Optimal. Leaf size=46 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt{d+e x}} \]
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Rubi [A] time = 0.0206646, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {648} \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0167825, size = 35, normalized size = 0.76 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)}}{c d \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 50, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( cdx+ae \right ) \sqrt{ex+d}}{cd\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05088, size = 24, normalized size = 0.52 \begin{align*} \frac{2 \, \sqrt{c d x + a e}}{c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57786, size = 107, normalized size = 2.33 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{c d e x + c d^{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{\sqrt{d + e x}}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \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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