Optimal. Leaf size=54 \[ a^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2} \]
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Rubi [A] time = 0.0356325, antiderivative size = 54, 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} \[ a^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2} \]
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^2\right )^{3/2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{3} \left (a+b x^2\right )^{3/2}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )\\ &=a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{b}\\ &=a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2}-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.020053, size = 50, normalized size = 0.93 \[ \frac{1}{3} \sqrt{a+b x^2} \left (4 a+b x^2\right )-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 52, normalized size = 1. \begin{align*}{\frac{1}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +a\sqrt{b{x}^{2}+a} \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 = 1.58719, size = 251, normalized size = 4.65 \begin{align*} \left [\frac{1}{2} \, a^{\frac{3}{2}} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \frac{1}{3} \,{\left (b x^{2} + 4 \, a\right )} \sqrt{b x^{2} + a}, \sqrt{-a} a \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + \frac{1}{3} \,{\left (b x^{2} + 4 \, a\right )} \sqrt{b x^{2} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.86413, size = 78, normalized size = 1.44 \begin{align*} \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{3} + \frac{a^{\frac{3}{2}} \log{\left (\frac{b x^{2}}{a} \right )}}{2} - a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )} + \frac{\sqrt{a} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37338, size = 65, normalized size = 1.2 \begin{align*} \frac{a^{2} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{1}{3} \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} + \sqrt{b x^{2} + a} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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