Optimal. Leaf size=51 \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
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Rubi [A] time = 0.0349198, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {369, 266, 50, 63, 208} \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\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{\sqrt{a+b \sqrt{c} x}}{x} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-4 \sqrt{a+b \sqrt{\frac{c}{x}}}-\operatorname{Subst}\left ((2 a) \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 )\\ &=-4 \sqrt{a+b \sqrt{\frac{c}{x}}}-\operatorname{Subst}\left (\frac{(4 a) \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 )\\ &=-4 \sqrt{a+b \sqrt{\frac{c}{x}}}+4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0465543, size = 51, normalized size = 1. \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 150, normalized size = 2.9 \begin{align*} 2\,{\frac{1}{bx\sqrt{a}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( \ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{3/2}ab+2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}x-2\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{\frac{1}{\sqrt{{\frac{c}{x}}}}}} \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 = 1.4072, size = 262, normalized size = 5.14 \begin{align*} \left [2 \, \sqrt{a} \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 ) - 4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}, -4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}{a}\right ) - 4 \, \sqrt{b \sqrt{\frac{c}{x}} + 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{\frac{c}{x}}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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