Optimal. Leaf size=67 \[ -\frac{\sqrt{a+b \sqrt{c x^2}}}{x}-\frac{b \sqrt{c x^2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{\sqrt{a} x} \]
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Rubi [A] time = 0.0289001, antiderivative size = 67, 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, 47, 63, 208} \[ -\frac{\sqrt{a+b \sqrt{c x^2}}}{x}-\frac{b \sqrt{c x^2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{\sqrt{a} x} \]
Antiderivative was successfully verified.
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Rule 368
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^2} \, dx &=\frac{\sqrt{c x^2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\sqrt{c x^2}\right )}{x}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{x}+\frac{\left (b \sqrt{c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{c x^2}\right )}{2 x}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{x}+\frac{\sqrt{c x^2} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{c x^2}}\right )}{x}\\ &=-\frac{\sqrt{a+b \sqrt{c x^2}}}{x}-\frac{b \sqrt{c x^2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{\sqrt{a} x}\\ \end{align*}
Mathematica [A] time = 0.0484649, size = 87, normalized size = 1.3 \[ -\frac{b \sqrt{c x^2} \sqrt{\frac{b \sqrt{c x^2}}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b \sqrt{c x^2}}{a}+1}\right )+a+b \sqrt{c x^2}}{x \sqrt{a+b \sqrt{c x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 54, normalized size = 0.8 \begin{align*} -{\frac{1}{x} \left ({\it Artanh} \left ({\sqrt{a+b\sqrt{c{x}^{2}}}{\frac{1}{\sqrt{a}}}} \right ) b\sqrt{c{x}^{2}}+\sqrt{a+b\sqrt{c{x}^{2}}}\sqrt{a} \right ){\frac{1}{\sqrt{a}}}} \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{\sqrt{c x^{2}} b + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56047, size = 392, normalized size = 5.85 \begin{align*} \left [\frac{b x \sqrt{\frac{c}{a}} \log \left (\frac{b c x^{2} - 2 \, \sqrt{\sqrt{c x^{2}} b + a} a x \sqrt{\frac{c}{a}} + 2 \, \sqrt{c x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt{\sqrt{c x^{2}} b + a}}{2 \, x}, -\frac{b x \sqrt{-\frac{c}{a}} \arctan \left (-\frac{{\left (a b c x^{2} \sqrt{-\frac{c}{a}} - \sqrt{c x^{2}} a^{2} \sqrt{-\frac{c}{a}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) + \sqrt{\sqrt{c x^{2}} b + a}}{x}\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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15603, size = 73, normalized size = 1.09 \begin{align*} \frac{\frac{b^{2} c \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b \sqrt{c} x + a} b \sqrt{c}}{x}}{b \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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