Optimal. Leaf size=93 \[ \frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2 (a x+1)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.166046, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6192, 6193, 43} \[ \frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2 (a x+1)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{\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^{\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) (1+a x)^2 \, dx}{a^3 \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 \left (-2 (1+a x)^2+(1+a x)^3\right ) \, dx}{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac{2 (1+a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 \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 = 0.0314056, size = 53, normalized size = 0.57 \[ -\frac{c (a x+1)^3 (3 a x-5) \sqrt{c-a^2 c x^2}}{12 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 68, normalized size = 0.7 \begin{align*}{\frac{x \left ( 3\,{x}^{3}{a}^{3}+4\,{a}^{2}{x}^{2}-6\,ax-12 \right ) }{ \left ( 12\,ax-12 \right ) \left ( ax+1 \right ) ^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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 = 1.56379, size = 96, normalized size = 1.03 \begin{align*} -\frac{{\left (3 \, a^{3} c x^{4} + 4 \, a^{2} c x^{3} - 6 \, a c x^{2} - 12 \, c x\right )} \sqrt{-a^{2} c}}{12 \, 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}}}{\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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