Optimal. Leaf size=136 \[ \frac{c^2 (a x+1)^6 \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-a^2 x^2}}-\frac{4 c^2 (a x+1)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}}+\frac{c^2 (a x+1)^4 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0949734, 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 = {6143, 6140, 43} \[ \frac{c^2 (a x+1)^6 \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-a^2 x^2}}-\frac{4 c^2 (a x+1)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}}+\frac{c^2 (a x+1)^4 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^2 (1+a x)^3 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int \left (4 (1+a x)^3-4 (1+a x)^4+(1+a x)^5\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{c^2 (1+a x)^4 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}-\frac{4 c^2 (1+a x)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}}+\frac{c^2 (1+a x)^6 \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0380002, size = 60, normalized size = 0.44 \[ \frac{c^2 (a x+1)^4 \left (5 a^2 x^2-14 a x+11\right ) \sqrt{c-a^2 c x^2}}{30 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 81, 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 ) ^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01905, size = 240, normalized size = 1.76 \begin{align*} \frac{1}{5} \, a^{4} c^{\frac{5}{2}} x^{5} - \frac{2}{3} \, a^{2} c^{\frac{5}{2}} x^{3} + c^{\frac{5}{2}} x + \frac{1}{6} \,{\left (\frac{4 \, a^{8} c^{5} \log \left (x^{2} - \frac{1}{a^{2}}\right )}{\left (a^{4} c\right )^{\frac{5}{2}}} + \frac{4 \, a^{6} c^{4} x^{2}}{\left (a^{4} c\right )^{\frac{3}{2}}} + \frac{2 \, a^{4} c^{3} x^{4}}{\sqrt{a^{4} c}} - \sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2} c^{2} x^{4} - 4 \, c^{3} \sqrt{\frac{1}{a^{4} c}} \log \left (x^{2} - \frac{1}{a^{2}}\right ) - \frac{7 \, \sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} c^{2}}{a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12908, size = 207, normalized size = 1.52 \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 x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{30 \,{\left (a^{2} x^{2} - 1\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{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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^{2} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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