Optimal. Leaf size=84 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(m+1) \sqrt{\frac{b}{a \sqrt{c x^3}}+1}} \]
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Rubi [A] time = 0.119183, antiderivative size = 84, 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} \[ \frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(m+1) \sqrt{\frac{b}{a \sqrt{c x^3}}+1}} \]
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}{\sqrt{c x^3}}} \, dx &=\operatorname{Subst}\left (\int \sqrt{a+\frac{b}{\sqrt{c} x^{3/2}}} (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}{\sqrt{c} x^{3/2}}} 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}{\sqrt{c} x^3}} 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^3}{\sqrt{c}}} \, 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}{\sqrt{c} x^{3/2}}} x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int x^{-1-2 (1+m)} \sqrt{1+\frac{b x^3}{a \sqrt{c}}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{\sqrt{1+\frac{b}{a \sqrt{c} x^{3/2}}}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (1+m);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(1+m) \sqrt{1+\frac{b}{a \sqrt{c x^3}}}}\\ \end{align*}
Mathematica [A] time = 0.109083, size = 89, normalized size = 1.06 \[ \frac{4 x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},\frac{2 m}{3}+\frac{1}{6};\frac{2 m}{3}+\frac{7}{6};-\frac{a \sqrt{c x^3}}{b}\right )}{(4 m+1) \sqrt{\frac{a \sqrt{c x^3}}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{a+{b{\frac{1}{\sqrt{c{x}^{3}}}}}}\, 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}{\sqrt{c x^{3}}}}\,{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}{\sqrt{c x^{3}}}}\, 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}{\sqrt{c x^{3}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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