Optimal. Leaf size=49 \[ \frac {(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6143, 6140, 32} \[ \frac {(a x+1)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{2 p \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^{2 p \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)^{2 p} \, dx\\ &=\frac {(1+a x)^{1+2 p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (1+2 p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 36, normalized size = 0.73 \[ \frac {(a x+1) \left (c-a^2 c x^2\right )^p e^{2 p \tanh ^{-1}(a x)}}{2 a p+a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 42, normalized size = 0.86 \[ \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}}{2 \, a p + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 38, normalized size = 0.78 \[ \frac {\left (a x +1\right ) {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (1+2 p \right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 34, normalized size = 0.69 \[ \frac {{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )} {\left (a x + 1\right )}^{2 \, p}}{a {\left (2 \, p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.06, size = 43, normalized size = 0.88 \[ \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^{p+1}}{a\,\left (2\,p+1\right )\,{\left (1-a\,x\right )}^p} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x & \text {for}\: a = 0 \\\int \frac {e^{- \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (- a^{2} c x^{2} + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{2 a p + a} + \frac {\left (- a^{2} c x^{2} + c\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}}{2 a p + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________