3.567 \(\int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}} \]

[Out]

-(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))

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Rubi [A]  time = 0.116012, antiderivative size = 122, normalized size of antiderivative = 1.69, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}}+\frac{\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{6 b^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^3)^(2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.5484, size = 114, normalized size = 1.58 \[ - \frac{\log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{3 b^{\frac{2}{3}}} + \frac{\log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{6 b^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{3 b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**3+a)**(2/3),x)

[Out]

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

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Mathematica [C]  time = 0.0262757, size = 52, normalized size = 0.72 \[ \frac{x^2 \left (\frac{a+b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^3)^(2/3),x]

[Out]

(x^2*((a + b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(a
+ b*x^3)^(2/3))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{x \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^3+a)^(2/3),x)

[Out]

int(x/(b*x^3+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^3 + a)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242715, size = 186, normalized size = 2.58 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} \log \left (\frac{b x +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{x}\right ) - \sqrt{3} \log \left (\frac{b^{2} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}} b x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b^{2}\right )^{\frac{2}{3}}}{x^{2}}\right ) + 6 \, \arctan \left (-\frac{\sqrt{3} b x - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{3 \, b x}\right )\right )}}{18 \, \left (-b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^3 + a)^(2/3),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(2*sqrt(3)*log((b*x + (b*x^3 + a)^(1/3)*(-b^2)^(1/3))/x) - sqrt(3)*
log((b^2*x^2 - (b*x^3 + a)^(1/3)*(-b^2)^(1/3)*b*x + (b*x^3 + a)^(2/3)*(-b^2)^(2/
3))/x^2) + 6*arctan(-1/3*(sqrt(3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-b^2)^(1/3)
)/(b*x)))/(-b^2)^(1/3)

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Sympy [A]  time = 3.65367, size = 37, normalized size = 0.51 \[ \frac{x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**3+a)**(2/3),x)

[Out]

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))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^3 + a)^(2/3),x, algorithm="giac")

[Out]

integrate(x/(b*x^3 + a)^(2/3), x)