Optimal. Leaf size=102 \[ \frac{x (d x)^m \sqrt{a+\frac{b c^3 x^9}{\left (c x^3\right )^{9/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{9} (m+1);\frac{1}{9} (7-2 m);-\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}\right )}{(m+1) \sqrt{\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}+1}} \]
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Rubi [A] time = 0.122616, 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} \[ \frac{x (d x)^m \sqrt{a+\frac{b c^3 x^9}{\left (c x^3\right )^{9/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{9} (m+1);\frac{1}{9} (7-2 m);-\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}\right )}{(m+1) \sqrt{\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}+1}} \]
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 (d x)^m \sqrt{a+\frac{b}{\left (c x^3\right )^{3/2}}} \, dx &=\operatorname{Subst}\left (\int \sqrt{a+\frac{b}{c^{3/2} x^{9/2}}} (d x)^m \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\left (x^{-m} (d x)^m\right ) \int \sqrt{a+\frac{b}{c^{3/2} x^{9/2}}} x^m \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int \sqrt{a+\frac{b}{c^{3/2} x^9}} x^{-1+2 (1+m)} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\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^9}{c^{3/2}}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\operatorname{Subst}\left (\frac{\left (2 \sqrt{a+\frac{b}{c^{3/2} x^{9/2}}} x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int x^{-1-2 (1+m)} \sqrt{1+\frac{b x^9}{a c^{3/2}}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{\sqrt{1+\frac{b}{a c^{3/2} x^{9/2}}}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{x (d x)^m \sqrt{a+\frac{b c^3 x^9}{\left (c x^3\right )^{9/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{9} (1+m);\frac{1}{9} (7-2 m);-\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}\right )}{(1+m) \sqrt{1+\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}}}\\ \end{align*}
Mathematica [F] time = 0.135631, size = 0, normalized size = 0. \[ \int (d x)^m \sqrt{a+\frac{b}{\left (c x^3\right )^{3/2}}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{a+{b \left ( c{x}^{3} \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 (c x^{3}\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 (c x^{3}\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 (c x^{3}\right )^{\frac{3}{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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