Optimal. Leaf size=65 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}} \]
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Rubi [A] time = 0.0445752, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {626, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}} \]
Antiderivative was successfully verified.
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Rule 626
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{c d^2}{e}+a e+\frac{c d x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}}\\ \end{align*}
Mathematica [A] time = 0.0161823, size = 65, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 48, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \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 = 1.83507, size = 325, normalized size = 5. \begin{align*} \left [\frac{\log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{c^{2} d^{3} - a c d e^{2}} \sqrt{e x + d}}{c d x + a e}\right )}{\sqrt{c^{2} d^{3} - a c d e^{2}}}, \frac{2 \, \sqrt{-c^{2} d^{3} + a c d e^{2}} \arctan \left (\frac{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}{c d e x + c d^{2}}\right )}{c^{2} d^{3} - a c d e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.71218, size = 48, normalized size = 0.74 \begin{align*} \frac{2 \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e^{2} - c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{a e^{2} - c d^{2}}{c d}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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