Optimal. Leaf size=52 \[ \frac {\sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {793, 217, 203} \[ \frac {\sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 793
Rubi steps
\begin {align*} \int \frac {x}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}+\frac {\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.94 \[ \frac {\frac {\sqrt {d^2-e^2 x^2}}{d+e x}+\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 67, normalized size = 1.29 \[ \frac {e x - 2 \, {\left (e x + d\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d + \sqrt {-e^{2} x^{2} + d^{2}}}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 1.42 \[ \frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}{\left (x +\frac {d}{e}\right ) e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 40, normalized size = 0.77 \[ \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e^{3} x + d e^{2}} + \frac {\arcsin \left (\frac {e x}{d}\right )}{e^{2}} \]
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 \[ \int \frac {x}{\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {x}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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