Optimal. Leaf size=62 \[ -\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {785, 780, 217, 203} \[ -\frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^2}-\frac {d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 785
Rubi steps
\begin {align*} \int \frac {x \sqrt {d^2-e^2 x^2}}{d+e x} \, dx &=\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 \operatorname {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*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.92 \[ \frac {(e x-2 d) \sqrt {d^2-e^2 x^2}-d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 60, normalized size = 0.97 \[ \frac {2 \, d^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 2 \, d\right )}}{2 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 43, normalized size = 0.69 \[ -\frac {1}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\relax (d) + \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e^{\left (-1\right )} - 2 \, d e^{\left (-2\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 140, normalized size = 2.26 \[ -\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, x}{2 e}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 56, normalized size = 0.90 \[ -\frac {d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} x}{2 \, e} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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