3.730 \(\int \frac{1}{x \left (a+b x^2\right )^{4/3}} \, dx\)

Optimal. Leaf size=104 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{2 a \sqrt [3]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.190817, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{2 a \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^2)^(4/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0278, size = 95, normalized size = 0.91 \[ \frac{3}{2 a \sqrt [3]{a + b x^{2}}} - \frac{\log{\left (x^{2} \right )}}{4 a^{\frac{4}{3}}} + \frac{3 \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{4 a^{\frac{4}{3}}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{2 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**2+a)**(4/3),x)

[Out]

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

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Mathematica [C]  time = 0.0482269, size = 55, normalized size = 0.53 \[ \frac{3-3 \sqrt [3]{\frac{a}{b x^2}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^2}\right )}{2 a \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{2}+a \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2+a)^(4/3),x)

[Out]

int(1/x/(b*x^2+a)^(4/3),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)^(4/3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22565, size = 173, normalized size = 1.66 \[ \frac{2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right )}}{3 \, a}\right ) -{\left (b x^{2} + a\right )}^{\frac{1}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) + 2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right ) + 6 \, a^{\frac{1}{3}}}{4 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(4/3)*x),x, algorithm="fricas")

[Out]

1/4*(2*sqrt(3)*(b*x^2 + a)^(1/3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3)*a^(2/3)
+ a)/a) - (b*x^2 + a)^(1/3)*log((b*x^2 + a)^(2/3)*a^(1/3) + (b*x^2 + a)^(1/3)*a
^(2/3) + a) + 2*(b*x^2 + a)^(1/3)*log((b*x^2 + a)^(1/3)*a^(2/3) - a) + 6*a^(1/3)
)/((b*x^2 + a)^(1/3)*a^(4/3))

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Sympy [A]  time = 4.18553, size = 41, normalized size = 0.39 \[ - \frac{\Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac{4}{3}} x^{\frac{8}{3}} \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**2+a)**(4/3),x)

[Out]

-gamma(4/3)*hyper((4/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b**(4/3)*x*
*(8/3)*gamma(7/3))

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GIAC/XCAS [A]  time = 0.575047, size = 136, normalized size = 1.31 \[ \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{2 \, a^{\frac{4}{3}}} - \frac{{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{4 \, a^{\frac{4}{3}}} + \frac{{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{2 \, a^{\frac{4}{3}}} + \frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(4/3)*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^(4/3)
- 1/4*ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) + 1/2*
ln(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(4/3) + 3/2/((b*x^2 + a)^(1/3)*a)