Optimal. Leaf size=172 \[ \frac{x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac{5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.199598, antiderivative size = 172, 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, 88} \[ \frac{x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac{5 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \log (a x+1)}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{x^3}{(-1+a x)^2 (1+a x)} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (\frac{1}{a^3}+\frac{1}{2 a^3 (-1+a x)^2}+\frac{5}{4 a^3 (-1+a x)}-\frac{1}{4 a^3 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}{2 a (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac{5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 a \left (c-a^2 c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0550632, size = 71, normalized size = 0.41 \[ \frac{x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)\right )}{4 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.152, size = 98, normalized size = 0.6 \begin{align*} -{\frac{-4\,{a}^{2}{x}^{2}+ax\ln \left ( ax+1 \right ) -5\,\ln \left ( ax-1 \right ) xa+4\,ax-\ln \left ( ax+1 \right ) +5\,\ln \left ( ax-1 \right ) +2}{ \left ( 4\,{a}^{2}{x}^{2}-4 \right ){c}^{2}{a}^{4}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \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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Fricas [A] time = 1.59521, size = 157, normalized size = 0.91 \begin{align*} \frac{{\left (4 \, a^{2} x^{2} - 4 \, a x -{\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt{-a^{2} c}}{4 \,{\left (a^{6} c^{2} x - a^{5} c^{2}\right )}} \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{x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \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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