Optimal. Leaf size=51 \[ 2 \sqrt{a+b \sqrt{c x^2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0239181, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {368, 50, 63, 208} \[ 2 \sqrt{a+b \sqrt{c x^2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 368
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sqrt{c x^2}\right )\\ &=2 \sqrt{a+b \sqrt{c x^2}}+a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )\\ &=2 \sqrt{a+b \sqrt{c x^2}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{c x^2}}\right )}{b}\\ &=2 \sqrt{a+b \sqrt{c x^2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0139572, size = 51, normalized size = 1. \[ 2 \sqrt{a+b \sqrt{c x^2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 40, normalized size = 0.8 \begin{align*} -2\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ) \sqrt{a}+2\,\sqrt{a+b\sqrt{c{x}^{2}}} \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 = 1.33027, size = 282, normalized size = 5.53 \begin{align*} \left [\sqrt{a} \log \left (\frac{b c x^{2} - 2 \, \sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a} \sqrt{a} + 2 \, \sqrt{c x^{2}} a}{x^{2}}\right ) + 2 \, \sqrt{\sqrt{c x^{2}} b + a}, 2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{\sqrt{c x^{2}} b + a} \sqrt{-a}}{a}\right ) + 2 \, \sqrt{\sqrt{c x^{2}} b + 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 \sqrt{c x^{2}}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2498, size = 51, normalized size = 1. \begin{align*} \frac{2 \, a \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b \sqrt{c} x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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