Optimal. Leaf size=46 \[ \frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.176097, antiderivative size = 46, 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} \[ \frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
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 = 0.0313415, size = 58, normalized size = 1.26 \[ -\frac{c \left (a^3 x^3+4 a^2 x^2+6 a x+4\right ) \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.116, 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 = 1.10909, size = 131, normalized size = 2.85 \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 = 1.59019, size = 90, normalized size = 1.96 \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 \frac{{\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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