Optimal. Leaf size=34 \[ \frac{\sqrt{1-a^2 x^2}}{a^2 (a x+1)}+\frac{\sin ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.0174339, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {793, 216} \[ \frac{\sqrt{1-a^2 x^2}}{a^2 (a x+1)}+\frac{\sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 793
Rule 216
Rubi steps
\begin{align*} \int \frac{x}{(1+a x) \sqrt{1-a^2 x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2}}{a^2 (1+a x)}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{\sqrt{1-a^2 x^2}}{a^2 (1+a x)}+\frac{\sin ^{-1}(a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0247696, size = 31, normalized size = 0.91 \[ \frac{\frac{\sqrt{1-a^2 x^2}}{a x+1}+\sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 65, normalized size = 1.9 \begin{align*}{\frac{1}{a}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{{a}^{3} \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 = 1.45976, size = 45, normalized size = 1.32 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac{\arcsin \left (a x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63347, size = 134, normalized size = 3.94 \begin{align*} \frac{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} + 1}{a^{3} x + a^{2}} \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}{\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 = 1.25238, size = 70, normalized size = 2.06 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a{\left | a \right |}} - \frac{2}{a{\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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