Optimal. Leaf size=55 \[ -\frac{\sqrt{1-a^2 x^2}}{a^3 (a x+1)}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sin ^{-1}(a x)}{a^3} \]
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Rubi [A] time = 0.0863924, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1639, 12, 793, 216} \[ -\frac{\sqrt{1-a^2 x^2}}{a^3 (a x+1)}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sin ^{-1}(a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 12
Rule 793
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2}{(1+a x) \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\int \frac{a^3 x}{(1+a x) \sqrt{1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\int \frac{x}{(1+a x) \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sqrt{1-a^2 x^2}}{a^3 (1+a x)}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sqrt{1-a^2 x^2}}{a^3 (1+a x)}-\frac{\sin ^{-1}(a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0551428, size = 37, normalized size = 0.67 \[ -\frac{\frac{\sqrt{1-a^2 x^2} (a x+2)}{a x+1}+\sin ^{-1}(a x)}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 84, normalized size = 1.5 \begin{align*} -{\frac{1}{{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{{a}^{4} \left ( x+{a}^{-1} \right ) }\sqrt{- \left ( x+{a}^{-1} \right ) ^{2}{a}^{2}+2\,a \left ( x+{a}^{-1} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47628, size = 70, normalized size = 1.27 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4} x + a^{3}} - \frac{\arcsin \left (a x\right )}{a^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62199, size = 151, normalized size = 2.75 \begin{align*} -\frac{2 \, a x - 2 \,{\left (a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (a x + 2\right )} + 2}{a^{4} x + a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2485, size = 95, normalized size = 1.73 \begin{align*} -\frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a^{2}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3}} + \frac{2}{a^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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