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

Optimal. Leaf size=59 \[ -\frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2} \]

[Out]

(2*a*Sqrt[a + b*x^3])/3 + (2*(a + b*x^3)^(3/2))/9 - (2*a^(3/2)*ArcTanh[Sqrt[a +
b*x^3]/Sqrt[a]])/3

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Rubi [A]  time = 0.0928763, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(3/2)/x,x]

[Out]

(2*a*Sqrt[a + b*x^3])/3 + (2*(a + b*x^3)^(3/2))/9 - (2*a^(3/2)*ArcTanh[Sqrt[a +
b*x^3]/Sqrt[a]])/3

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Rubi in Sympy [A]  time = 8.95038, size = 53, normalized size = 0.9 \[ - \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3} + \frac{2 a \sqrt{a + b x^{3}}}{3} + \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(3/2)*atanh(sqrt(a + b*x**3)/sqrt(a))/3 + 2*a*sqrt(a + b*x**3)/3 + 2*(a +
b*x**3)**(3/2)/9

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Mathematica [A]  time = 0.176983, size = 61, normalized size = 1.03 \[ \frac{1}{3} \sqrt{a+b x^3} \left (\frac{2}{3} \left (4 a+b x^3\right )-\frac{2 a \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 48, normalized size = 0.8 \[{\frac{2\,b{x}^{3}}{9}\sqrt{b{x}^{3}+a}}+{\frac{8\,a}{9}\sqrt{b{x}^{3}+a}}-{\frac{2}{3}{a}^{{\frac{3}{2}}}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(3/2)/x,x)

[Out]

2/9*b*x^3*(b*x^3+a)^(1/2)+8/9*a*(b*x^3+a)^(1/2)-2/3*a^(3/2)*arctanh((b*x^3+a)^(1
/2)/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((b*x^3 + a)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230769, size = 1, normalized size = 0.02 \[ \left [\frac{1}{3} \, a^{\frac{3}{2}} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + \frac{2}{9} \,{\left (b x^{3} + 4 \, a\right )} \sqrt{b x^{3} + a}, -\frac{2}{3} \, \sqrt{-a} a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) + \frac{2}{9} \,{\left (b x^{3} + 4 \, a\right )} \sqrt{b x^{3} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*a^(3/2)*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2/9*(b*x^3 + 4
*a)*sqrt(b*x^3 + a), -2/3*sqrt(-a)*a*arctan(sqrt(b*x^3 + a)/sqrt(-a)) + 2/9*(b*x
^3 + 4*a)*sqrt(b*x^3 + a)]

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Sympy [A]  time = 7.00038, size = 83, normalized size = 1.41 \[ \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{9} + \frac{a^{\frac{3}{2}} \log{\left (\frac{b x^{3}}{a} \right )}}{3} - \frac{2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3} + \frac{2 \sqrt{a} b x^{3} \sqrt{1 + \frac{b x^{3}}{a}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260912, size = 68, normalized size = 1.15 \[ \frac{2 \, a^{2} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2}{9} \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{b x^{3} + a} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*a^2*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + 2/9*(b*x^3 + a)^(3/2) + 2/3*
sqrt(b*x^3 + a)*a