Optimal. Leaf size=92 \[ -\frac {11 \sin ^{-1}(a x)}{2 a^3}+\frac {(a x+1)^3}{a^3 \sqrt {1-a^2 x^2}}+\frac {(a x+3)^2 \sqrt {1-a^2 x^2}}{3 a^3}+\frac {(3 a x+28) \sqrt {1-a^2 x^2}}{6 a^3} \]
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Rubi [A] time = 0.65, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6124, 1633, 1593, 12, 852, 1635, 1654, 780, 216} \[ \frac {(a x+1)^3}{a^3 \sqrt {1-a^2 x^2}}+\frac {(a x+3)^2 \sqrt {1-a^2 x^2}}{3 a^3}+\frac {(3 a x+28) \sqrt {1-a^2 x^2}}{6 a^3}-\frac {11 \sin ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 780
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1654
Rule 6124
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1+a x)^2}{(1-a x) \sqrt {1-a^2 x^2}} \, dx\\ &=-\left (a \int \frac {\sqrt {1-a^2 x^2} \left (-\frac {x^2}{a}-x^3\right )}{(1-a x)^2} \, dx\right )\\ &=-\left (a \int \frac {\left (-\frac {1}{a}-x\right ) x^2 \sqrt {1-a^2 x^2}}{(1-a x)^2} \, dx\right )\\ &=a^2 \int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{a^2 (1-a x)^3} \, dx\\ &=\int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3} \, dx\\ &=\int \frac {x^2 (1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {(1+a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\int \frac {\left (\frac {3}{a^2}+\frac {x}{a}\right ) (1+a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {(1+a x)^3}{a^3 \sqrt {1-a^2 x^2}}+\frac {(3+a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}+\frac {1}{3} \int \frac {\left (\frac {3}{a^2}+\frac {x}{a}\right ) (-5-3 a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {(1+a x)^3}{a^3 \sqrt {1-a^2 x^2}}+\frac {(3+a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}+\frac {(28+3 a x) \sqrt {1-a^2 x^2}}{6 a^3}-\frac {11 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=\frac {(1+a x)^3}{a^3 \sqrt {1-a^2 x^2}}+\frac {(3+a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}+\frac {(28+3 a x) \sqrt {1-a^2 x^2}}{6 a^3}-\frac {11 \sin ^{-1}(a x)}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 58, normalized size = 0.63 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (2 a^3 x^3+7 a^2 x^2+19 a x-52\right )}{a x-1}-33 \sin ^{-1}(a x)}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 85, normalized size = 0.92 \[ \frac {52 \, a x + 66 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} x^{3} + 7 \, a^{2} x^{2} + 19 \, a x - 52\right )} \sqrt {-a^{2} x^{2} + 1} - 52}{6 \, {\left (a^{4} x - a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 87, normalized size = 0.95 \[ \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left (x {\left (\frac {2 \, x}{a} + \frac {9}{a^{2}}\right )} + \frac {28}{a^{3}}\right )} - \frac {11 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a^{2} {\left | a \right |}} + \frac {8}{a^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 122, normalized size = 1.33 \[ -\frac {a \,x^{4}}{3 \sqrt {-a^{2} x^{2}+1}}-\frac {13 x^{2}}{3 a \sqrt {-a^{2} x^{2}+1}}+\frac {26}{3 a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {3 x^{3}}{2 \sqrt {-a^{2} x^{2}+1}}+\frac {11 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 100, normalized size = 1.09 \[ -\frac {a x^{4}}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13 \, x^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1} a} + \frac {11 \, x}{2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {11 \, \arcsin \left (a x\right )}{2 \, a^{3}} + \frac {26}{3 \, \sqrt {-a^{2} x^{2} + 1} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 143, normalized size = 1.55 \[ \frac {4\,\sqrt {1-a^2\,x^2}}{a^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,a\,\sqrt {-a^2}}-\frac {4\,\sqrt {-a^2}}{a^3}+\frac {a\,x^2}{3\,\sqrt {-a^2}}-\frac {3\,x\,\sqrt {-a^2}}{2\,a^2}\right )}{\sqrt {-a^2}}-\frac {11\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^2\,\sqrt {-a^2}} \]
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 \[ \int \frac {x^{2} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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