Optimal. Leaf size=43 \[ \frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
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Rubi [A] time = 0.0101861, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5147, 261} \[ \frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 5147
Rule 261
Rubi steps
\begin{align*} \int \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\sqrt{-e} \int \frac{x}{\sqrt{d+e x^2}} \, dx\\ &=\frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0195625, size = 43, normalized size = 1. \[ \frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 84, normalized size = 2. \begin{align*} x\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) +{\frac{{x}^{2}}{3\,d}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{2}{3\,e}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{1}{3\,de}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00229, size = 104, normalized size = 2.42 \begin{align*} x \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} \sqrt{-e}}{3 \, d e} + \frac{{\left ({\left (e x^{2} + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x^{2} + d} d\right )} \sqrt{-e}}{3 \, d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36114, size = 96, normalized size = 2.23 \begin{align*} \frac{e x \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \sqrt{e x^{2} + d} \sqrt{-e}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.768269, size = 39, normalized size = 0.91 \begin{align*} \begin{cases} i x \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )} - \frac{i \sqrt{d + e x^{2}}}{\sqrt{e}} & \text{for}\: e \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13894, size = 55, normalized size = 1.28 \begin{align*} x \arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right ) - \sqrt{-x^{2} e^{2} - d e} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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