Optimal. Leaf size=124 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x}}-\frac{2 \sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2}} \]
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Rubi [A] time = 0.186361, antiderivative size = 124, 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 = {864, 874, 205} \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x}}-\frac{2 \sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2}} \]
Antiderivative was successfully verified.
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Rule 864
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)} \, dx &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g \sqrt{d+e x}}-\frac{\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right ) \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}{e^2 g}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g \sqrt{d+e x}}-\frac{\left (2 e^2 (c d f-a e g)\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 )}{g}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g \sqrt{d+e x}}-\frac{2 \sqrt{c d f-a e g} \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 )}{g^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.135251, size = 101, normalized size = 0.81 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (\sqrt{g}-\frac{\sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right )}{\sqrt{a e+c d x}}\right )}{g^{3/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.38, size = 153, normalized size = 1.2 \begin{align*} -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}{\sqrt{ex+d}\sqrt{cdx+ae}g\sqrt{ \left ( aeg-cdf \right ) g}} \left ({\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) aeg-{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) cdf-\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g} \right ) } \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{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70312, size = 702, normalized size = 5.66 \begin{align*} \left [\frac{{\left (e x + d\right )} \sqrt{-\frac{c d f - a e g}{g}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} g \sqrt{-\frac{c d f - a e g}{g}} -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{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} \sqrt{e x + d}}{e g x + d g}, \frac{2 \,{\left ({\left (e x + d\right )} \sqrt{\frac{c d f - a e g}{g}} \arctan \left (\frac{\sqrt{e x + d} \sqrt{\frac{c d f - a e g}{g}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\right ) + \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}\right )}}{e g x + d g}\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{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt{d + e x} \left (f + g 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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