Integrand size = 20, antiderivative size = 86 \[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\frac {B x^2 \left (a+b x^3\right )^{5/3}}{7 b}+\frac {(7 A b-2 a B) x^2 \left (a+b x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{14 b \left (1+\frac {b x^3}{a}\right )^{2/3}} \] Output:
1/7*B*x^2*(b*x^3+a)^(5/3)/b+1/14*(7*A*b-2*B*a)*x^2*(b*x^3+a)^(2/3)*hyperge om([-2/3, 2/3],[5/3],-b*x^3/a)/b/(1+b*x^3/a)^(2/3)
Time = 7.74 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\frac {\left (a+b x^3\right )^{2/3} \left (5 A x^2 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )+2 B x^5 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {5}{3},\frac {8}{3},-\frac {b x^3}{a}\right )\right )}{10 \left (1+\frac {b x^3}{a}\right )^{2/3}} \] Input:
Integrate[x*(a + b*x^3)^(2/3)*(A + B*x^3),x]
Output:
((a + b*x^3)^(2/3)*(5*A*x^2*Hypergeometric2F1[-2/3, 2/3, 5/3, -((b*x^3)/a) ] + 2*B*x^5*Hypergeometric2F1[-2/3, 5/3, 8/3, -((b*x^3)/a)]))/(10*(1 + (b* x^3)/a)^(2/3))
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.
\(\displaystyle \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(7 A b-2 a B) \int x \left (b x^3+a\right )^{2/3}dx}{7 b}+\frac {B x^2 \left (a+b x^3\right )^{5/3}}{7 b}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\left (a+b x^3\right )^{2/3} (7 A b-2 a B) \int x \left (\frac {b x^3}{a}+1\right )^{2/3}dx}{7 b \left (\frac {b x^3}{a}+1\right )^{2/3}}+\frac {B x^2 \left (a+b x^3\right )^{5/3}}{7 b}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^2 \left (a+b x^3\right )^{2/3} (7 A b-2 a B) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{14 b \left (\frac {b x^3}{a}+1\right )^{2/3}}+\frac {B x^2 \left (a+b x^3\right )^{5/3}}{7 b}\) |
Input:
Int[x*(a + b*x^3)^(2/3)*(A + B*x^3),x]
Output:
(B*x^2*(a + b*x^3)^(5/3))/(7*b) + ((7*A*b - 2*a*B)*x^2*(a + b*x^3)^(2/3)*H ypergeometric2F1[-2/3, 2/3, 5/3, -((b*x^3)/a)])/(14*b*(1 + (b*x^3)/a)^(2/3 ))
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]
\[\int x \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (B \,x^{3}+A \right )d x\]
Input:
int(x*(b*x^3+a)^(2/3)*(B*x^3+A),x)
Output:
int(x*(b*x^3+a)^(2/3)*(B*x^3+A),x)
\[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} x \,d x } \] Input:
integrate(x*(b*x^3+a)^(2/3)*(B*x^3+A),x, algorithm="fricas")
Output:
integral((B*x^4 + A*x)*(b*x^3 + a)^(2/3), x)
Result contains complex when optimal does not.
Time = 1.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\frac {A a^{\frac {2}{3}} x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {B a^{\frac {2}{3}} x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate(x*(b*x**3+a)**(2/3)*(B*x**3+A),x)
Output:
A*a**(2/3)*x**2*gamma(2/3)*hyper((-2/3, 2/3), (5/3,), b*x**3*exp_polar(I*p i)/a)/(3*gamma(5/3)) + B*a**(2/3)*x**5*gamma(5/3)*hyper((-2/3, 5/3), (8/3, ), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3))
\[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} x \,d x } \] Input:
integrate(x*(b*x^3+a)^(2/3)*(B*x^3+A),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(2/3)*x, x)
\[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} x \,d x } \] Input:
integrate(x*(b*x^3+a)^(2/3)*(B*x^3+A),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(2/3)*x, x)
Timed out. \[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\int x\,\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{2/3} \,d x \] Input:
int(x*(A + B*x^3)*(a + b*x^3)^(2/3),x)
Output:
int(x*(A + B*x^3)*(a + b*x^3)^(2/3), x)
\[ \int x \left (a+b x^3\right )^{2/3} \left (A+B x^3\right ) \, dx=\frac {9 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,x^{2}}{28}+\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} b \,x^{5}}{7}+\frac {5 \left (\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) a^{2}}{14} \] Input:
int(x*(b*x^3+a)^(2/3)*(B*x^3+A),x)
Output:
(9*(a + b*x**3)**(2/3)*a*x**2 + 4*(a + b*x**3)**(2/3)*b*x**5 + 10*int(((a + b*x**3)**(2/3)*x)/(a + b*x**3),x)*a**2)/28