Optimal. Leaf size=55 \[ \frac{4}{3} \sqrt{a+b \sqrt{c x^3}}-\frac{4}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0347186, 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} \[ \frac{4}{3} \sqrt{a+b \sqrt{c x^3}}-\frac{4}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right ) \]
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 \sqrt{c x^3}}}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x^{3/2}}}{x} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x}}{x} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{3} \sqrt{a+b \sqrt{c x^3}}+\operatorname{Subst}\left (\frac{1}{3} (2 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{3} \sqrt{a+b \sqrt{c x^3}}+\operatorname{Subst}\left (\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b \sqrt{c}}+\frac{x^2}{b \sqrt{c}}} \, dx,x,\sqrt{a+b \sqrt{c} x^{3/2}}\right )}{3 b \sqrt{c}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{3} \sqrt{a+b \sqrt{c x^3}}-\frac{4}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0432338, size = 55, normalized size = 1. \[ \frac{4}{3} \sqrt{a+b \sqrt{c x^3}}-\frac{4}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 40, normalized size = 0.7 \begin{align*} -{\frac{4}{3}{\it Artanh} \left ({\sqrt{a+b\sqrt{c{x}^{3}}}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}}+{\frac{4}{3}\sqrt{a+b\sqrt{c{x}^{3}}}} \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 \sqrt{c x^{3}}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13353, size = 124, normalized size = 2.25 \begin{align*} \frac{4 \,{\left (\frac{a c \arctan \left (\frac{\sqrt{\sqrt{c x} b c x + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c}} + \sqrt{\sqrt{c x} b c x + a c} - \frac{a c \arctan \left (\frac{\sqrt{a c}}{\sqrt{-a c}}\right ) + \sqrt{a c} \sqrt{-a c}}{\sqrt{-a c}}\right )}{\left | c \right |}}{3 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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