Optimal. Leaf size=125 \[ \frac{2 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt{d+e x}} \]
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Rubi [A] time = 0.0917179, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {794, 648} \[ \frac{2 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 648
Rubi steps
\begin{align*} \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 &=\frac{2 g \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}+\frac{1}{3} \left (3 f-\frac{d g}{e}-\frac{2 a e g}{c d}\right ) \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 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e \sqrt{d+e x}}+\frac{2 g \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d e}\\ \end{align*}
Mathematica [A] time = 0.0469351, size = 53, normalized size = 0.42 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} (c d (3 f+g x)-2 a e g)}{3 c^2 d^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 67, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -xcdg+2\,aeg-3\,cdf \right ) }{3\,{c}^{2}{d}^{2}}\sqrt{ex+d}{\frac{1}{\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 = 1.11433, size = 88, normalized size = 0.7 \begin{align*} \frac{2 \, \sqrt{c d x + a e} f}{c d} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} g}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00958, size = 158, normalized size = 1.26 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt{e x + d}}{3 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\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{d + e x} \left (f + g x\right )}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}{\left (g x + f\right )}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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