Optimal. Leaf size=202 \[ -\frac{3 \sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{3}{2}-p} (c-a c x)^p \text{Hypergeometric2F1}\left (1-p,\frac{3}{2}-p,2-p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{a^2 p \left (1-p^2\right ) x \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{x \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^p}{(p+1) \sqrt{1-\frac{1}{a x}}}+\frac{3 \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{a p (p+1) \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.206049, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6176, 6181, 94, 132} \[ -\frac{3 \sqrt{\frac{1}{a x}+1} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{3}{2}-p} (c-a c x)^p \, _2F_1\left (1-p,\frac{3}{2}-p;2-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{a^2 p \left (1-p^2\right ) x \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{x \left (\frac{1}{a x}+1\right )^{3/2} (c-a c x)^p}{(p+1) \sqrt{1-\frac{1}{a x}}}+\frac{3 \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{a p (p+1) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 132
Rubi steps
\begin{align*} \int e^{3 \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^{3 \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{3}{2}+p} \left (1+\frac{x}{a}\right )^{3/2} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt{1-\frac{1}{a x}}}-\frac{\left (3 \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^{-1-p} \left (1-\frac{x}{a}\right )^{-\frac{3}{2}+p} \sqrt{1+\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{a (1+p)}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} (c-a c x)^p}{a p (1+p) \sqrt{1-\frac{1}{a x}}}+\frac{\left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt{1-\frac{1}{a x}}}-\frac{\left (3 \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^{-p} \left (1-\frac{x}{a}\right )^{-\frac{3}{2}+p}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^2 p (1+p)}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} (c-a c x)^p}{a p (1+p) \sqrt{1-\frac{1}{a x}}}+\frac{\left (1+\frac{1}{a x}\right )^{3/2} x (c-a c x)^p}{(1+p) \sqrt{1-\frac{1}{a x}}}-\frac{3 \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{3}{2}-p} \sqrt{1+\frac{1}{a x}} (c-a c x)^p \, _2F_1\left (1-p,\frac{3}{2}-p;2-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{a^2 p \left (1-p^2\right ) \left (1-\frac{1}{a x}\right )^{3/2} x}\\ \end{align*}
Mathematica [A] time = 0.136312, size = 155, normalized size = 0.77 \[ \frac{\sqrt{\frac{1}{a x}+1} \left (\frac{a x-1}{a x+1}\right )^{-p} (c-a c x)^p \left (3 \sqrt{\frac{a x-1}{a x+1}} \text{Hypergeometric2F1}\left (1-p,\frac{3}{2}-p,2-p,\frac{2}{a x+1}\right )+(p-1) (a x+1) (a p x+p+3) \left (\frac{a x-1}{a x+1}\right )^p\right )}{a (p-1) p (p+1) \sqrt{1-\frac{1}{a x}} (a x+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.409, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -acx+c \right ) ^{p} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}}\, 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}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{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^{2} x^{2} + 2 \, a x + 1\right )}{\left (-a c x + c\right )}^{p} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} x^{2} - 2 \, 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 \frac{{\left (-a c x + c\right )}^{p}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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