Optimal. Leaf size=102 \[ \frac{x (d x)^m \sqrt{\frac{b c^3}{a x^3 \left (\frac{c}{x}\right )^{3/2}}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b c^3}{a \left (\frac{c}{x}\right )^{3/2} x^3}\right )}{(m+1) \sqrt{a+\frac{b c^3}{x^3 \left (\frac{c}{x}\right )^{3/2}}}} \]
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Rubi [A] time = 0.0968887, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {369, 343, 341, 339, 365, 364} \[ \frac{x (d x)^m \sqrt{\frac{b c^3}{a x^3 \left (\frac{c}{x}\right )^{3/2}}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b c^3}{a \left (\frac{c}{x}\right )^{3/2} x^3}\right )}{(m+1) \sqrt{a+\frac{b c^3}{x^3 \left (\frac{c}{x}\right )^{3/2}}}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 343
Rule 341
Rule 339
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\sqrt{a+b \left (\frac{c}{x}\right )^{3/2}}} \, dx &=\operatorname{Subst}\left (\int \frac{(d x)^m}{\sqrt{a+\frac{b c^{3/2}}{x^{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 \frac{x^m}{\sqrt{a+\frac{b c^{3/2}}{x^{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 \frac{x^{-1+2 (1+m)}}{\sqrt{a+\frac{b c^{3/2}}{x^3}}} \, dx,x,\sqrt{x}\right ),\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 \frac{x^{-1-2 (1+m)}}{\sqrt{a+b c^{3/2} x^3}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (\frac{\left (2 \sqrt{1+\frac{b c^{3/2}}{a x^{3/2}}} x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-2 (1+m)}}{\sqrt{1+\frac{b c^{3/2} x^3}{a}}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{\sqrt{a+\frac{b c^{3/2}}{x^{3/2}}}},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{\sqrt{1+\frac{b c^3}{a \left (\frac{c}{x}\right )^{3/2} x^3}} x (d x)^m \, _2F_1\left (\frac{1}{2},-\frac{2}{3} (1+m);\frac{1}{3} (1-2 m);-\frac{b c^3}{a \left (\frac{c}{x}\right )^{3/2} x^3}\right )}{(1+m) \sqrt{a+\frac{b c^3}{\left (\frac{c}{x}\right )^{3/2} x^3}}}\\ \end{align*}
Mathematica [F] time = 0.172472, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\sqrt{a+b \left (\frac{c}{x}\right )^{3/2}}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m}{\frac{1}{\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 \frac{\left (d x\right )^{m}}{\sqrt{b \left (\frac{c}{x}\right )^{\frac{3}{2}} + a}}\,{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 \frac{\left (d x\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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{\sqrt{b \left (\frac{c}{x}\right )^{\frac{3}{2}} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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