Optimal. Leaf size=82 \[ \frac{x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.212945, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6192, 6193, 43} \[ \frac{x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
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 = 0.0330308, size = 56, normalized size = 0.68 \[ \frac{x^m \sqrt{c-a^2 c x^2} (a m x+a x+m+2)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 62, normalized size = 0.8 \begin{align*}{\frac{{x}^{1+m} \left ( amx+ax+m+2 \right ) }{ \left ( ax+1 \right ) \left ( 2+m \right ) \left ( 1+m \right ) }\sqrt{-{a}^{2}c{x}^{2}+c}{\frac{1}{\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 = 1.08742, size = 73, normalized size = 0.89 \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 = 1.96989, size = 166, normalized size = 2.02 \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 \frac{\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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