Optimal. Leaf size=94 \[ \frac{x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-p-\frac{3}{2}} (c-a c x)^p \text{Hypergeometric2F1}\left (-p-\frac{3}{2},-p-1,-p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{(p+1) \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.125535, 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} \[ \frac{x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-p-\frac{3}{2}} (c-a c x)^p \, _2F_1\left (-p-\frac{3}{2},-p-1;-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{(p+1) \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
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 \frac{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 (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{-\frac{3}{2}-p} \left (1-\frac{1}{a x}\right )^{3/2} x (c-a c x)^p \, _2F_1\left (-\frac{3}{2}-p,-1-p;-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{(1+p) \sqrt{1+\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0613283, size = 96, normalized size = 1.02 \[ \frac{\sqrt{1-\frac{1}{a x}} (a x+1) \left (\frac{a x-1}{a x+1}\right )^{-p-\frac{1}{2}} (c-a c x)^p \text{Hypergeometric2F1}\left (-p-\frac{3}{2},-p-1,-p,\frac{2}{a x+1}\right )}{a (p+1) \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.365, 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{\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 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(-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} \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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