Optimal. Leaf size=60 \[ -\frac{(1-3 a x) \sqrt{1-a^2 x^2}}{24 a^3 c^5 (1-a x)^6 (a x+1)^3 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.23957, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 81} \[ -\frac{(1-3 a x) \sqrt{1-a^2 x^2}}{24 a^3 c^5 (1-a x)^6 (a x+1)^3 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 81
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{3 \tanh ^{-1}(a x)} x^2}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^5 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^2}{(1-a x)^7 (1+a x)^4} \, dx}{c^5 \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-3 a x) \sqrt{1-a^2 x^2}}{24 a^3 c^5 (1-a x)^6 (1+a x)^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.094181, size = 59, normalized size = 0.98 \[ \frac{(3 a x-1) \sqrt{1-a^2 x^2}}{24 a^3 c^5 (a x-1)^6 (a x+1)^3 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 49, normalized size = 0.8 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) ^{4} \left ( 3\,ax-1 \right ) }{24\,{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{11}{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} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{11}{2}}{\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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Fricas [B] time = 2.23565, size = 393, normalized size = 6.55 \begin{align*} -\frac{{\left (a^{6} x^{9} - 3 \, a^{5} x^{8} + 8 \, a^{3} x^{6} - 6 \, a^{2} x^{5} - 6 \, a x^{4} + 8 \, x^{3}\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{24 \,{\left (a^{11} c^{6} x^{11} - 3 \, a^{10} c^{6} x^{10} - a^{9} c^{6} x^{9} + 11 \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} - 14 \, a^{6} c^{6} x^{6} + 14 \, a^{5} c^{6} x^{5} + 6 \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} + a^{2} c^{6} x^{2} + 3 \, a c^{6} x - c^{6}\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} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{11}{2}}{\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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