Optimal. Leaf size=95 \[ \frac{(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{2 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.179096, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6192, 6193, 43} \[ \frac{(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{2 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{5/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (c-a^2 c x^2\right )^{5/2} \int (-1+a x)^4 (1+a x) \, dx}{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (c-a^2 c x^2\right )^{5/2} \int \left (2 (-1+a x)^4+(-1+a x)^5\right ) \, dx}{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{2 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}+\frac{(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}\\ \end{align*}
Mathematica [A] time = 0.0392302, size = 55, normalized size = 0.58 \[ \frac{c^2 (a x-1)^5 (5 a x+7) \sqrt{c-a^2 c x^2}}{30 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 84, normalized size = 0.9 \begin{align*}{\frac{x \left ( 5\,{x}^{5}{a}^{5}-18\,{x}^{4}{a}^{4}+15\,{x}^{3}{a}^{3}+20\,{a}^{2}{x}^{2}-45\,ax+30 \right ) }{ \left ( 30\,ax+30 \right ) \left ( ax-1 \right ) ^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10204, size = 189, normalized size = 1.99 \begin{align*} \frac{{\left (5 \, a^{7} \sqrt{-c} c^{2} x^{7} - 13 \, a^{6} \sqrt{-c} c^{2} x^{6} - 3 \, a^{5} \sqrt{-c} c^{2} x^{5} + 35 \, a^{4} \sqrt{-c} c^{2} x^{4} - 25 \, a^{3} \sqrt{-c} c^{2} x^{3} - 15 \, a^{2} \sqrt{-c} c^{2} x^{2} - 30 \, \sqrt{-c} c^{2}\right )}{\left (a x - 1\right )}^{2}}{30 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62129, size = 154, normalized size = 1.62 \begin{align*} \frac{{\left (5 \, a^{5} c^{2} x^{6} - 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} + 30 \, c^{2} x\right )} \sqrt{-a^{2} c}}{30 \, a} \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{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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