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 = 0.24, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 81
Rule 6150
Rule 6153
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*}
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Mathematica [A] time = 0.12, 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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fricas [B] time = 0.54, size = 193, normalized size = 3.22 \[ -\frac {{\left (a^{6} x^{9} - 3 \, a^{5} x^{8} + 8 \, a^{3} x^{6} - 6 \, a^{2} x^{5} - 6 \, a x^{4} + 8 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{24 \, {\left (a^{11} c^{6} x^{11} - 3 \, a^{10} c^{6} x^{10} - a^{9} c^{6} x^{9} + 11 \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} - 14 \, a^{6} c^{6} x^{6} + 14 \, a^{5} c^{6} x^{5} + 6 \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} + a^{2} c^{6} x^{2} + 3 \, a c^{6} x - c^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {11}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 0.82 \[ -\frac {\left (a x -1\right ) \left (a x +1\right )^{4} \left (3 a x -1\right )}{24 a^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a^{2} c \,x^{2}+c \right )^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {11}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 186, normalized size = 3.10 \[ -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {1}{24\,a^{12}\,c^6}-\frac {x}{8\,a^{11}\,c^6}\right )}{\frac {\sqrt {1-a^2\,x^2}}{a^9}+x^9\,\sqrt {1-a^2\,x^2}-\frac {3\,x\,\sqrt {1-a^2\,x^2}}{a^8}-\frac {3\,x^8\,\sqrt {1-a^2\,x^2}}{a}+\frac {8\,x^6\,\sqrt {1-a^2\,x^2}}{a^3}-\frac {6\,x^5\,\sqrt {1-a^2\,x^2}}{a^4}-\frac {6\,x^4\,\sqrt {1-a^2\,x^2}}{a^5}+\frac {8\,x^3\,\sqrt {1-a^2\,x^2}}{a^6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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