Optimal. Leaf size=140 \[ \frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{\sqrt{g} (c d f-a e g)^{3/2}} \]
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Rubi [A] time = 0.19132, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {872, 874, 205} \[ \frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{\sqrt{g} (c d f-a e g)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 872
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{\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} (f+g x)}+\frac{(c d) \int \frac{\sqrt{d+e x}}{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 (c d f-a e g)}\\ &=\frac{\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} (f+g x)}+\frac{\left (c d e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{c d f-a e g}\\ &=\frac{\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} (f+g x)}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d f-a e g} \sqrt{d+e x}}\right )}{\sqrt{g} (c d f-a e g)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.111714, size = 136, normalized size = 0.97 \[ \frac{\sqrt{d+e x} \left (\sqrt{g} (a e+c d x) \sqrt{c d f-a e g}+c d (f+g x) \sqrt{a e+c d x} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right )\right )}{\sqrt{g} (f+g x) \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.297, size = 168, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( aeg-cdf \right ) \left ( gx+f \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ({\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) xcdg+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) cdf-\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71952, size = 1478, normalized size = 10.56 \begin{align*} \left [\frac{{\left (c d e g x^{2} + c d^{2} f +{\left (c d e f + c d^{2} g\right )} x\right )} \sqrt{-c d f g + a e g^{2}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d f g + a e g^{2}} \sqrt{e x + d}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f g - a e g^{2}\right )} \sqrt{e x + d}}{2 \,{\left (c^{2} d^{3} f^{3} g - 2 \, a c d^{2} e f^{2} g^{2} + a^{2} d e^{2} f g^{3} +{\left (c^{2} d^{2} e f^{2} g^{2} - 2 \, a c d e^{2} f g^{3} + a^{2} e^{3} g^{4}\right )} x^{2} +{\left (c^{2} d^{2} e f^{3} g + a^{2} d e^{2} g^{4} +{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g^{2} -{\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{3}\right )} x\right )}}, -\frac{{\left (c d e g x^{2} + c d^{2} f +{\left (c d e f + c d^{2} g\right )} x\right )} \sqrt{c d f g - a e g^{2}} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d f g - a e g^{2}} \sqrt{e x + d}}{c d e g x^{2} + a d e g +{\left (c d^{2} + a e^{2}\right )} g x}\right ) - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f g - a e g^{2}\right )} \sqrt{e x + d}}{c^{2} d^{3} f^{3} g - 2 \, a c d^{2} e f^{2} g^{2} + a^{2} d e^{2} f g^{3} +{\left (c^{2} d^{2} e f^{2} g^{2} - 2 \, a c d e^{2} f g^{3} + a^{2} e^{3} g^{4}\right )} x^{2} +{\left (c^{2} d^{2} e f^{3} g + a^{2} d e^{2} g^{4} +{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g^{2} -{\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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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