Optimal. Leaf size=125 \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{15 c^2 d^2 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.0951645, 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} \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{15 c^2 d^2 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}+\frac{1}{5} \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx\\ &=\frac{2 \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 c d (d+e x)^{3/2}}+\frac{2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0534116, size = 54, normalized size = 0.43 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+3 g x)-2 a e g)}{15 c^2 d^2 (d+e x)^{3/2}} \]
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 ( -3\,xcdg+2\,aeg-5\,cdf \right ) }{15\,{c}^{2}{d}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10425, size = 88, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f}{3 \, c d} + \frac{2 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} g}{15 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61304, size = 221, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} g x^{2} + 5 \, a c d e f - 2 \, a^{2} e^{2} g +{\left (5 \, c^{2} d^{2} f + a c d e g\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}}{15 \,{\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{\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )}{\sqrt{d + e x}}\, 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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