\(\int \frac {x}{(-b+a x^3)^{2/3}} \, dx\) [1941]

Optimal result

Integrand size = 15, antiderivative size = 136 \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{-b+a x^3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{3 a^{2/3}}+\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 a^{2/3}} \]

[Out]

-1/3*arctan(3^(1/2)*a^(1/3)*x/(a^(1/3)*x+2*(a*x^3-b)^(1/3)))*3^(1/2)/a^(2/3)-1/3*ln(-a^(1/3)*x+(a*x^3-b)^(1/3)
)/a^(2/3)+1/6*ln(a^(2/3)*x^2+a^(1/3)*x*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/a^(2/3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {337} \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (\sqrt [3]{a} x-\sqrt [3]{a x^3-b}\right )}{2 a^{2/3}} \]

[In]

Int[x/(-b + a*x^3)^(2/3),x]

[Out]

-(ArcTan[(1 + (2*a^(1/3)*x)/(-b + a*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*a^(2/3))) - Log[a^(1/3)*x - (-b + a*x^3)^(1/
3)]/(2*a^(2/3))

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (\sqrt [3]{a} x-\sqrt [3]{-b+a x^3}\right )}{2 a^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{-b+a x^3}}\right )-2 \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )+\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 a^{2/3}} \]

[In]

Integrate[x/(-b + a*x^3)^(2/3),x]

[Out]

(-2*Sqrt[3]*ArcTan[(Sqrt[3]*a^(1/3)*x)/(a^(1/3)*x + 2*(-b + a*x^3)^(1/3))] - 2*Log[-(a^(1/3)*x) + (-b + a*x^3)
^(1/3)] + Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(6*a^(2/3))

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (a^{\frac {1}{3}} x +2 \left (a \,x^{3}-b \right )^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}} x}\right )-2 \ln \left (\frac {-a^{\frac {1}{3}} x +\left (a \,x^{3}-b \right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {a^{\frac {2}{3}} x^{2}+a^{\frac {1}{3}} x \left (a \,x^{3}-b \right )^{\frac {1}{3}}+\left (a \,x^{3}-b \right )^{\frac {2}{3}}}{x^{2}}\right )}{6 a^{\frac {2}{3}}}\)	\(108\)

[In]

int(x/(a*x^3-b)^(2/3),x,method=_RETURNVERBOSE)

[Out]

1/6*(2*3^(1/2)*arctan(1/3*3^(1/2)/a^(1/3)/x*(a^(1/3)*x+2*(a*x^3-b)^(1/3)))-2*ln((-a^(1/3)*x+(a*x^3-b)^(1/3))/x
)+ln((a^(2/3)*x^2+a^(1/3)*x*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/x^2))/a^(2/3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=\frac {2 \, \sqrt {3} a \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a x - 2 \, \sqrt {3} {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2} x}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-a^{2}\right )^{\frac {2}{3}} x - {\left (a x^{3} - b\right )}^{\frac {1}{3}} a}{x}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-a^{2}\right )^{\frac {1}{3}} a x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}} x - {\left (a x^{3} - b\right )}^{\frac {2}{3}} a}{x^{2}}\right )}{6 \, a^{2}} \]

[In]

integrate(x/(a*x^3-b)^(2/3),x, algorithm="fricas")

[Out]

1/6*(2*sqrt(3)*a*sqrt(-(-a^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-a^2)^(1/3)*a*x - 2*sqrt(3)*(a*x^3 - b)^(1/3)*(-a^2
)^(2/3))*sqrt(-(-a^2)^(1/3))/(a^2*x)) - 2*(-a^2)^(2/3)*log(-((-a^2)^(2/3)*x - (a*x^3 - b)^(1/3)*a)/x) + (-a^2)
^(2/3)*log(-((-a^2)^(1/3)*a*x^2 - (a*x^3 - b)^(1/3)*(-a^2)^(2/3)*x - (a*x^3 - b)^(2/3)*a)/x^2))/a^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.30 \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=\frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 b^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )} \]

[In]

integrate(x/(a*x**3-b)**(2/3),x)

[Out]

x**2*exp(-2*I*pi/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), a*x**3/b)/(3*b**(2/3)*gamma(5/3))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} + \frac {\log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{\frac {2}{3}}} - \frac {\log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{3 \, a^{\frac {2}{3}}} \]

[In]

integrate(x/(a*x^3-b)^(2/3),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(a^(1/3) + 2*(a*x^3 - b)^(1/3)/x)/a^(1/3))/a^(2/3) + 1/6*log(a^(2/3) + (a*x^3 -
b)^(1/3)*a^(1/3)/x + (a*x^3 - b)^(2/3)/x^2)/a^(2/3) - 1/3*log(-a^(1/3) + (a*x^3 - b)^(1/3)/x)/a^(2/3)

Giac [F]

\[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=\int { \frac {x}{{\left (a x^{3} - b\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(x/(a*x^3-b)^(2/3),x, algorithm="giac")

[Out]

integrate(x/(a*x^3 - b)^(2/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx=\int \frac {x}{{\left (a\,x^3-b\right )}^{2/3}} \,d x \]

[In]

int(x/(a*x^3 - b)^(2/3),x)

[Out]

int(x/(a*x^3 - b)^(2/3), x)