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 = 0.0221245, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, 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 793
Rule 217
Rule 203
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*}
Mathematica [A] time = 0.0318285, 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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Maple [A] time = 0.053, size = 74, normalized size = 1.4 \begin{align*}{\frac{1}{e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{1}{{e}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63051, size = 143, normalized size = 2.75 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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