}dx}{\sqrt {\frac {b x^n}{a}+1}}+\frac {2 B x^3 \left (a+b x^n\right )^{7/2}}{b (7 n+6)}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {a^2 x^3 \sqrt {a+b x^n} \left (A-\frac {6 a B}{7 b n+6 b}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{n},\frac {n+3}{n},-\frac {b x^n}{a}\right )}{3 \sqrt {\frac {b x^n}{a}+1}}+\frac {2 B x^3 \left (a+b x^n\right )^{7/2}}{b (7 n+6)}\) |
Input:
Int[x^2*(a + b*x^n)^(5/2)*(A + B*x^n),x]
Output:
(2*B*x^3*(a + b*x^n)^(7/2))/(b*(6 + 7*n)) + (a^2*(A - (6*a*B)/(6*b + 7*b*n ))*x^3*Sqrt[a + b*x^n]*Hypergeometric2F1[-5/2, 3/n, (3 + n)/n, -((b*x^n)/a )])/(3*Sqrt[1 + (b*x^n)/a])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
\[\int x^{2} \left (a +b \,x^{n}\right )^{\frac {5}{2}} \left (A +B \,x^{n}\right )d x\]
Input:
int(x^2*(a+b*x^n)^(5/2)*(A+B*x^n),x)
Output:
int(x^2*(a+b*x^n)^(5/2)*(A+B*x^n),x)
Exception generated. \[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(a+b*x^n)^(5/2)*(A+B*x^n),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Result contains complex when optimal does not.
Time = 18.91 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.81 \[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {A a^{2} a^{\frac {3}{n}} a^{\frac {1}{2} - \frac {3}{n}} x^{3} \Gamma \left (\frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{n} \\ 1 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {3}{n}\right )} + \frac {2 A a a^{- \frac {1}{2} - \frac {3}{n}} a^{1 + \frac {3}{n}} b x^{n + 3} \Gamma \left (1 + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {3}{n} \\ 2 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {3}{n}\right )} + \frac {A a^{- \frac {3}{2} - \frac {3}{n}} a^{2 + \frac {3}{n}} b^{2} x^{2 n + 3} \Gamma \left (2 + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 + \frac {3}{n} \\ 3 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {3}{n}\right )} + \frac {B a^{2} a^{- \frac {1}{2} - \frac {3}{n}} a^{1 + \frac {3}{n}} x^{n + 3} \Gamma \left (1 + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 + \frac {3}{n} \\ 2 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {3}{n}\right )} + \frac {2 B a a^{- \frac {3}{2} - \frac {3}{n}} a^{2 + \frac {3}{n}} b x^{2 n + 3} \Gamma \left (2 + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 2 + \frac {3}{n} \\ 3 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {3}{n}\right )} + \frac {B a^{- \frac {5}{2} - \frac {3}{n}} a^{3 + \frac {3}{n}} b^{2} x^{3 n + 3} \Gamma \left (3 + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 3 + \frac {3}{n} \\ 4 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 + \frac {3}{n}\right )} \] Input:
integrate(x**2*(a+b*x**n)**(5/2)*(A+B*x**n),x)
Output:
A*a**2*a**(3/n)*a**(1/2 - 3/n)*x**3*gamma(3/n)*hyper((-1/2, 3/n), (1 + 3/n ,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 3/n)) + 2*A*a*a**(-1/2 - 3/n)*a **(1 + 3/n)*b*x**(n + 3)*gamma(1 + 3/n)*hyper((-1/2, 1 + 3/n), (2 + 3/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(2 + 3/n)) + A*a**(-3/2 - 3/n)*a**(2 + 3/n)*b**2*x**(2*n + 3)*gamma(2 + 3/n)*hyper((-1/2, 2 + 3/n), (3 + 3/n,), b *x**n*exp_polar(I*pi)/a)/(n*gamma(3 + 3/n)) + B*a**2*a**(-1/2 - 3/n)*a**(1 + 3/n)*x**(n + 3)*gamma(1 + 3/n)*hyper((-1/2, 1 + 3/n), (2 + 3/n,), b*x** n*exp_polar(I*pi)/a)/(n*gamma(2 + 3/n)) + 2*B*a*a**(-3/2 - 3/n)*a**(2 + 3/ n)*b*x**(2*n + 3)*gamma(2 + 3/n)*hyper((-1/2, 2 + 3/n), (3 + 3/n,), b*x**n *exp_polar(I*pi)/a)/(n*gamma(3 + 3/n)) + B*a**(-5/2 - 3/n)*a**(3 + 3/n)*b* *2*x**(3*n + 3)*gamma(3 + 3/n)*hyper((-1/2, 3 + 3/n), (4 + 3/n,), b*x**n*e xp_polar(I*pi)/a)/(n*gamma(4 + 3/n))
\[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\int { {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*x^n)^(5/2)*(A+B*x^n),x, algorithm="maxima")
Output:
integrate((B*x^n + A)*(b*x^n + a)^(5/2)*x^2, x)
\[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\int { {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*x^n)^(5/2)*(A+B*x^n),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(b*x^n + a)^(5/2)*x^2, x)
