3.403 \(\int \frac{1}{x \sqrt [3]{-a+b x}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}+\frac{\log (x)}{2 \sqrt [3]{a}} \]

[Out]

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

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Rubi [A]  time = 0.073957, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}+\frac{\log (x)}{2 \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-a + b*x)^(1/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.85057, size = 73, normalized size = 0.89 \[ \frac{\log{\left (x \right )}}{2 \sqrt [3]{a}} - \frac{3 \log{\left (\sqrt [3]{a} + \sqrt [3]{- a + b x} \right )}}{2 \sqrt [3]{a}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{- a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{\sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0747954, size = 100, normalized size = 1.22 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b x-a}+(b x-a)^{2/3}\right )-2 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b x-a}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2 \sqrt [3]{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 83, normalized size = 1. \[ -{1\ln \left ( \sqrt [3]{a}+\sqrt [3]{bx-a} \right ){\frac{1}{\sqrt [3]{a}}}}+{\frac{1}{2}\ln \left ( \left ( bx-a \right ) ^{{\frac{2}{3}}}-\sqrt [3]{bx-a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{a}}}}+{\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx-a}}{\sqrt [3]{a}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x-a)^(1/3),x)

[Out]

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

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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(1/((b*x - a)^(1/3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219363, size = 131, normalized size = 1.6 \[ -\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right )}}{3 \, a}\right ) + \log \left ({\left (b x - a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 2 \, \log \left ({\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + a\right )}{2 \, \left (-a\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.30774, size = 160, normalized size = 1.95 \[ - \frac{2 e^{\frac{10 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{- \frac{a}{b} + x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} - \frac{2 \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{- \frac{a}{b} + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} - \frac{2 e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{- \frac{a}{b} + x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x-a)**(1/3),x)

[Out]

-2*exp(10*I*pi/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))
*gamma(2/3)/(3*a**(1/3)*gamma(5/3)) - 2*log(1 - b**(1/3)*(-a/b + x)**(1/3)*exp_p
olar(I*pi)/a**(1/3))*gamma(2/3)/(3*a**(1/3)*gamma(5/3)) - 2*exp(2*I*pi/3)*log(1
- b**(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(3*a**(1/3
)*gamma(5/3))

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GIAC/XCAS [A]  time = 0.538719, size = 151, normalized size = 1.84 \[ -\frac{\sqrt{3} \left (-a\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x - a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{a} + \frac{\left (-a\right )^{\frac{2}{3}}{\rm ln}\left ({\left (b x - a\right )}^{\frac{2}{3}} +{\left (b x - a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right )}{2 \, a} - \frac{\left (-a\right )^{\frac{2}{3}}{\rm ln}\left ({\left |{\left (b x - a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}} \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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