Optimal. Leaf size=136 \[ \frac{(a x+1)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 (a x+1)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.175529, antiderivative size = 136, 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} \[ \frac{(a x+1)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 (a x+1)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{(a x+1)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{\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^{\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)^2 (1+a x)^3 \, 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 (4 (1+a x)^3-4 (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{(1+a x)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}-\frac{4 (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 = 0.0405933, size = 63, normalized size = 0.46 \[ \frac{c^2 (a x+1)^4 \left (5 a^2 x^2-14 a x+11\right ) \sqrt{c-a^2 c x^2}}{30 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 84, normalized size = 0.6 \begin{align*}{\frac{x \left ( 5\,{x}^{5}{a}^{5}+6\,{x}^{4}{a}^{4}-15\,{x}^{3}{a}^{3}-20\,{a}^{2}{x}^{2}+15\,ax+30 \right ) }{30\, \left ( ax-1 \right ) ^{2} \left ( ax+1 \right ) ^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{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{5}{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 = 1.60498, size = 153, normalized size = 1.12 \begin{align*} \frac{{\left (5 \, a^{5} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} + 15 \, 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 \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{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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