Optimal. Leaf size=94 \[ \frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-p-\frac{1}{2}} (c-a c x)^p \text{Hypergeometric2F1}\left (-p-1,-p-\frac{1}{2},-p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{p+1} \]
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Rubi [A] time = 0.121731, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6176, 6181, 132} \[ \frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-p-\frac{1}{2}} (c-a c x)^p \, _2F_1\left (-p-1,-p-\frac{1}{2};-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{p+1} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 132
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx &=\left (\left (1-\frac{1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^p x^p \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{x}\right )^p (c-a c x)^p\right ) \operatorname{Subst}\left (\int \frac{x^{-2-p} \left (1-\frac{x}{a}\right )^{\frac{1}{2}+p}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-\frac{1}{2}-p} \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x (c-a c x)^p \, _2F_1\left (-1-p,-\frac{1}{2}-p;-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{1+p}\\ \end{align*}
Mathematica [A] time = 0.0446656, size = 76, normalized size = 0.81 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (\frac{a x-1}{a x+1}\right )^{-p-\frac{1}{2}} (c-a c x)^p \text{Hypergeometric2F1}\left (-p-1,-p-\frac{1}{2},-p,\frac{2}{a x+1}\right )}{p+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.356, size = 0, normalized size = 0. \begin{align*} \int \left ( -acx+c \right ) ^{p}\sqrt{{\frac{ax-1}{ax+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{\left (-a c x + c\right )}^{p} \sqrt{\frac{a x - 1}{a x + 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 ({\left (-a c x + c\right )}^{p} \sqrt{\frac{a x - 1}{a x + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (-a c x + c\right )}^{p} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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