Optimal. Leaf size=80 \[ \frac{x^m \sqrt{c-\frac{c}{a^2 x^2}}}{a m \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.244871, antiderivative size = 80, 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 = {6197, 6193, 43} \[ \frac{x^m \sqrt{c-\frac{c}{a^2 x^2}}}{a m \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^m \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^m \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int x^{-1+m} (1+a x) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \left (x^{-1+m}+a x^m\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^m}{a m \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^{1+m}}{(1+m) \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.03545, size = 53, normalized size = 0.66 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{a x^{m+1}}{m+1}+\frac{x^m}{m}\right )}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.13, size = 63, normalized size = 0.8 \begin{align*}{\frac{{x}^{1+m} \left ( axm+m+1 \right ) }{ \left ( ax+1 \right ) \left ( 1+m \right ) m}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}{\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 = 1.24026, size = 59, normalized size = 0.74 \begin{align*} \frac{{\left (a \sqrt{c} m x + \sqrt{c}{\left (m + 1\right )}\right )}{\left (a x + 1\right )} x^{m}}{{\left (m^{2} + m\right )} a^{2} x +{\left (m^{2} + m\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75595, size = 154, normalized size = 1.92 \begin{align*} -\frac{{\left (a m x^{2} +{\left (m + 1\right )} x\right )} x^{m} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{m^{2} -{\left (a m^{2} + a m\right )} x + m} \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{c - \frac{c}{a^{2} x^{2}}} 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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