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

Optimal. Leaf size=43 \[ \frac{2}{3} \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0699796, antiderivative size = 43, 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 \[ \frac{2}{3} \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^3]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.95605, size = 37, normalized size = 0.86 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3} + \frac{2 \sqrt{a + b x^{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101235, size = 48, normalized size = 1.12 \[ \frac{2}{3} \sqrt{a+b x^3} \left (1-\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^3]/x,x]

[Out]

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

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Maple [A]  time = 0.158, size = 32, normalized size = 0.7 \[ -{\frac{2}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}}+{\frac{2}{3}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230027, size = 1, normalized size = 0.02 \[ \left [\frac{1}{3} \, \sqrt{a} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + \frac{2}{3} \, \sqrt{b x^{3} + a}, -\frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) + \frac{2}{3} \, \sqrt{b x^{3} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.01579, size = 76, normalized size = 1.77 \[ - \frac{2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3} + \frac{2 a}{3 \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 \sqrt{b} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269739, size = 49, normalized size = 1.14 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2}{3} \, \sqrt{b x^{3} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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