Optimal. Leaf size=65 \[ -\frac{2 \sqrt{2} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},p+\frac{1}{2},p+\frac{3}{2},\frac{1}{2} (1-a x)\right )}{a c (2 p+1) \sqrt{1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0490697, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6130, 23, 69} \[ -\frac{2 \sqrt{2} (c-a c x)^{p+1} \, _2F_1\left (-\frac{1}{2},p+\frac{1}{2};p+\frac{3}{2};\frac{1}{2} (1-a x)\right )}{a c (2 p+1) \sqrt{1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
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{c-a c x} \int \sqrt{1+a x} (c-a c x)^{-\frac{1}{2}+p} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{2} (c-a c x)^{1+p} \, _2F_1\left (-\frac{1}{2},\frac{1}{2}+p;\frac{3}{2}+p;\frac{1}{2} (1-a x)\right )}{a c (1+2 p) \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0307093, size = 53, normalized size = 0.82 \[ -\frac{2 \sqrt{2-2 a x} (c-a c x)^p \text{Hypergeometric2F1}\left (-\frac{1}{2},p+\frac{1}{2},p+\frac{3}{2},\frac{1}{2}-\frac{a x}{2}\right )}{2 a p+a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.378, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) \left ( -acx+c \right ) ^{p}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, 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 )}{\left (-a c x + c\right )}^{p}}{\sqrt{-a^{2} x^{2} + 1}}\,{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 c x + c\right )}^{p}}{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 )\right )^{p} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 )}{\left (-a c x + c\right )}^{p}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]