Optimal. Leaf size=56 \[ \frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c} \]
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Rubi [A] time = 0.0380851, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c} \]
Antiderivative was successfully verified.
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Rule 369
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^2} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}}{x^2} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int 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 (2 \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b \sqrt{c} x}}{b \sqrt{c}}+\frac{\left (a+b \sqrt{c} x\right )^{3/2}}{b \sqrt{c}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.0335908, size = 43, normalized size = 0.77 \[ \frac{4 \left (2 a-3 b \sqrt{\frac{c}{x}}\right ) \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{15 b^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 70, normalized size = 1.3 \begin{align*} -{\frac{4}{15\,cx{b}^{2}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 3\,b\sqrt{{\frac{c}{x}}}-2\,a \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932362, size = 58, normalized size = 1.04 \begin{align*} -\frac{4 \,{\left (\frac{3 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43465, size = 104, normalized size = 1.86 \begin{align*} -\frac{4 \,{\left (a b x \sqrt{\frac{c}{x}} + 3 \, b^{2} c - 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{15 \, b^{2} c x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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