Optimal. Leaf size=31 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0326503, antiderivative size = 31, 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 = {369, 266, 63, 208} \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (\frac{4 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b \sqrt{c}}+\frac{x^2}{b \sqrt{c}}} \, dx,x,\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}\right )}{b \sqrt{c}},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0430557, size = 31, normalized size = 1. \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 203, normalized size = 6.6 \begin{align*} -{\frac{1}{b}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( -b\sqrt{{\frac{c}{x}}}\sqrt{x}\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) -b\sqrt{{\frac{c}{x}}}\sqrt{x}\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) +2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}-2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{\frac{1}{\sqrt{{\frac{c}{x}}}}}{\frac{1}{\sqrt{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 = 1.44002, size = 194, normalized size = 6.26 \begin{align*} \left [\frac{2 \, \log \left (2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a} x \sqrt{\frac{c}{x}} + 2 \, a x \sqrt{\frac{c}{x}} + b c\right )}{\sqrt{a}}, -\frac{4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}{a}\right )}{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{1}{x \sqrt{a + b \sqrt{\frac{c}{x}}}}\, 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{1}{\sqrt{b \sqrt{\frac{c}{x}} + a} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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