3.424 \(\int \frac{1}{x \left (a^3+b^3 x\right )^{2/3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0621119, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{\log (x)}{2 a^2}+\frac{3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a^3 + b^3*x)^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.37111, size = 65, normalized size = 0.9 \[ - \frac{\log{\left (x \right )}}{2 a^{2}} + \frac{3 \log{\left (a - \sqrt [3]{a^{3} + b^{3} x} \right )}}{2 a^{2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} + \frac{2 \sqrt [3]{a^{3} + b^{3} x}}{3}\right )}{a} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b**3*x+a**3)**(2/3),x)

[Out]

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

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Mathematica [A]  time = 0.0285316, size = 95, normalized size = 1.32 \[ -\frac{-2 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )+\log \left (a \sqrt [3]{a^3+b^3 x}+\left (a^3+b^3 x\right )^{2/3}+a^2\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a^3 + b^3*x)^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 88, normalized size = 1.2 \[{\frac{1}{{a}^{2}}\ln \left ( \sqrt [3]{{b}^{3}x+{a}^{3}}-a \right ) }-{\frac{1}{2\,{a}^{2}}\ln \left ( \left ({b}^{3}x+{a}^{3} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{{b}^{3}x+{a}^{3}}a+{a}^{2} \right ) }-{\frac{\sqrt{3}}{{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( a+2\,\sqrt [3]{{b}^{3}x+{a}^{3}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b^3*x+a^3)^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.47607, size = 117, normalized size = 1.62 \[ -\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{\log \left (a^{2} +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{\log \left (-a +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^3*x + a^3)^(2/3)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22109, size = 111, normalized size = 1.54 \[ -\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right ) + \log \left (a^{2} +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right ) - 2 \, \log \left (-a +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^3*x + a^3)^(2/3)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.37894, size = 134, normalized size = 1.86 \[ \frac{\log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b**3*x+a**3)**(2/3),x)

[Out]

log(1 - b*(a**3/b**3 + x)**(1/3)/a)*gamma(1/3)/(3*a**2*gamma(4/3)) + exp(4*I*pi/
3)*log(1 - b*(a**3/b**3 + x)**(1/3)*exp_polar(2*I*pi/3)/a)*gamma(1/3)/(3*a**2*ga
mma(4/3)) + exp(2*I*pi/3)*log(1 - b*(a**3/b**3 + x)**(1/3)*exp_polar(4*I*pi/3)/a
)*gamma(1/3)/(3*a**2*gamma(4/3))

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GIAC/XCAS [A]  time = 0.230379, size = 119, normalized size = 1.65 \[ -\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{{\rm ln}\left (a^{2} +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{{\rm ln}\left ({\left | -a +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^3*x + a^3)^(2/3)*x),x, algorithm="giac")

[Out]

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