Optimal. Leaf size=36 \[ \frac{2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)} \]
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Rubi [A] time = 0.0298995, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6174} \[ \frac{2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx &=\frac{2 e^{n \coth ^{-1}(a x)} (1+a x) (c-a c x)^{n/2}}{a (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0172169, size = 58, normalized size = 1.61 \[ -\frac{x \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n}{2}+1} (c-a c x)^{n/2}}{-\frac{n}{2}-1} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.04, size = 34, normalized size = 0.9 \begin{align*} 2\,{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( ax+1 \right ) \left ( -acx+c \right ) ^{n/2}}{a \left ( 2+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a c x + c\right )}^{\frac{1}{2} \, n} \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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a c x + c\right )}^{\frac{1}{2} \, n} \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.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a c x + c\right )}^{\frac{1}{2} \, n} \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.
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