Optimal. Leaf size=60 \[ \frac {(3 a x+1) \sqrt {1-a^2 x^2}}{24 a^3 c^5 (1-a x)^3 (a x+1)^6 \sqrt {c-a^2 c x^2}} \]
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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} \[ \frac {(3 a x+1) \sqrt {1-a^2 x^2}}{24 a^3 c^5 (1-a x)^3 (a x+1)^6 \sqrt {c-a^2 c x^2}} \]
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)^4 (1+a x)^7} \, 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)^3 (1+a x)^6 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 59, normalized size = 0.98 \[ -\frac {(3 a x+1) \sqrt {1-a^2 x^2}}{24 a^3 c^5 (a x-1)^3 (a x+1)^6 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 192, normalized size = 3.20 \[ \frac {{\left (a^{6} x^{9} + 3 \, a^{5} x^{8} - 8 \, a^{3} x^{6} - 6 \, a^{2} x^{5} + 6 \, a x^{4} + 8 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{24 \, {\left (a^{11} c^{6} x^{11} + 3 \, a^{10} c^{6} x^{10} - a^{9} c^{6} x^{9} - 11 \, a^{8} c^{6} x^{8} - 6 \, a^{7} c^{6} x^{7} + 14 \, a^{6} c^{6} x^{6} + 14 \, a^{5} c^{6} x^{5} - 6 \, a^{4} c^{6} x^{4} - 11 \, a^{3} c^{6} x^{3} - a^{2} c^{6} x^{2} + 3 \, a c^{6} x + c^{6}\right )}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {11}{2}} {\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ -\frac {\left (a x -1\right ) \left (3 a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{24 \left (a x +1\right )^{2} a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 93, normalized size = 1.55 \[ -\frac {3 \, a x + 1}{24 \, {\left (a^{12} c^{\frac {11}{2}} x^{9} + 3 \, a^{11} c^{\frac {11}{2}} x^{8} - 8 \, a^{9} c^{\frac {11}{2}} x^{6} - 6 \, a^{8} c^{\frac {11}{2}} x^{5} + 6 \, a^{7} c^{\frac {11}{2}} x^{4} + 8 \, a^{6} c^{\frac {11}{2}} x^{3} - 3 \, a^{4} c^{\frac {11}{2}} x - a^{3} c^{\frac {11}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 187, normalized size = 3.12 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}+3\,a\,x\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}}{24\,a^{14}\,c^6\,x^{11}+72\,a^{13}\,c^6\,x^{10}-24\,a^{12}\,c^6\,x^9-264\,a^{11}\,c^6\,x^8-144\,a^{10}\,c^6\,x^7+336\,a^9\,c^6\,x^6+336\,a^8\,c^6\,x^5-144\,a^7\,c^6\,x^4-264\,a^6\,c^6\,x^3-24\,a^5\,c^6\,x^2+72\,a^4\,c^6\,x+24\,a^3\,c^6} \]
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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