Optimal. Leaf size=269 \[ \frac{31 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{9 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x)^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{6 a^6 x^5 (1-a x)^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{49 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{\left (1-a^2 x^2\right )^{5/2} \log (a x+1)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.218608, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6160, 6150, 88} \[ \frac{31 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{9 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 x^5 (1-a x)^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{6 a^6 x^5 (1-a x)^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}+\frac{49 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{\left (1-a^2 x^2\right )^{5/2} \log (a x+1)}{16 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \frac{e^{3 \tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \frac{x^5}{(1-a x)^4 (1+a x)} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2} \int \left (\frac{1}{a^5}+\frac{1}{2 a^5 (-1+a x)^4}+\frac{9}{4 a^5 (-1+a x)^3}+\frac{31}{8 a^5 (-1+a x)^2}+\frac{49}{16 a^5 (-1+a x)}-\frac{1}{16 a^5 (1+a x)}\right ) \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}+\frac{\left (1-a^2 x^2\right )^{5/2}}{6 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)^3}-\frac{9 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)^2}+\frac{31 \left (1-a^2 x^2\right )^{5/2}}{8 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 (1-a x)}+\frac{49 \left (1-a^2 x^2\right )^{5/2} \log (1-a x)}{16 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (1-a^2 x^2\right )^{5/2} \log (1+a x)}{16 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ \end{align*}
Mathematica [A] time = 0.0817772, size = 113, normalized size = 0.42 \[ \frac{\sqrt{1-a^2 x^2} \left (2 \left (24 a^4 x^4-72 a^3 x^3-21 a^2 x^2+135 a x-70\right )+147 (a x-1)^3 \log (1-a x)-3 (a x-1)^3 \log (a x+1)\right )}{48 a^2 c^2 x (a x-1)^3 \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 176, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -48\,{x}^{4}{a}^{4}+3\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -147\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+144\,{x}^{3}{a}^{3}-9\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+441\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+42\,{a}^{2}{x}^{2}+9\,ax\ln \left ( ax+1 \right ) -441\,\ln \left ( ax-1 \right ) xa-270\,ax-3\,\ln \left ( ax+1 \right ) +147\,\ln \left ( ax-1 \right ) +140 \right ) \left ( ax+1 \right ) ^{2}}{ \left ( 48\,ax-48 \right ){a}^{6}{x}^{5}}\sqrt{-{a}^{2}{x}^{2}+1} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \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 x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1} a^{6} x^{6} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{7} c^{3} x^{7} - 3 \, a^{6} c^{3} x^{6} + a^{5} c^{3} x^{5} + 5 \, a^{4} c^{3} x^{4} - 5 \, a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2} + 3 \, a c^{3} x - 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{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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