Integrand size = 28, antiderivative size = 62 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=-\frac {3 \sqrt {1-a^2 x^2}}{2 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3 \arcsin (a x)}{2 a} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {809, 679, 222} \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac {3 \sqrt {1-a^2 x^2}}{2 a}+\frac {3 \arcsin (a x)}{2 a} \]
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Rule 222
Rule 679
Rule 809
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=\frac {(-4-a x) \sqrt {1-a^2 x^2}}{2 a}+\frac {3 \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )}{a} \]
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Time = 0.36 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\left (a x +4\right ) \left (a^{2} x^{2}-1\right )}{2 a \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\) | \(60\) |
default | \(-\frac {x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {2 \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{a}\) | \(120\) |
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Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.77 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 4\right )} + 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \]
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Time = 4.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=- \begin {cases} - \frac {- \sqrt {- a^{2} x^{2} + 1} + \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases} - \begin {cases} - \frac {- \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} - \sqrt {- a^{2} x^{2} + 1} + \frac {\operatorname {asin}{\left (a x \right )}}{2}}{a} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.68 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.55 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (x + \frac {4}{a}\right )} + \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} \]
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Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx=\frac {\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2}+\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a}{\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2}\right )}{\sqrt {-a^2}} \]
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