Optimal. Leaf size=81 \[ -\frac{2}{3} a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b} \]
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Rubi [A] time = 0.0566899, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 80, 50, 63, 208} \[ -\frac{2}{3} a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (A+B x)}{x} \, dx,x,x^3\right )\\ &=\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b}+\frac{1}{3} A \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^3\right )\\ &=\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b}+\frac{1}{3} (a A) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^3\right )\\ &=\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b}+\frac{1}{3} \left (a^2 A\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b}+\frac{\left (2 a^2 A\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 A \sqrt{a+b x^3}+\frac{2}{9} A \left (a+b x^3\right )^{3/2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b}-\frac{2}{3} a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0671314, size = 80, normalized size = 0.99 \[ \frac{2 \left (-15 a^{3/2} A b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+5 A b \left (a+b x^3\right )^{3/2}+15 a A b \sqrt{a+b x^3}+3 B \left (a+b x^3\right )^{5/2}\right )}{45 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 66, normalized size = 0.8 \begin{align*}{\frac{2\,B}{15\,b} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}}+A \left ({\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 ) } \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 = 1.74219, size = 409, normalized size = 5.05 \begin{align*} \left [\frac{15 \, A a^{\frac{3}{2}} b \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (3 \, B b^{2} x^{6} +{\left (6 \, B a b + 5 \, A b^{2}\right )} x^{3} + 3 \, B a^{2} + 20 \, A a b\right )} \sqrt{b x^{3} + a}}{45 \, b}, \frac{2 \,{\left (15 \, A \sqrt{-a} a b \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, B b^{2} x^{6} +{\left (6 \, B a b + 5 \, A b^{2}\right )} x^{3} + 3 \, B a^{2} + 20 \, A a b\right )} \sqrt{b x^{3} + a}\right )}}{45 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 39.0241, size = 82, normalized size = 1.01 \begin{align*} \frac{2 A a^{2} \operatorname{atan}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{- a}} \right )}}{3 \sqrt{- a}} + \frac{2 A a \sqrt{a + b x^{3}}}{3} + \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{9} + \frac{2 B \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26816, size = 108, normalized size = 1.33 \begin{align*} \frac{2 \, A a^{2} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} B b^{4} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{5} + 15 \, \sqrt{b x^{3} + a} A a b^{5}\right )}}{45 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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