Optimal. Leaf size=57 \[ \frac {x^2 (a x+1)}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{3 a^3 c^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 796, 12, 261} \[ \frac {x^2 (a x+1)}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{3 a^3 c^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 261
Rule 796
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^2 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {x^2 (1+a x)}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {2 a x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {x^2 (1+a x)}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a c^2}\\ &=\frac {x^2 (1+a x)}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{3 a^3 c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.79 \[ \frac {-a^2 x^2-2 a x+2}{3 a^3 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 91, normalized size = 1.60 \[ -\frac {2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 2 \, a x - {\left (a^{2} x^{2} + 2 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{3 \, {\left (a^{6} c^{2} x^{3} - a^{5} c^{2} x^{2} - a^{4} c^{2} x + a^{3} c^{2}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{2}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 41, normalized size = 0.72 \[ -\frac {a^{2} x^{2}+2 a x -2}{3 \left (a x -1\right ) c^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3}} \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{2}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 102, normalized size = 1.79 \[ -\frac {12\,a^7\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}-36\,a^7\,c^2\,{\left (1-a^2\,x^2\right )}^{5/2}+12\,a^8\,c^2\,x\,{\left (1-a^2\,x^2\right )}^{3/2}-12\,a^8\,c^2\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{36\,a^{10}\,c^4\,{\left (a^2\,x^2-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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