Optimal. Leaf size=104 \[ \frac{x \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^p \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \text{Hypergeometric2F1}\left (\frac{1}{2} (n-2 p),-p-1,-p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{p+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.127591, antiderivative size = 104, 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 = {6176, 6181, 132} \[ \frac{x \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^p \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \, _2F_1\left (\frac{1}{2} (n-2 p),-p-1;-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{p+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6176
Rule 6181
Rule 132
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx &=\left (\left (1-\frac{1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^p x^p \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{x}\right )^p (c-a c x)^p\right ) \operatorname{Subst}\left (\int x^{-2-p} \left (1-\frac{x}{a}\right )^{-\frac{n}{2}+p} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^p \, _2F_1\left (\frac{1}{2} (n-2 p),-1-p;-p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{1+p}\\ \end{align*}
Mathematica [A] time = 0.0476149, size = 104, normalized size = 1. \[ \frac{(a x+1) \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} (c-a c x)^p \left (\frac{a x-1}{a x+1}\right )^{\frac{1}{2} (n-2 p)} \text{Hypergeometric2F1}\left (-p-1,\frac{n}{2}-p,-p,\frac{2}{a x+1}\right )}{a (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+c \right ) ^{p}\, 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{\left (-a c x + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{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 ({\left (-a c x + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, 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 \left (- c \left (a x - 1\right )\right )^{p} e^{n \operatorname{acoth}{\left (a x \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{\left (-a c x + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]