Optimal. Leaf size=102 \[ \frac{x (d x)^m \sqrt{a+\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{c^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{a c^3}\right )}{(m+1) \sqrt{\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{a c^3}+1}} \]
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Rubi [A] time = 0.0874423, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {369, 343, 341, 365, 364} \[ \frac{x (d x)^m \sqrt{a+\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{c^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{a c^3}\right )}{(m+1) \sqrt{\frac{b x^3 \left (\frac{c}{x}\right )^{3/2}}{a c^3}+1}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 343
Rule 341
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{\left (\frac{c}{x}\right )^{3/2}}} (d x)^m \, dx &=\operatorname{Subst}\left (\int (d x)^m \sqrt{a+\frac{b x^{3/2}}{c^{3/2}}} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\left (x^{-m} (d x)^m\right ) \int x^m \sqrt{a+\frac{b x^{3/2}}{c^{3/2}}} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int x^{-1+2 (1+m)} \sqrt{a+\frac{b x^3}{c^{3/2}}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\frac{\left (2 x^{-m} (d x)^m \sqrt{a+\frac{b x^{3/2}}{c^{3/2}}}\right ) \operatorname{Subst}\left (\int x^{-1+2 (1+m)} \sqrt{1+\frac{b x^3}{a c^{3/2}}} \, dx,x,\sqrt{x}\right )}{\sqrt{1+\frac{b x^{3/2}}{a c^{3/2}}}},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{x (d x)^m \sqrt{a+\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{c^3}} \, _2F_1\left (-\frac{1}{2},\frac{2 (1+m)}{3};\frac{1}{3} (5+2 m);-\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{a c^3}\right )}{(1+m) \sqrt{1+\frac{b \left (\frac{c}{x}\right )^{3/2} x^3}{a c^3}}}\\ \end{align*}
Mathematica [F] time = 0.178989, size = 0, normalized size = 0. \[ \int \sqrt{a+\frac{b}{\left (\frac{c}{x}\right )^{3/2}}} (d x)^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{a+{b \left ({\frac{c}{x}} \right ) ^{-{\frac{3}{2}}}}}\, dx \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 \left (d x\right )^{m} \sqrt{a + \frac{b}{\left (\frac{c}{x}\right )^{\frac{3}{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \left (d x\right )^{m} \sqrt{a + \frac{b}{\left (\frac{c}{x}\right )^{\frac{3}{2}}}}\, 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 \left (d x\right )^{m} \sqrt{a + \frac{b}{\left (\frac{c}{x}\right )^{\frac{3}{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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