Optimal. Leaf size=65 \[ \frac{4 \sqrt{2} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (-\frac{3}{2},p-\frac{1}{2},p+\frac{1}{2},\frac{1}{2} (1-a x)\right )}{a c (1-2 p) (1-a x)^{3/2}} \]
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Rubi [A] time = 0.0518962, 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} \[ \frac{4 \sqrt{2} (c-a c x)^{p+1} \, _2F_1\left (-\frac{3}{2},p-\frac{1}{2};p+\frac{1}{2};\frac{1}{2} (1-a x)\right )}{a c (1-2 p) (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 69
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac{(1+a x)^{3/2} (c-a c x)^p}{(1-a x)^{3/2}} \, dx\\ &=\frac{(c-a c x)^{3/2} \int (1+a x)^{3/2} (c-a c x)^{-\frac{3}{2}+p} \, dx}{(1-a x)^{3/2}}\\ &=\frac{4 \sqrt{2} (c-a c x)^{1+p} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2}+p;\frac{1}{2}+p;\frac{1}{2} (1-a x)\right )}{a c (1-2 p) (1-a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0235826, size = 58, normalized size = 0.89 \[ \frac{4 \sqrt{2} (c-a c x)^p \text{Hypergeometric2F1}\left (-\frac{3}{2},p-\frac{1}{2},p+\frac{1}{2},\frac{1}{2}-\frac{a x}{2}\right )}{(a-2 a p) \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.47, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) ^{3} \left ( -acx+c \right ) ^{p} \left ( -{a}^{2}{x}^{2}+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 x + 1\right )}^{3}{\left (-a c x + c\right )}^{p}}{{\left (-a^{2} x^{2} + 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{\sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )}{\left (-a c x + c\right )}^{p}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 x + 1\right )}^{3}{\left (-a c x + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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