Optimal. Leaf size=65 \[ -\frac{(1-a x)^{3/2} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (\frac{3}{2},p+\frac{5}{2},p+\frac{7}{2},\frac{1}{2} (1-a x)\right )}{\sqrt{2} a c (2 p+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0521735, 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} \[ -\frac{(1-a x)^{3/2} (c-a c x)^{p+1} \, _2F_1\left (\frac{3}{2},p+\frac{5}{2};p+\frac{7}{2};\frac{1}{2} (1-a x)\right )}{\sqrt{2} a c (2 p+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
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{(1-a x)^{3/2} \int \frac{(c-a c x)^{\frac{3}{2}+p}}{(1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{(1-a x)^{3/2} (c-a c x)^{1+p} \, _2F_1\left (\frac{3}{2},\frac{5}{2}+p;\frac{7}{2}+p;\frac{1}{2} (1-a x)\right )}{\sqrt{2} a c (5+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0275208, size = 60, normalized size = 0.92 \[ -\frac{(1-a x)^{5/2} (c-a c x)^p \text{Hypergeometric2F1}\left (\frac{3}{2},p+\frac{5}{2},p+\frac{7}{2},\frac{1}{2}-\frac{a x}{2}\right )}{\sqrt{2} a (2 p+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.474, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p}}{ \left ( ax+1 \right ) ^{3}} \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^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (-a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{3}}\,{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 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.
[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 (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\left (a x + 1\right )^{3}}\, 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^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (-a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]