Optimal. Leaf size=184 \[ -\frac{\sqrt{1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4 c^2 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.23539, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6153, 6150, 88, 207} \[ -\frac{\sqrt{1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4 c^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 88
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^3}{(1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (-\frac{1}{4 a^3 (-1+a x)^3}-\frac{1}{2 a^3 (-1+a x)^2}-\frac{1}{8 a^3 (1+a x)^2}-\frac{3}{8 a^3 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (3 \sqrt{1-a^2 x^2}\right ) \int \frac{1}{-1+a^2 x^2} \, dx}{8 a^3 c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2}}{8 a^4 c^2 (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0575734, size = 85, normalized size = 0.46 \[ \frac{\sqrt{1-a^2 x^2} \left (5 a^2 x^2-a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)-2\right )}{8 a^4 c^2 (a x-1)^2 (a x+1) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 166, normalized size = 0.9 \begin{align*} -{\frac{3\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-3\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+3\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+10\,{a}^{2}{x}^{2}-3\,ax\ln \left ( ax+1 \right ) +3\,\ln \left ( ax-1 \right ) xa-2\,ax+3\,\ln \left ( ax+1 \right ) -3\,\ln \left ( ax-1 \right ) -4}{ \left ( 16\,{a}^{2}{x}^{2}-16 \right ){c}^{3}{a}^{4} \left ( ax-1 \right ) ^{2} \left ( ax+1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \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*} a \int -\frac{x^{4}}{{\left (a^{4} c^{\frac{5}{2}} x^{4} - 2 \, a^{2} c^{\frac{5}{2}} x^{2} + c^{\frac{5}{2}}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}\,{d x} + \frac{1}{4 \,{\left (a^{8} c^{\frac{5}{2}} x^{4} - 2 \, a^{6} c^{\frac{5}{2}} x^{2} + a^{4} c^{\frac{5}{2}}\right )}} + \frac{1}{2 \,{\left (a^{6} c^{\frac{5}{2}} x^{2} - a^{4} c^{\frac{5}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05747, size = 945, normalized size = 5.14 \begin{align*} \left [\frac{3 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{32 \,{\left (a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right )}}, \frac{3 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} a \sqrt{-c} x}{a^{4} c x^{4} - c}\right ) - 2 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{16 \,{\left (a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, dx \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 )} x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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