Optimal. Leaf size=116 \[ -\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2} \]
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Rubi [A] time = 0.0686195, antiderivative size = 116, 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} \[ -\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2} \]
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^3} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}}{x^3} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int x^3 \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^3 \sqrt{a+b \sqrt{c} x}}{b^3 c^{3/2}}+\frac{3 a^2 \left (a+b \sqrt{c} x\right )^{3/2}}{b^3 c^{3/2}}-\frac{3 a \left (a+b \sqrt{c} x\right )^{5/2}}{b^3 c^{3/2}}+\frac{\left (a+b \sqrt{c} x\right )^{7/2}}{b^3 c^{3/2}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}\\ \end{align*}
Mathematica [A] time = 0.0500101, size = 75, normalized size = 0.65 \[ \frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (-24 a^2 b x \sqrt{\frac{c}{x}}+16 a^3 x+30 a b^2 c-35 b^3 c \sqrt{\frac{c}{x}}\right )}{315 b^4 c^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 97, normalized size = 0.8 \begin{align*} -{\frac{4}{315\,{c}^{2}{x}^{2}{b}^{4}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 35\, \left ({\frac{c}{x}} \right ) ^{3/2}x{b}^{3}+24\,\sqrt{{\frac{c}{x}}}x{a}^{2}b-30\,ac{b}^{2}-16\,{a}^{3}x \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 = 0.931915, size = 115, normalized size = 0.99 \begin{align*} -\frac{4 \,{\left (\frac{35 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48611, size = 170, normalized size = 1.47 \begin{align*} -\frac{4 \,{\left (35 \, b^{4} c^{2} - 6 \, a^{2} b^{2} c x - 16 \, a^{4} x^{2} +{\left (5 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{315 \, b^{4} c^{2} x^{2}} \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^{3}}\, 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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