Optimal. Leaf size=83 \[ \frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.226649, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6192, 6193, 43} \[ \frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^{1+m} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int x^m (-1+a x) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (-x^m+a x^{1+m}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=-\frac{x^m \sqrt{c-a^2 c x^2}}{a (1+m) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{1+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0338995, size = 58, normalized size = 0.7 \[ \frac{x^m \sqrt{c-a^2 c x^2} (m (a x-1)+a x-2)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 64, normalized size = 0.8 \begin{align*}{\frac{{x}^{1+m} \left ( amx+ax-m-2 \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) \left ( ax-1 \right ) }\sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15089, size = 77, normalized size = 0.93 \begin{align*} \frac{{\left (a \sqrt{-c}{\left (m + 1\right )} x^{2} - \sqrt{-c}{\left (m + 2\right )} x\right )}{\left (a x - 1\right )} x^{m}}{{\left (m^{2} + 3 \, m + 2\right )} a x - m^{2} - 3 \, m - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10874, size = 166, normalized size = 2. \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left ({\left (a m + a\right )} x^{2} -{\left (m + 2\right )} x\right )} x^{m} \sqrt{\frac{a x - 1}{a x + 1}}}{m^{2} -{\left (a m^{2} + 3 \, a m + 2 \, a\right )} x + 3 \, m + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{-a^{2} c x^{2} + c} x^{m} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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