Optimal. Leaf size=65 \[ -\frac{\sqrt{2} \sqrt{1-a x} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},p+\frac{3}{2},p+\frac{5}{2},\frac{1}{2} (1-a x)\right )}{a c (2 p+3)} \]
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Rubi [A] time = 0.0493803, antiderivative size = 65, 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 = {6130, 23, 69} \[ -\frac{\sqrt{2} \sqrt{1-a x} (c-a c x)^{p+1} \, _2F_1\left (\frac{1}{2},p+\frac{3}{2};p+\frac{5}{2};\frac{1}{2} (1-a x)\right )}{a c (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 69
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac{\sqrt{1-a x} (c-a c x)^p}{\sqrt{1+a x}} \, dx\\ &=\frac{\sqrt{1-a x} \int \frac{(c-a c x)^{\frac{1}{2}+p}}{\sqrt{1+a x}} \, dx}{\sqrt{c-a c x}}\\ &=-\frac{\sqrt{2} \sqrt{1-a x} (c-a c x)^{1+p} \, _2F_1\left (\frac{1}{2},\frac{3}{2}+p;\frac{5}{2}+p;\frac{1}{2} (1-a x)\right )}{a c (3+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0232447, size = 59, normalized size = 0.91 \[ \frac{\sqrt{2-2 a x} (a x-1) (c-a c x)^p \text{Hypergeometric2F1}\left (\frac{1}{2},p+\frac{3}{2},p+\frac{5}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.451, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p}}{ax+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{\sqrt{-a^{2} x^{2} + 1}{\left (-a c x + c\right )}^{p}}{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{\sqrt{-a^{2} x^{2} + 1}{\left (-a c x + c\right )}^{p}}{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{- \left (a x - 1\right ) \left (a x + 1\right )}}{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{\sqrt{-a^{2} x^{2} + 1}{\left (-a c x + c\right )}^{p}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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