Optimal. Leaf size=248 \[ \frac{8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac{4 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt{d+e x}}-\frac{10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac{4 c^3 (d+e x)^{9/2}}{9 e^6} \]
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Rubi [A] time = 0.127214, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {771} \[ \frac{8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac{4 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt{d+e x}}-\frac{10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac{4 c^3 (d+e x)^{9/2}}{9 e^6} \]
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 )^2}{(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 )^2}{e^5 (d+e x)^{3/2}}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^5 \sqrt{d+e x}}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) \sqrt{d+e x}}{e^5}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^{5/2}}{e^5}+\frac{2 c^3 (d+e x)^{7/2}}{e^5}\right ) \, dx\\ &=\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 \sqrt{d+e x}}+\frac{4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt{d+e x}}{e^6}-\frac{2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3/2}}{3 e^6}+\frac{8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac{10 c^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^6}+\frac{4 c^3 (d+e x)^{9/2}}{9 e^6}\\ \end{align*}
Mathematica [A] time = 0.35737, size = 287, normalized size = 1.16 \[ \frac{252 c e^2 \left (5 a^2 e^2 (2 d+e x)+5 a b e \left (-8 d^2-4 d e x+e^2 x^2\right )+2 b^2 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )-210 b e^3 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )-18 c^2 e \left (5 b \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )-28 a e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )+20 c^3 \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )}{315 e^6 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 359, normalized size = 1.5 \begin{align*} -{\frac{-140\,{c}^{3}{x}^{5}{e}^{5}-450\,b{c}^{2}{e}^{5}{x}^{4}+200\,{c}^{3}d{e}^{4}{x}^{4}-504\,a{c}^{2}{e}^{5}{x}^{3}-504\,{b}^{2}c{e}^{5}{x}^{3}+720\,b{c}^{2}d{e}^{4}{x}^{3}-320\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}-1260\,abc{e}^{5}{x}^{2}+1008\,a{c}^{2}d{e}^{4}{x}^{2}-210\,{b}^{3}{e}^{5}{x}^{2}+1008\,{b}^{2}cd{e}^{4}{x}^{2}-1440\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}+640\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}-1260\,{a}^{2}c{e}^{5}x-1260\,a{b}^{2}{e}^{5}x+5040\,abcd{e}^{4}x-4032\,a{c}^{2}{d}^{2}{e}^{3}x+840\,{b}^{3}d{e}^{4}x-4032\,{b}^{2}c{d}^{2}{e}^{3}x+5760\,b{c}^{2}{d}^{3}{e}^{2}x-2560\,{c}^{3}{d}^{4}ex+630\,b{a}^{2}{e}^{5}-2520\,{a}^{2}cd{e}^{4}-2520\,a{b}^{2}d{e}^{4}+10080\,abc{d}^{2}{e}^{3}-8064\,a{c}^{2}{d}^{3}{e}^{2}+1680\,{b}^{3}{d}^{2}{e}^{3}-8064\,{b}^{2}c{d}^{3}{e}^{2}+11520\,b{c}^{2}{d}^{4}e-5120\,{c}^{3}{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04948, size = 427, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (\frac{70 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 225 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 252 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 630 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39148, size = 707, normalized size = 2.85 \begin{align*} \frac{2 \,{\left (70 \, c^{3} e^{5} x^{5} + 2560 \, c^{3} d^{5} - 5760 \, b c^{2} d^{4} e - 315 \, a^{2} b e^{5} + 4032 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 840 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 1260 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 25 \,{\left (4 \, c^{3} d e^{4} - 9 \, b c^{2} e^{5}\right )} x^{4} + 4 \,{\left (40 \, c^{3} d^{2} e^{3} - 90 \, b c^{2} d e^{4} + 63 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} -{\left (320 \, c^{3} d^{3} e^{2} - 720 \, b c^{2} d^{2} e^{3} + 504 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} - 105 \,{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \,{\left (640 \, c^{3} d^{4} e - 1440 \, b c^{2} d^{3} e^{2} + 1008 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 210 \,{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 315 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{7} x + d e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 63.3351, size = 316, normalized size = 1.27 \begin{align*} \frac{4 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (10 b c^{2} e - 20 c^{3} d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (8 a c^{2} e^{2} + 8 b^{2} c e^{2} - 40 b c^{2} d e + 40 c^{3} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{3 e^{6}} + \frac{\sqrt{d + e x} \left (4 a^{2} c e^{4} + 4 a b^{2} e^{4} - 24 a b c d e^{3} + 24 a c^{2} d^{2} e^{2} - 4 b^{3} d e^{3} + 24 b^{2} c d^{2} e^{2} - 40 b c^{2} d^{3} e + 20 c^{3} d^{4}\right )}{e^{6}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26711, size = 621, normalized size = 2.5 \begin{align*} \frac{2}{315} \,{\left (70 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{48} - 450 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{48} + 1260 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{48} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{48} + 3150 \, \sqrt{x e + d} c^{3} d^{4} e^{48} + 225 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{49} - 1260 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{49} + 3150 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{49} - 6300 \, \sqrt{x e + d} b c^{2} d^{3} e^{49} + 252 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{50} + 252 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} e^{50} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{50} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d e^{50} + 3780 \, \sqrt{x e + d} b^{2} c d^{2} e^{50} + 3780 \, \sqrt{x e + d} a c^{2} d^{2} e^{50} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{51} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a b c e^{51} - 630 \, \sqrt{x e + d} b^{3} d e^{51} - 3780 \, \sqrt{x e + d} a b c d e^{51} + 630 \, \sqrt{x e + d} a b^{2} e^{52} + 630 \, \sqrt{x e + d} a^{2} c e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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