Optimal. Leaf size=143 \[ \frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2}-p} (c-a c x)^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,1-p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{a p (p+1) \sqrt{1-\frac{1}{a x}}}+\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{p+1} \]
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Rubi [A] time = 0.157184, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 94, 132} \[ \frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2}-p} (c-a c x)^p \, _2F_1\left (\frac{1}{2}-p,-p;1-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{a p (p+1) \sqrt{1-\frac{1}{a x}}}+\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{p+1} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
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 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{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x (c-a c x)^p}{1+p}-\frac{\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^{-1-p} \left (1-\frac{x}{a}\right )^{-\frac{1}{2}+p}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a (1+p)}\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x (c-a c x)^p}{1+p}+\frac{\left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2}-p} \sqrt{1+\frac{1}{a x}} (c-a c x)^p \, _2F_1\left (\frac{1}{2}-p,-p;1-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{a p (1+p) \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0849435, size = 131, normalized size = 0.92 \[ \frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a x-1}{a x+1}\right )^{-p} (c-a c x)^p \left (\sqrt{\frac{a x-1}{a x+1}} \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,1-p,\frac{2}{a x+1}\right )+p (a x-1) \left (\frac{a x-1}{a x+1}\right )^p\right )}{a p (p+1) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.392, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -acx+c \right ) ^{p}{\frac{1}{\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 \frac{{\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 (\frac{{\left (a x + 1\right )}{\left (-a c x + c\right )}^{p} \sqrt{\frac{a x - 1}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right )\right )^{p}}{\sqrt{\frac{a x - 1}{a x + 1}}}\, 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 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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