Optimal. Leaf size=47 \[ \frac{(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.168354, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6192, 6193, 32} \[ \frac{(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 32
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (c-a^2 c x^2\right )^{3/2} \int (-1+a x)^3 \, dx}{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ \end{align*}
Mathematica [A] time = 0.0318155, size = 58, normalized size = 1.23 \[ -\frac{c \left (a^3 x^3-4 a^2 x^2+6 a x-4\right ) \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.042, size = 60, normalized size = 1.3 \begin{align*}{\frac{x \left ({x}^{3}{a}^{3}-4\,{a}^{2}{x}^{2}+6\,ax-4 \right ) }{4\, \left ( ax-1 \right ) ^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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 [B] time = 1.11477, size = 131, normalized size = 2.79 \begin{align*} -\frac{{\left (a^{5} \sqrt{-c} c x^{5} - 3 \, a^{4} \sqrt{-c} c x^{4} + 2 \, a^{3} \sqrt{-c} c x^{3} + 2 \, a^{2} \sqrt{-c} c x^{2} + 4 \, \sqrt{-c} c\right )}{\left (a x - 1\right )}^{2}}{4 \,{\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 = 1.61588, size = 90, normalized size = 1.91 \begin{align*} -\frac{{\left (a^{3} c x^{4} - 4 \, a^{2} c x^{3} + 6 \, a c x^{2} - 4 \, c x\right )} \sqrt{-a^{2} c}}{4 \, 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{3}{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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