Timed out. \[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\int x^2\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^{5/2} \,d x \] Input:
int(x^2*(A + B*x^n)*(a + b*x^n)^(5/2),x)
Output:
int(x^2*(A + B*x^n)*(a + b*x^n)^(5/2), x)
\[ \int x^2 \left (a+b x^n\right )^{5/2} \left (A+B x^n\right ) \, dx=\frac {30 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n^{3} x^{3}+276 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n^{2} x^{3}+648 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} n \,x^{3}+432 x^{3 n} \sqrt {x^{n} b +a}\, b^{3} x^{3}+132 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n^{3} x^{3}+1164 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n^{2} x^{3}+2448 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} n \,x^{3}+1296 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2} x^{3}+244 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,n^{3} x^{3}+1920 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,n^{2} x^{3}+2952 x^{n} \sqrt {x^{n} b +a}\, a^{2} b n \,x^{3}+1296 x^{n} \sqrt {x^{n} b +a}\, a^{2} b \,x^{3}+352 \sqrt {x^{n} b +a}\, a^{3} n^{3} x^{3}+1032 \sqrt {x^{n} b +a}\, a^{3} n^{2} x^{3}+1152 \sqrt {x^{n} b +a}\, a^{3} n \,x^{3}+432 \sqrt {x^{n} b +a}\, a^{3} x^{3}+3675 \left (\int \frac {\sqrt {x^{n} b +a}\, x^{2}}{35 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+1032 x^{n} b \,n^{2}+1152 x^{n} b n +432 x^{n} b +35 a \,n^{4}+352 a \,n^{3}+1032 a \,n^{2}+1152 a n +432 a}d x \right ) a^{4} n^{8}+36960 \left (\int \frac {\sqrt {x^{n} b +a}\, x^{2}}{35 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+1032 x^{n} b \,n^{2}+1152 x^{n} b n +432 x^{n} b +35 a \,n^{4}+352 a \,n^{3}+1032 a \,n^{2}+1152 a n +432 a}d x \right ) a^{4} n^{7}+108360 \left (\int \frac {\sqrt {x^{n} b +a}\, x^{2}}{35 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+1032 x^{n} b \,n^{2}+1152 x^{n} b n +432 x^{n} b +35 a \,n^{4}+352 a \,n^{3}+1032 a \,n^{2}+1152 a n +432 a}d x \right ) a^{4} n^{6}+120960 \left (\int \frac {\sqrt {x^{n} b +a}\, x^{2}}{35 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+1032 x^{n} b \,n^{2}+1152 x^{n} b n +432 x^{n} b +35 a \,n^{4}+352 a \,n^{3}+1032 a \,n^{2}+1152 a n +432 a}d x \right ) a^{4} n^{5}+45360 \left (\int \frac {\sqrt {x^{n} b +a}\, x^{2}}{35 x^{n} b \,n^{4}+352 x^{n} b \,n^{3}+1032 x^{n} b \,n^{2}+1152 x^{n} b n +432 x^{n} b +35 a \,n^{4}+352 a \,n^{3}+1032 a \,n^{2}+1152 a n +432 a}d x \right ) a^{4} n^{4}}{105 n^{4}+1056 n^{3}+3096 n^{2}+3456 n +1296} \] Input:
int(x^2*(a+b*x^n)^(5/2)*(A+B*x^n),x)
Output:
(30*x**(3*n)*sqrt(x**n*b + a)*b**3*n**3*x**3 + 276*x**(3*n)*sqrt(x**n*b + a)*b**3*n**2*x**3 + 648*x**(3*n)*sqrt(x**n*b + a)*b**3*n*x**3 + 432*x**(3* n)*sqrt(x**n*b + a)*b**3*x**3 + 132*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n**3* x**3 + 1164*x**(2*n)*sqrt(x**n*b + a)*a*b**2*n**2*x**3 + 2448*x**(2*n)*sqr t(x**n*b + a)*a*b**2*n*x**3 + 1296*x**(2*n)*sqrt(x**n*b + a)*a*b**2*x**3 + 244*x**n*sqrt(x**n*b + a)*a**2*b*n**3*x**3 + 1920*x**n*sqrt(x**n*b + a)*a **2*b*n**2*x**3 + 2952*x**n*sqrt(x**n*b + a)*a**2*b*n*x**3 + 1296*x**n*sqr t(x**n*b + a)*a**2*b*x**3 + 352*sqrt(x**n*b + a)*a**3*n**3*x**3 + 1032*sqr t(x**n*b + a)*a**3*n**2*x**3 + 1152*sqrt(x**n*b + a)*a**3*n*x**3 + 432*sqr t(x**n*b + a)*a**3*x**3 + 3675*int((sqrt(x**n*b + a)*x**2)/(35*x**n*b*n**4 + 352*x**n*b*n**3 + 1032*x**n*b*n**2 + 1152*x**n*b*n + 432*x**n*b + 35*a* n**4 + 352*a*n**3 + 1032*a*n**2 + 1152*a*n + 432*a),x)*a**4*n**8 + 36960*i nt((sqrt(x**n*b + a)*x**2)/(35*x**n*b*n**4 + 352*x**n*b*n**3 + 1032*x**n*b *n**2 + 1152*x**n*b*n + 432*x**n*b + 35*a*n**4 + 352*a*n**3 + 1032*a*n**2 + 1152*a*n + 432*a),x)*a**4*n**7 + 108360*int((sqrt(x**n*b + a)*x**2)/(35* x**n*b*n**4 + 352*x**n*b*n**3 + 1032*x**n*b*n**2 + 1152*x**n*b*n + 432*x** n*b + 35*a*n**4 + 352*a*n**3 + 1032*a*n**2 + 1152*a*n + 432*a),x)*a**4*n** 6 + 120960*int((sqrt(x**n*b + a)*x**2)/(35*x**n*b*n**4 + 352*x**n*b*n**3 + 1032*x**n*b*n**2 + 1152*x**n*b*n + 432*x**n*b + 35*a*n**4 + 352*a*n**3 + 1032*a*n**2 + 1152*a*n + 432*a),x)*a**4*n**5 + 45360*int((sqrt(x**n*b +...