Optimal. Leaf size=62 \[ -\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac {3 \sqrt {1-a^2 x^2}}{2 a}+\frac {3 \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {795, 665, 216} \begin {gather*} -\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac {3 \sqrt {1-a^2 x^2}}{2 a}+\frac {3 \sin ^{-1}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 665
Rule 795
Rubi steps
\begin {align*} \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx &=-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3}{2} \int \frac {\sqrt {1-a^2 x^2}}{1-a x} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2}}{2 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3}{2} \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2}}{2 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3 \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 91, normalized size = 1.47 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (6 \sqrt {a x+1} \sin ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )-\sqrt {1-a x} \left (a^2 x^2+5 a x+4\right )\right )}{2 a \sqrt {1-a x} (a x+1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 72, normalized size = 1.16 \begin {gather*} \frac {\sqrt {1-a^2 x^2} (-a x-4)}{2 a}+\frac {3 \sqrt {-a^2} \log \left (\sqrt {1-a^2 x^2}-\sqrt {-a^2} x\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 48, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 4\right )} + 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 34, normalized size = 0.55 \begin {gather*} -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (x + \frac {4}{a}\right )} + \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 118, normalized size = 1.90 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2}+1}\, x}{2}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 42, normalized size = 0.68 \begin {gather*} -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 55, normalized size = 0.89 \begin {gather*} \frac {\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2}+\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a}{\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.08, size = 76, normalized size = 1.23 \begin {gather*} - \begin {cases} - \frac {- \sqrt {- a^{2} x^{2} + 1} + \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases} - \begin {cases} - \frac {- \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} - \sqrt {- a^{2} x^{2} + 1} + \frac {\operatorname {asin}{\left (a x \right )}}{2}}{a} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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