Optimal. Leaf size=126 \[ \frac{2 \sqrt{d+e x} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} (2 c d-b e)}{e^4}+\frac{4 c^2 (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A] time = 0.0678039, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{2 \sqrt{d+e x} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} (2 c d-b e)}{e^4}+\frac{4 c^2 (d+e x)^{5/2}}{5 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^{3/2}}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 \sqrt{d+e x}}-\frac{3 c (2 c d-b e) \sqrt{d+e x}}{e^3}+\frac{2 c^2 (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt{d+e x}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{3/2}}{e^4}+\frac{4 c^2 (d+e x)^{5/2}}{5 e^4}\\ \end{align*}
Mathematica [A] time = 0.108237, size = 106, normalized size = 0.84 \[ \frac{2 \left (5 c e \left (2 a e (2 d+e x)+b \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+5 b e^2 (-a e+2 b d+b e x)+2 c^2 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )}{5 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 123, normalized size = 1. \begin{align*} -{\frac{-4\,{c}^{2}{x}^{3}{e}^{3}-10\,bc{e}^{3}{x}^{2}+8\,{c}^{2}d{e}^{2}{x}^{2}-20\,ac{e}^{3}x-10\,{b}^{2}{e}^{3}x+40\,bcd{e}^{2}x-32\,{c}^{2}{d}^{2}ex+10\,ab{e}^{3}-40\,acd{e}^{2}-20\,{b}^{2}d{e}^{2}+80\,b{d}^{2}ce-64\,{c}^{2}{d}^{3}}{5\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00919, size = 174, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (\frac{2 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} - 5 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 5 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{5 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )}}{\sqrt{e x + d} e^{3}}\right )}}{5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34043, size = 275, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (2 \, c^{2} e^{3} x^{3} + 32 \, c^{2} d^{3} - 40 \, b c d^{2} e - 5 \, a b e^{3} + 10 \,{\left (b^{2} + 2 \, a c\right )} d e^{2} -{\left (4 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} x^{2} +{\left (16 \, c^{2} d^{2} e - 20 \, b c d e^{2} + 5 \,{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{5 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.0123, size = 128, normalized size = 1.02 \begin{align*} \frac{4 c^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 b c e - 12 c^{2} d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (4 a c e^{2} + 2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{e^{4}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41537, size = 220, normalized size = 1.75 \begin{align*} \frac{2}{5} \,{\left (2 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} e^{16} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e^{16} + 30 \, \sqrt{x e + d} c^{2} d^{2} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b c e^{17} - 30 \, \sqrt{x e + d} b c d e^{17} + 5 \, \sqrt{x e + d} b^{2} e^{18} + 10 \, \sqrt{x e + d} a c e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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