Optimal. Leaf size=111 \[ \frac{2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]
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Rubi [A] time = 0.0661105, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx &=\int \left (\frac{(-e f+d g) \left (c f^2+a g^2\right )}{g^3 (f+g x)^{3/2}}+\frac{a e g^2+c f (3 e f-2 d g)}{g^3 \sqrt{f+g x}}+\frac{c (-3 e f+d g) \sqrt{f+g x}}{g^3}+\frac{c e (f+g x)^{3/2}}{g^3}\right ) \, dx\\ &=\frac{2 (e f-d g) \left (c f^2+a g^2\right )}{g^4 \sqrt{f+g x}}+\frac{2 \left (a e g^2+c f (3 e f-2 d g)\right ) \sqrt{f+g x}}{g^4}-\frac{2 c (3 e f-d g) (f+g x)^{3/2}}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4}\\ \end{align*}
Mathematica [A] time = 0.0860907, size = 92, normalized size = 0.83 \[ \frac{30 a g^2 (-d g+2 e f+e g x)+10 c d g \left (-8 f^2-4 f g x+g^2 x^2\right )+6 c e \left (8 f^2 g x+16 f^3-2 f g^2 x^2+g^3 x^3\right )}{15 g^4 \sqrt{f+g x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 101, normalized size = 0.9 \begin{align*} -{\frac{-6\,ce{x}^{3}{g}^{3}-10\,cd{g}^{3}{x}^{2}+12\,cef{g}^{2}{x}^{2}-30\,ae{g}^{3}x+40\,cdf{g}^{2}x-48\,ce{f}^{2}gx+30\,ad{g}^{3}-60\,aef{g}^{2}+80\,cd{f}^{2}g-96\,ce{f}^{3}}{15\,{g}^{4}}{\frac{1}{\sqrt{gx+f}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998517, size = 151, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (g x + f\right )}^{\frac{5}{2}} c e - 5 \,{\left (3 \, c e f - c d g\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, c e f^{2} - 2 \, c d f g + a e g^{2}\right )} \sqrt{g x + f}}{g^{3}} + \frac{15 \,{\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\right )}}{\sqrt{g x + f} g^{3}}\right )}}{15 \, g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72792, size = 252, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (3 \, c e g^{3} x^{3} + 48 \, c e f^{3} - 40 \, c d f^{2} g + 30 \, a e f g^{2} - 15 \, a d g^{3} -{\left (6 \, c e f g^{2} - 5 \, c d g^{3}\right )} x^{2} +{\left (24 \, c e f^{2} g - 20 \, c d f g^{2} + 15 \, a e g^{3}\right )} x\right )} \sqrt{g x + f}}{15 \,{\left (g^{5} x + f g^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.7237, size = 112, normalized size = 1.01 \begin{align*} \frac{2 c e \left (f + g x\right )^{\frac{5}{2}}}{5 g^{4}} + \frac{\left (f + g x\right )^{\frac{3}{2}} \left (2 c d g - 6 c e f\right )}{3 g^{4}} + \frac{\sqrt{f + g x} \left (2 a e g^{2} - 4 c d f g + 6 c e f^{2}\right )}{g^{4}} - \frac{2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )}{g^{4} \sqrt{f + g x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12359, size = 193, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (c d f^{2} g + a d g^{3} - c f^{3} e - a f g^{2} e\right )}}{\sqrt{g x + f} g^{4}} + \frac{2 \,{\left (5 \,{\left (g x + f\right )}^{\frac{3}{2}} c d g^{17} - 30 \, \sqrt{g x + f} c d f g^{17} + 3 \,{\left (g x + f\right )}^{\frac{5}{2}} c g^{16} e - 15 \,{\left (g x + f\right )}^{\frac{3}{2}} c f g^{16} e + 45 \, \sqrt{g x + f} c f^{2} g^{16} e + 15 \, \sqrt{g x + f} a g^{18} e\right )}}{15 \, g^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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