3.706 \(\int \frac{1}{x \sqrt [3]{a+b x^2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 \sqrt [3]{a}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}} \]

[Out]

(Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(2*a^(1/3))
- Log[x]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(4*a^(1/3))

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Rubi [A]  time = 0.144326, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 \sqrt [3]{a}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^2)^(1/3)),x]

[Out]

(Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(2*a^(1/3))
- Log[x]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(4*a^(1/3))

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Rubi in Sympy [A]  time = 7.83303, size = 80, normalized size = 0.93 \[ - \frac{\log{\left (x^{2} \right )}}{4 \sqrt [3]{a}} + \frac{3 \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{4 \sqrt [3]{a}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{2 \sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2)/(4*a**(1/3)) + 3*log(a**(1/3) - (a + b*x**2)**(1/3))/(4*a**(1/3)) + s
qrt(3)*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**2)**(1/3)/3)/a**(1/3))/(2*a**(1/3)
)

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Mathematica [C]  time = 0.0380073, size = 48, normalized size = 0.56 \[ -\frac{3 \sqrt [3]{\frac{a}{b x^2}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^2}\right )}{2 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(1 + a/(b*x^2))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -(a/(b*x^2))])/(2*(a
+ b*x^2)^(1/3))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2+a)^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223546, size = 117, normalized size = 1.36 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right )}}{3 \, a}\right ) - \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) + 2 \, \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right )}{4 \, a^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3)*a^(2/3) + a)/a) - log((b*
x^2 + a)^(2/3)*a^(1/3) + (b*x^2 + a)^(1/3)*a^(2/3) + a) + 2*log((b*x^2 + a)^(1/3
)*a^(2/3) - a))/a^(1/3)

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Sympy [A]  time = 3.64169, size = 41, normalized size = 0.48 \[ - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 \sqrt [3]{b} x^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(1/3)*hyper((1/3, 1/3), (4/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b**(1/3)*x*
*(2/3)*gamma(4/3))

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GIAC/XCAS [A]  time = 0.605376, size = 117, normalized size = 1.36 \[ \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{2 \, a^{\frac{1}{3}}} - \frac{{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{4 \, a^{\frac{1}{3}}} + \frac{{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{2 \, a^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(1/3)
- 1/4*ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(1/3) + 1/2*
ln(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(1/3)