3.2017 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0578388, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^3]*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.14286, size = 24, normalized size = 0.89 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0398385, size = 59, normalized size = 2.19 \[ \frac{2 \sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{3 \sqrt{a} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.016, size = 480, normalized size = 17.8 \[ -4\,{\frac{ \left ( a{x}^{3}+b \right ) \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) ^{2}}{x{a}^{2}\sqrt{x \left ( a{x}^{3}+b \right ) } \left ( i\sqrt{3}-3 \right ) }\sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{a}^{2}b}+2\,ax+\sqrt [3]{-{a}^{2}b}}{ \left ( i\sqrt{3}+1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{a}^{2}b}-2\,ax-\sqrt [3]{-{a}^{2}b}}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}} \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( i\sqrt{3}-1 \right ) }{ \left ( i\sqrt{3}+1 \right ) \left ( i\sqrt{3}-3 \right ) }}} \right ) -{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}},{\frac{i\sqrt{3}-1}{i\sqrt{3}-3}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( i\sqrt{3}-1 \right ) }{ \left ( i\sqrt{3}+1 \right ) \left ( i\sqrt{3}-3 \right ) }}} \right ) \right ){\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}{\frac{1}{\sqrt{{\frac{x \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) \left ( i\sqrt{3}\sqrt [3]{-{a}^{2}b}+2\,ax+\sqrt [3]{-{a}^{2}b} \right ) \left ( i\sqrt{3}\sqrt [3]{-{a}^{2}b}-2\,ax-\sqrt [3]{-{a}^{2}b} \right ) }{{a}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b/x^3)^(1/2),x)

[Out]

-4/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)*(I*3^(1/2)-1)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2
)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(-a*x+(-a^2*b)^(1/3))^2*((I*3^(1/2)*(-a^2*b)^(
1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2
)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2
)/a^2*(EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),
((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))-EllipticPi((-(I
*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1
/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2)))/(x*(a*x
^3+b))^(1/2)/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1
/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)

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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/(sqrt(a + b/x^3)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.369302, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right )}{6 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*log(-(8*a^2*x^6 + 8*a*b*x^3 + b^2)*sqrt(a) - 4*(2*a^2*x^6 + a*b*x^3)*sqrt((
a*x^3 + b)/x^3))/sqrt(a), -1/3*sqrt(-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x
^3)/(2*a*x^3 + b))/a]

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Sympy [A]  time = 4.49235, size = 24, normalized size = 0.89 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(3*sqrt(a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(a + b/x^3)*x), x)