Integrand size = 20, antiderivative size = 114 \[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {B x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac {(3 A b-2 a B) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3}}-\frac {(3 A b-2 a B) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{6 b^{5/3}} \] Output:
1/3*B*x^2*(b*x^3+a)^(1/3)/b-1/9*(3*A*b-2*B*a)*arctan(1/3*(1+2*b^(1/3)*x/(b *x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(5/3)-1/6*(3*A*b-2*B*a)*ln(b^(1/3)*x-(b* x^3+a)^(1/3))/b^(5/3)
Time = 0.70 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.46 \[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {6 b^{2/3} B x^2 \sqrt [3]{a+b x^3}-2 \sqrt {3} (3 A b-2 a B) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )+2 (-3 A b+2 a B) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+(3 A b-2 a B) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{18 b^{5/3}} \] Input:
Integrate[(x*(A + B*x^3))/(a + b*x^3)^(2/3),x]
Output:
(6*b^(2/3)*B*x^2*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(3*A*b - 2*a*B)*ArcTan[(Sqr t[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] + 2*(-3*A*b + 2*a*B)*Lo g[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] + (3*A*b - 2*a*B)*Log[b^(2/3)*x^2 + b^ (1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(18*b^(5/3))
Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {959, 853}
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.
\(\displaystyle \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(3 A b-2 a B) \int \frac {x}{\left (b x^3+a\right )^{2/3}}dx}{3 b}+\frac {B x^2 \sqrt [3]{a+b x^3}}{3 b}\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {(3 A b-2 a B) \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}}\right )}{3 b}+\frac {B x^2 \sqrt [3]{a+b x^3}}{3 b}\) |
Input:
Int[(x*(A + B*x^3))/(a + b*x^3)^(2/3),x]
Output:
(B*x^2*(a + b*x^3)^(1/3))/(3*b) + ((3*A*b - 2*a*B)*(-(ArcTan[(1 + (2*b^(1/ 3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(2/3))) - Log[b^(1/3)*x - (a + b*x^3)^(1/3)]/(2*b^(2/3))))/(3*b)
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(91)=182\).
Time = 1.24 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.99
method | result | size |
pseudoelliptic | \(\frac {6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} B +6 A \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}\, b -4 B \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}\, a -6 A \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) b +3 A \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) b +4 B \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a -2 B \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a}{18 b^{\frac {5}{3}}}\) | \(227\) |
Input:
int(x*(B*x^3+A)/(b*x^3+a)^(2/3),x,method=_RETURNVERBOSE)
Output:
1/18*(6*(b*x^3+a)^(1/3)*b^(2/3)*x^2*B+6*A*arctan(1/3*3^(1/2)*(b^(1/3)*x+2* (b*x^3+a)^(1/3))/b^(1/3)/x)*3^(1/2)*b-4*B*arctan(1/3*3^(1/2)*(b^(1/3)*x+2* (b*x^3+a)^(1/3))/b^(1/3)/x)*3^(1/2)*a-6*A*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/ x)*b+3*A*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*b +4*B*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a-2*B*ln((b^(2/3)*x^2+b^(1/3)*(b*x ^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*a)/b^(5/3)
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (91) = 182\).
Time = 0.10 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.91 \[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} B b^{2} x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (2 \, B a b - 3 \, A b^{2}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b^{2}\right )^{\frac {1}{3}} b x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{b^{2} x}\right ) + 2 \, {\left (2 \, B a - 3 \, A b\right )} \left (-b^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) - {\left (2 \, B a - 3 \, A b\right )} \left (-b^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right )}{18 \, b^{3}} \] Input:
integrate(x*(B*x^3+A)/(b*x^3+a)^(2/3),x, algorithm="fricas")
Output:
1/18*(6*(b*x^3 + a)^(1/3)*B*b^2*x^2 - 6*sqrt(1/3)*(2*B*a*b - 3*A*b^2)*sqrt (-(-b^2)^(1/3))*arctan(-sqrt(1/3)*((-b^2)^(1/3)*b*x - 2*(b*x^3 + a)^(1/3)* (-b^2)^(2/3))*sqrt(-(-b^2)^(1/3))/(b^2*x)) + 2*(2*B*a - 3*A*b)*(-b^2)^(2/3 )*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) - (2*B*a - 3*A*b)*(-b^2)^ (2/3)*log(-((-b^2)^(1/3)*b*x^2 - (b*x^3 + a)^(1/3)*(-b^2)^(2/3)*x - (b*x^3 + a)^(2/3)*b)/x^2))/b^3
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {A x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )} + \frac {B x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate(x*(B*x**3+A)/(b*x**3+a)**(2/3),x)
Output:
A*x**2*gamma(2/3)*hyper((2/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*a **(2/3)*gamma(5/3)) + B*x**5*gamma(5/3)*hyper((2/3, 5/3), (8/3,), b*x**3*e xp_polar(I*pi)/a)/(3*a**(2/3)*gamma(8/3))
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (91) = 182\).
Time = 0.12 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.12 \[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=-\frac {1}{9} \, B {\left (\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {5}{3}}} + \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {5}{3}}} - \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {5}{3}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a}{{\left (b^{2} - \frac {{\left (b x^{3} + a\right )} b}{x^{3}}\right )} x}\right )} + \frac {1}{6} \, A {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {2}{3}}} + \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {2}{3}}} - \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {2}{3}}}\right )} \] Input:
integrate(x*(B*x^3+A)/(b*x^3+a)^(2/3),x, algorithm="maxima")
Output:
-1/9*B*(2*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b ^(1/3))/b^(5/3) + a*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a )^(2/3)/x^2)/b^(5/3) - 2*a*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(5/3) + 3 *(b*x^3 + a)^(1/3)*a/((b^2 - (b*x^3 + a)*b/x^3)*x)) + 1/6*A*(2*sqrt(3)*arc tan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(2/3) + log(b ^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(2/3) - 2* log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(2/3))
\[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(x*(B*x^3+A)/(b*x^3+a)^(2/3),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*x/(b*x^3 + a)^(2/3), x)
Timed out. \[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\int \frac {x\,\left (B\,x^3+A\right )}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \] 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)
\[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{2/3}} \, dx=\left (\int \frac {x^{4}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) b +\left (\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) a \] Input:
int(x*(B*x^3+A)/(b*x^3+a)^(2/3),x)
Output:
int(x**4/(a + b*x**3)**(2/3),x)*b + int(x/(a + b*x**3)**(2/3),x)*a