Optimal. Leaf size=37 \[ \frac{(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0346184, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6130, 23, 32} \[ \frac{(1-a x)^{-p} (a x+1)^{p+1} (c-a c x)^p}{a (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6130
Rule 23
Rule 32
Rubi steps
\begin{align*} \int e^{2 p \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int (1-a x)^{-p} (1+a x)^p (c-a c x)^p \, dx\\ &=\left ((1-a x)^{-p} (c-a c x)^p\right ) \int (1+a x)^p \, dx\\ &=\frac{(1-a x)^{-p} (1+a x)^{1+p} (c-a c x)^p}{a (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0236089, size = 32, normalized size = 0.86 \[ \frac{(a x+1) (c-a c x)^p e^{2 p \tanh ^{-1}(a x)}}{a (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax+1 \right ){{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{p}}{a \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.988049, size = 41, normalized size = 1.11 \begin{align*} \frac{{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )}{\left (a x + 1\right )}^{p}}{a{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.90134, size = 81, normalized size = 2.19 \begin{align*} \frac{{\left (a x + 1\right )}{\left (-a c x + c\right )}^{p} \left (\frac{a x + 1}{a x - 1}\right )^{p}}{a p + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]