Optimal. Leaf size=59 \[ -\frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2} \]
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Rubi [A] time = 0.0343941, 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, Rules used = {266, 50, 63, 208} \[ -\frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2}}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^3\right )\\ &=\frac{2}{9} \left (a+b x^3\right )^{3/2}+\frac{1}{3} a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^3\right )\\ &=\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2}+\frac{1}{3} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 b}\\ &=\frac{2}{3} a \sqrt{a+b x^3}+\frac{2}{9} \left (a+b x^3\right )^{3/2}-\frac{2}{3} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.030422, size = 51, normalized size = 0.86 \[ \frac{2}{9} \left (\sqrt{a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 48, normalized size = 0.8 \begin{align*}{\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 ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5212, size = 258, normalized size = 4.37 \begin{align*} \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}}{a}\right ) + \frac{2}{9} \,{\left (b x^{3} + 4 \, a\right )} \sqrt{b x^{3} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.30488, size = 83, normalized size = 1.41 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1304, size = 68, normalized size = 1.15 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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