Optimal. Leaf size=132 \[ \frac{2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac{6 c (d+e x)^{7/2} (2 c d-b e)}{7 e^4}+\frac{4 c^2 (d+e x)^{9/2}}{9 e^4} \]
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Rubi [A] time = 0.0700871, antiderivative size = 132, 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} \[ \frac{2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac{6 c (d+e x)^{7/2} (2 c d-b e)}{7 e^4}+\frac{4 c^2 (d+e x)^{9/2}}{9 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (b+2 c x) \sqrt{d+e x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}{e^3}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{e^3}-\frac{3 c (2 c d-b e) (d+e x)^{5/2}}{e^3}+\frac{2 c^2 (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^4}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{5/2}}{5 e^4}-\frac{6 c (2 c d-b e) (d+e x)^{7/2}}{7 e^4}+\frac{4 c^2 (d+e x)^{9/2}}{9 e^4}\\ \end{align*}
Mathematica [A] time = 0.128145, size = 110, normalized size = 0.83 \[ \frac{2 (d+e x)^{3/2} \left (3 c e \left (14 a e (3 e x-2 d)+3 b \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+21 b e^2 (5 a e-2 b d+3 b e x)+c^2 \left (48 d^2 e x-32 d^3-60 d e^2 x^2+70 e^3 x^3\right )\right )}{315 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 123, normalized size = 0.9 \begin{align*}{\frac{140\,{c}^{2}{x}^{3}{e}^{3}+270\,bc{e}^{3}{x}^{2}-120\,{c}^{2}d{e}^{2}{x}^{2}+252\,ac{e}^{3}x+126\,{b}^{2}{e}^{3}x-216\,bcd{e}^{2}x+96\,{c}^{2}{d}^{2}ex+210\,ab{e}^{3}-168\,acd{e}^{2}-84\,{b}^{2}d{e}^{2}+144\,b{d}^{2}ce-64\,{c}^{2}{d}^{3}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977821, size = 163, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (70 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{2} - 135 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\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 )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25928, size = 383, normalized size = 2.9 \begin{align*} \frac{2 \,{\left (70 \, c^{2} e^{4} x^{4} - 32 \, c^{2} d^{4} + 72 \, b c d^{3} e + 105 \, a b d e^{3} - 42 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 5 \,{\left (2 \, c^{2} d e^{3} + 27 \, b c e^{4}\right )} x^{3} - 3 \,{\left (4 \, c^{2} d^{2} e^{2} - 9 \, b c d e^{3} - 21 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} +{\left (16 \, c^{2} d^{3} e - 36 \, b c d^{2} e^{2} + 105 \, a b e^{4} + 21 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.64201, size = 155, normalized size = 1.17 \begin{align*} \frac{2 \left (\frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 b c e - 6 c^{2} d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 3 b c d^{2} e - 2 c^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.6333, size = 230, normalized size = 1.74 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b^{2} e^{\left (-1\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a c e^{\left (-1\right )} + 9 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b c e^{\left (-2\right )} + 2 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} c^{2} e^{\left (-3\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a b\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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