Optimal. Leaf size=55 \[ \frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}-\frac{4}{9} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0575646, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {369, 266, 50, 63, 208} \[ \frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}-\frac{4}{9} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b c^{3/2} x^{9/2}}}{x} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{9} \operatorname{Subst}\left (\int \frac{\sqrt{a+b c^{3/2} x}}{x} \, dx,x,x^{9/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}+\operatorname{Subst}\left (\frac{1}{9} (2 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b c^{3/2} x}} \, dx,x,x^{9/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}+\operatorname{Subst}\left (\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b c^{3/2}}+\frac{x^2}{b c^{3/2}}} \, dx,x,\sqrt{a+b c^{3/2} x^{9/2}}\right )}{9 b c^{3/2}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}-\frac{4}{9} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0587139, size = 55, normalized size = 1. \[ \frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}-\frac{4}{9} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}\, dx \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24533, size = 159, normalized size = 2.89 \begin{align*} \frac{4 \,{\left (\frac{a c^{2} \arctan \left (\frac{\sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}}}{\sqrt{-a c} c}\right )}{\sqrt{-a c}} + \sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}} - \frac{a c^{2} \arctan \left (\frac{\sqrt{a c^{3}}}{\sqrt{-a c} c}\right ) + \sqrt{a c^{3}} \sqrt{-a c}}{\sqrt{-a c}}\right )}{\left | c \right |}}{9 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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