Integrand size = 25, antiderivative size = 62 \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {799, 794, 223, 209} \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2}-\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2} \]
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Rule 209
Rule 223
Rule 794
Rule 799
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {x \left (d^2 e-d e^2 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d e} \\ & = -\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e} \\ & = -\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \\ & = -\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {(-2 d+e x) \sqrt {d^2-e^2 x^2}}{2 e^2}+\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
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Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {\left (-e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}-\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e \sqrt {e^{2}}}\) | \(64\) |
default | \(\frac {\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}}{e}-\frac {d \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{2}}\) | \(135\) |
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.97 \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {2 \, d^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 2 \, d\right )}}{2 \, e^{2}} \]
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\[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\int \frac {x \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} x}{2 \, e} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{e^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, e {\left | e \right |}} + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {x}{e} - \frac {2 \, d}{e^{2}}\right )} \]
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Timed out. \[ \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\int \frac {x\,\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \]
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