Optimal. Leaf size=55 \[ \frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0429101, 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 = {368, 266, 50, 63, 208} \[ \frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 368
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^3}}{x} \, dx,x,\sqrt{c x^2}\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\left (c x^2\right )^{3/2}\right )\\ &=\frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}+\frac{1}{3} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right )\\ &=\frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \left (c x^2\right )^{3/2}}\right )}{3 b}\\ &=\frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0239491, size = 55, normalized size = 1. \[ \frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{a+b \left ( c{x}^{2} \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 [A] time = 1.35816, size = 331, normalized size = 6.02 \begin{align*} \left [\frac{1}{3} \, \sqrt{a} \log \left (\frac{b c^{2} x^{4} - 2 \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a} \sqrt{c x^{2}} \sqrt{a} + 2 \, \sqrt{c x^{2}} a}{x^{4}}\right ) + \frac{2}{3} \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a}, \frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{\sqrt{c x^{2}} b c x^{2} + a} \sqrt{-a}}{a}\right ) + \frac{2}{3} \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a}\right ] \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^{2}\right )^{\frac{3}{2}}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14873, size = 57, normalized size = 1.04 \begin{align*} \frac{2 \, a \arctan \left (\frac{\sqrt{b c^{\frac{3}{2}} x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2}{3} \, \sqrt{b c^{\frac{3}{2}} x^{3} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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