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

Optimal. Leaf size=80 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0639694, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.91084, size = 73, normalized size = 0.91 \[ - \frac{\log{\left (x \right )}}{2 a^{\frac{2}{3}}} + \frac{3 \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{2 a^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0260645, size = 48, normalized size = 0.6 \[ -\frac{3 \left (\frac{a+b x}{b x}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )}{2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 76, normalized size = 1. \[{1\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{1}{2}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)^(2/3),x)

[Out]

1/a^(2/3)*ln((b*x+a)^(1/3)-a^(1/3))-1/2/a^(2/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a
^(1/3)+a^(2/3))-1/a^(2/3)*3^(1/2)*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)^(2/3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217864, size = 122, normalized size = 1.52 \[ -\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right ) + \log \left (a^{2} +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 2 \, \log \left (-a +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )}{2 \,{\left (a^{2}\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.38623, size = 150, normalized size = 1.88 \[ \frac{\log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} + \frac{e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} + \frac{e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)**(2/3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.524042, size = 105, normalized size = 1.31 \[ -\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}}} - \frac{{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{2 \, a^{\frac{2}{3}}} + \frac{{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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