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

Optimal. Leaf size=46 \[ \frac{2}{3 a \sqrt{a+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}} \]

[Out]

2/(3*a*Sqrt[a + b*x^3]) - (2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2))

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Rubi [A]  time = 0.0749307, antiderivative size = 46, 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 \[ \frac{2}{3 a \sqrt{a+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3)^(3/2)),x]

[Out]

2/(3*a*Sqrt[a + b*x^3]) - (2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2))

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Rubi in Sympy [A]  time = 7.52561, size = 39, normalized size = 0.85 \[ \frac{2}{3 a \sqrt{a + b x^{3}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(3*a*sqrt(a + b*x**3)) - 2*atanh(sqrt(a + b*x**3)/sqrt(a))/(3*a**(3/2))

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Mathematica [A]  time = 0.134817, size = 51, normalized size = 1.11 \[ \frac{2-2 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{3 a \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.035, size = 39, normalized size = 0.9 \[{\frac{2}{3\,a}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^(3/2),x)

[Out]

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

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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 + a)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238317, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x^{3} + a} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \, \sqrt{a}}{3 \, \sqrt{b x^{3} + a} a^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{b x^{3} + a} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) + \sqrt{-a}\right )}}{3 \, \sqrt{b x^{3} + a} \sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.03013, size = 184, normalized size = 4. \[ \frac{2 a^{3} \sqrt{1 + \frac{b x^{3}}{a}}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{2} b x^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{2} b x^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*sqrt(1 + b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) + a**3*log(b*x**3/a)/
(3*a**(9/2) + 3*a**(7/2)*b*x**3) - 2*a**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(9/2
) + 3*a**(7/2)*b*x**3) + a**2*b*x**3*log(b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x*
*3) - 2*a**2*b*x**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(9/2) + 3*a**(7/2)*b*x**3)

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GIAC/XCAS [A]  time = 0.212907, size = 55, normalized size = 1.2 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a} + \frac{2}{3 \, \sqrt{b x^{3} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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