Optimal. Leaf size=57 \[ \frac{2 x^{m+1} \sqrt{c-\frac{c}{a x}} \text{Hypergeometric2F1}\left (-\frac{1}{2},m+\frac{1}{2},m+\frac{3}{2},-a x\right )}{(2 m+1) \sqrt{1-a x}} \]
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Rubi [A] time = 0.229812, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6134, 6128, 848, 64} \[ \frac{2 x^{m+1} \sqrt{c-\frac{c}{a x}} \, _2F_1\left (-\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};-a x\right )}{(2 m+1) \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6128
Rule 848
Rule 64
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^m \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{\tanh ^{-1}(a x)} x^{-\frac{1}{2}+m} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{-\frac{1}{2}+m} \sqrt{1-a^2 x^2}}{\sqrt{1-a x}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int x^{-\frac{1}{2}+m} \sqrt{1+a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{2 \sqrt{c-\frac{c}{a x}} x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1}{2}+m;\frac{3}{2}+m;-a x\right )}{(1+2 m) \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0268785, size = 56, normalized size = 0.98 \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a x}} \text{Hypergeometric2F1}\left (-\frac{1}{2},m+\frac{1}{2},m+\frac{3}{2},-a x\right )}{\left (m+\frac{1}{2}\right ) \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.296, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ){x}^{m}\sqrt{c-{\frac{c}{ax}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, 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 (a x + 1\right )} \sqrt{c - \frac{c}{a x}} x^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{m} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}, x\right ) \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{x^{m} \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 (a x + 1\right )} \sqrt{c - \frac{c}{a x}} x^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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