Optimal. Leaf size=61 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)} \]
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Rubi [A] time = 0.0652654, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {860} \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{(f+g x)^{3/2} \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 f-a e g) \sqrt{d+e x} \sqrt{f+g x}}\\ \end{align*}
Mathematica [A] time = 0.0342655, size = 50, normalized size = 0.82 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 63, normalized size = 1. \begin{align*} -2\,{\frac{ \left ( cdx+ae \right ) \sqrt{ex+d}}{\sqrt{gx+f} \left ( aeg-cdf \right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70672, size = 242, normalized size = 3.97 \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} \sqrt{g x + f}}{c d^{2} f^{2} - a d e f g +{\left (c d e f g - a e^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} - a d e g^{2} +{\left (c d^{2} - a e^{2}\right )} f g\right )} x} \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 )} \left (f + g x\right )^{\frac{3}{2}}}\, 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{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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