Optimal. Leaf size=93 \[ \frac{2 c^2 (a x+1)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}}-\frac{c^2 (a x+1)^6 \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0911275, antiderivative size = 93, 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 = {6143, 6140, 43} \[ \frac{2 c^2 (a x+1)^5 \sqrt{c-a^2 c x^2}}{5 a \sqrt{1-a^2 x^2}}-\frac{c^2 (a x+1)^6 \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{3 \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^{3 \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) (1+a x)^4 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int \left (2 (1+a x)^4-(1+a x)^5\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 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 = 0.0379155, size = 52, normalized size = 0.56 \[ -\frac{c^2 (a x+1)^5 (5 a x-7) \sqrt{c-a^2 c x^2}}{30 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 81, normalized size = 0.9 \begin{align*}{\frac{x \left ( 5\,{x}^{5}{a}^{5}+18\,{x}^{4}{a}^{4}+15\,{x}^{3}{a}^{3}-20\,{a}^{2}{x}^{2}-45\,ax-30 \right ) }{ \left ( 30\,ax-30 \right ) \left ( ax+1 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.105, size = 320, normalized size = 3.44 \begin{align*} -\frac{1}{3} \, a^{2} c^{\frac{5}{2}} x^{3} + \frac{1}{12} \,{\left (\frac{2 \, a^{4} c^{3} x^{8}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac{5 \, a^{2} c^{3} x^{6}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac{3 \, c^{3} x^{4}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} + c^{\frac{5}{2}} x - \frac{1}{5} \,{\left (3 \, a^{2} c^{\frac{5}{2}} x^{5} - 5 \, c^{\frac{5}{2}} x^{3}\right )} a^{2} + \frac{3}{4} \,{\left (\frac{a^{4} c^{3} x^{6}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac{3 \, a^{2} c^{3} x^{4}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac{2 \, c^{3}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51615, size = 207, normalized size = 2.23 \begin{align*} \frac{{\left (5 \, a^{5} c^{2} x^{6} + 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} - 45 \, 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 )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{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^{2} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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