Optimal. Leaf size=46 \[ \frac{\sqrt{2} b \sinh ^{-1}\left (\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.621526, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2130, 215} \[ \frac{\sqrt{2} b \sinh ^{-1}\left (\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 2130
Rule 215
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx &=\frac{\left (\sqrt{2} b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,a x+b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right )}{a}\\ &=\frac{\sqrt{2} b \sinh ^{-1}\left (\frac{a x+b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [B] time = 1.07563, size = 148, normalized size = 3.22 \[ \frac{\sqrt{2} x \sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \tanh ^{-1}\left (\frac{\sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt{2} a x}\right )}{\sqrt{\frac{a \left (a x^2-1\right )}{b^2}} \left (x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{a{x}^{2}+bx\sqrt{-{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}{\frac{1}{\sqrt{-{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2} + \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} b x}}{\sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 71.2977, size = 385, normalized size = 8.37 \begin{align*} \left [\frac{\sqrt{2} b \log \left (-4 \, a x^{2} - 4 \, b x \sqrt{\frac{a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt{a x^{2} + b x \sqrt{\frac{a^{2} x^{2} - a}{b^{2}}}}{\left (\sqrt{2} \sqrt{a} x + \frac{\sqrt{2} b \sqrt{\frac{a^{2} x^{2} - a}{b^{2}}}}{\sqrt{a}}\right )} + 1\right )}{2 \, \sqrt{a}}, -\sqrt{2} b \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{a x^{2} + b x \sqrt{\frac{a^{2} x^{2} - a}{b^{2}}}} \sqrt{-\frac{1}{a}}}{2 \, x}\right )\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{x \left (a x + b \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}}\right )}}{x \sqrt{\frac{a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2} + \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} b x}}{\sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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