Optimal. Leaf size=307 \[ -\frac{3 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^5 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \log (x)}{\left (c-a^2 c x^2\right )^{5/2}}+\frac{23 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \log (1-a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}-\frac{7 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \log (a x+1)}{16 \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.292859, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6192, 6193, 88} \[ -\frac{3 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (a x+1) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^5 x^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \log (x)}{\left (c-a^2 c x^2\right )^{5/2}}+\frac{23 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \log (1-a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}-\frac{7 a^6 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \log (a x+1)}{16 \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^7} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{\left (a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{x^2 (-1+a x)^3 (1+a x)^2} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{\left (a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (-\frac{1}{x^2}-\frac{a}{x}+\frac{a^2}{4 (-1+a x)^3}-\frac{3 a^2}{4 (-1+a x)^2}+\frac{23 a^2}{16 (-1+a x)}-\frac{a^2}{8 (1+a x)^2}-\frac{7 a^2}{16 (1+a x)}\right ) \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^4}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{4 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}+\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (c-a^2 c x^2\right )^{5/2}}+\frac{23 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \log (1-a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}-\frac{7 a^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \log (1+a x)}{16 \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0888769, size = 99, normalized size = 0.32 \[ \frac{a^5 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{12 a}{a x-1}+\frac{2 a}{a x+1}-\frac{2 a}{(a x-1)^2}-16 a \log (x)+23 a \log (1-a x)-7 a \log (a x+1)+\frac{16}{x}\right )}{16 \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.143, size = 225, normalized size = 0.7 \begin{align*}{\frac{16\,{a}^{4}\ln \left ( x \right ){x}^{4}+7\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-23\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-16\,{a}^{3}\ln \left ( x \right ){x}^{3}-7\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +23\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-30\,{x}^{3}{a}^{3}-16\,{a}^{2}\ln \left ( x \right ){x}^{2}-7\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+23\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+22\,{a}^{2}{x}^{2}+16\,a\ln \left ( x \right ) x+7\,ax\ln \left ( ax+1 \right ) -23\,\ln \left ( ax-1 \right ) xa+28\,ax-16}{ \left ( 16\,ax-16 \right ) \left ({a}^{2}{x}^{2}-1 \right ){c}^{3}x \left ( ax+1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\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{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{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 = 1.76535, size = 360, normalized size = 1.17 \begin{align*} -\frac{{\left (30 \, a^{3} x^{3} - 22 \, a^{2} x^{2} - 28 \, a x - 7 \,{\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (a x + 1\right ) + 23 \,{\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (a x - 1\right ) - 16 \,{\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (x\right ) + 16\right )} \sqrt{-a^{2} c}}{16 \,{\left (a^{4} c^{3} x^{4} - a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2} + a c^{3} x\right )}} \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{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{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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