3.2273 \(\int \frac{1}{x \left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{a^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.140263, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{a^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.60732, size = 83, normalized size = 0.98 \[ - \frac{\log{\left (x^{\frac{3}{2}} \right )}}{3 a^{\frac{2}{3}}} + \frac{\log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{\frac{3}{2}}} \right )}}{a^{\frac{2}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{\frac{3}{2}}}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0350378, size = 52, normalized size = 0.61 \[ -\frac{\left (\frac{a}{b x^{3/2}}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^{3/2}}\right )}{\left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 85, normalized size = 1. \[{\frac{2}{3}\ln \left ( \sqrt [3]{a+b{x}^{{\frac{3}{2}}}}-\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{1}{3}\ln \left ( \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a+b{x}^{{\frac{3}{2}}}}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{a+b{x}^{3/2}}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^(3/2))^(2/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 5.60182, size = 41, normalized size = 0.48 \[ - \frac{2 \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{\frac{3}{2}}}} \right )}}{3 b^{\frac{2}{3}} x \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^(3/2) + a)^(2/3)*x), x)