Optimal. Leaf size=76 \[ -\frac{4 a c (d x)^m \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} \left (-\frac{b}{a \sqrt{\frac{c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac{1}{2},-2 m-1;\frac{3}{2};\frac{b}{a \sqrt{\frac{c}{x}}}+1\right )}{b^2} \]
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Rubi [A] time = 0.0667351, antiderivative size = 76, 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, 67, 65} \[ -\frac{4 a c (d x)^m \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} \left (-\frac{b}{a \sqrt{\frac{c}{x}}}\right )^{-2 m} \, _2F_1\left (\frac{1}{2},-2 m-1;\frac{3}{2};\frac{b}{a \sqrt{\frac{c}{x}}}+1\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 369
Rule 343
Rule 341
Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}}} \, dx &=\operatorname{Subst}\left (\int \frac{(d x)^m}{\sqrt{a+\frac{b \sqrt{x}}{\sqrt{c}}}} \, 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 \sqrt{x}}{\sqrt{c}}}} \, 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 x}{\sqrt{c}}}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (\frac{\left (2 a \sqrt{c} \left (-\frac{b \sqrt{x}}{a \sqrt{c}}\right )^{-2 m} (d x)^m\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{b x}{a \sqrt{c}}\right )^{-1+2 (1+m)}}{\sqrt{a+\frac{b x}{\sqrt{c}}}} \, dx,x,\sqrt{x}\right )}{b},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\frac{4 a c \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} \left (-\frac{b}{a \sqrt{\frac{c}{x}}}\right )^{-2 m} (d x)^m \, _2F_1\left (\frac{1}{2},-1-2 m;\frac{3}{2};1+\frac{b}{a \sqrt{\frac{c}{x}}}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.364293, size = 116, normalized size = 1.53 \[ \frac{a^2 c (d x)^m \left (\frac{a \sqrt{\frac{c}{x}}}{a \sqrt{\frac{c}{x}}+b}\right )^{2 m-\frac{1}{2}} \, _2F_1\left (2 m+2,2 m+\frac{5}{2};2 m+3;\frac{b}{\sqrt{\frac{c}{x}} a+b}\right )}{(m+1) \sqrt{a+\frac{b}{\sqrt{\frac{c}{x}}}} \left (a \sqrt{\frac{c}{x}}+b\right )^2} \]
Antiderivative was successfully verified.
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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{\frac{1}{\sqrt{{\frac{c}{x}}}}}}}}}}\, 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{a + \frac{b}{\sqrt{\frac{c}{x}}}}}\,{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 + \frac{b}{\sqrt{\frac{c}{x}}}}}\, 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{a + \frac{b}{\sqrt{\frac{c}{x}}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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