Optimal. Leaf size=52 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e} \]
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Rubi [A] time = 0.0241019, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {661, 208} \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e} \]
Antiderivative was successfully verified.
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Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \sqrt{d^2-e^2 x^2}} \, dx &=(2 e) \operatorname{Subst}\left (\int \frac{1}{-2 d e^2+e^2 x^2} \, dx,x,\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{d+e x}}\right )\\ &=-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e}\\ \end{align*}
Mathematica [A] time = 0.0515172, size = 52, normalized size = 1. \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 58, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{2}}{e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87935, size = 333, normalized size = 6.4 \begin{align*} \left [\frac{\sqrt{2} \log \left (-\frac{e^{2} x^{2} - 2 \, d e x + 2 \, \sqrt{2} \sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d} \sqrt{d} - 3 \, d^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt{d} e}, -\frac{\sqrt{2} \sqrt{-\frac{1}{d}} \arctan \left (\frac{\sqrt{2} \sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d} d \sqrt{-\frac{1}{d}}}{e^{2} x^{2} - d^{2}}\right )}{e}\right ] \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{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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