Optimal. Leaf size=85 \[ \frac{2^{p+\frac{5}{2}} (1-a x)^{p-\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (-p-\frac{3}{2},p-\frac{1}{2},p+\frac{1}{2},\frac{1}{2} (1-a x)\right )}{a (1-2 p)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0740719, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6143, 6140, 69} \[ \frac{2^{p+\frac{5}{2}} (1-a x)^{p-\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-\frac{3}{2},p-\frac{1}{2};p+\frac{1}{2};\frac{1}{2} (1-a x)\right )}{a (1-2 p)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6143
Rule 6140
Rule 69
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{-\frac{3}{2}+p} (1+a x)^{\frac{3}{2}+p} \, dx\\ &=\frac{2^{\frac{5}{2}+p} (1-a x)^{-\frac{1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac{3}{2}-p,-\frac{1}{2}+p;\frac{1}{2}+p;\frac{1}{2} (1-a x)\right )}{a (1-2 p)}\\ \end{align*}
Mathematica [A] time = 0.0291076, size = 83, normalized size = 0.98 \[ \frac{2^{p+\frac{5}{2}} (1-a x)^{p-\frac{1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (-p-\frac{3}{2},p-\frac{1}{2},p+\frac{1}{2},\frac{1}{2} (1-a x)\right )}{a-2 a p} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.395, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) ^{3} \left ( -{a}^{2}c{x}^{2}+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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (-a^{2} c x^{2} + 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.
[In]
[Out]
________________________________________________________________________________________
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^{2} c x^{2} + 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (a x - 1\right ) \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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (-a^{2} c x^{2} + 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.
[In]
[Out]