Optimal. Leaf size=250 \[ \frac{8 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{4 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{10 c^2 (d+e x)^{9/2} (2 c d-b e)}{9 e^6}+\frac{4 c^3 (d+e x)^{11/2}}{11 e^6} \]
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Rubi [A] time = 0.125853, antiderivative size = 250, 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} \[ \frac{8 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{4 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac{10 c^2 (d+e x)^{9/2} (2 c d-b e)}{9 e^6}+\frac{4 c^3 (d+e x)^{11/2}}{11 e^6} \]
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}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{e^5}+\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 ) (d+e x)^{3/2}}{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)^{5/2}}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^{7/2}}{e^5}+\frac{2 c^3 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}{e^6}+\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 ) (d+e x)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 e^6}-\frac{10 c^2 (2 c d-b e) (d+e x)^{9/2}}{9 e^6}+\frac{4 c^3 (d+e x)^{11/2}}{11 e^6}\\ \end{align*}
Mathematica [A] time = 0.393182, size = 290, normalized size = 1.16 \[ \frac{2 \sqrt{d+e x} \left (-66 c e^2 \left (-35 a^2 e^2 (e x-2 d)-21 a b e \left (8 d^2-4 d e x+3 e^2 x^2\right )+6 b^2 \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right )\right )+231 b e^3 \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+11 c^2 e \left (36 a e \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+5 b \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )-10 c^3 \left (96 d^3 e^2 x^2-80 d^2 e^3 x^3-128 d^4 e x+256 d^5+70 d e^4 x^4-63 e^5 x^5\right )\right )}{3465 e^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 359, normalized size = 1.4 \begin{align*}{\frac{1260\,{c}^{3}{x}^{5}{e}^{5}+3850\,b{c}^{2}{e}^{5}{x}^{4}-1400\,{c}^{3}d{e}^{4}{x}^{4}+3960\,a{c}^{2}{e}^{5}{x}^{3}+3960\,{b}^{2}c{e}^{5}{x}^{3}-4400\,b{c}^{2}d{e}^{4}{x}^{3}+1600\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}+8316\,abc{e}^{5}{x}^{2}-4752\,a{c}^{2}d{e}^{4}{x}^{2}+1386\,{b}^{3}{e}^{5}{x}^{2}-4752\,{b}^{2}cd{e}^{4}{x}^{2}+5280\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}-1920\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+4620\,{a}^{2}c{e}^{5}x+4620\,a{b}^{2}{e}^{5}x-11088\,abcd{e}^{4}x+6336\,a{c}^{2}{d}^{2}{e}^{3}x-1848\,{b}^{3}d{e}^{4}x+6336\,{b}^{2}c{d}^{2}{e}^{3}x-7040\,b{c}^{2}{d}^{3}{e}^{2}x+2560\,{c}^{3}{d}^{4}ex+6930\,b{a}^{2}{e}^{5}-9240\,{a}^{2}cd{e}^{4}-9240\,a{b}^{2}d{e}^{4}+22176\,abc{d}^{2}{e}^{3}-12672\,a{c}^{2}{d}^{3}{e}^{2}+3696\,{b}^{3}{d}^{2}{e}^{3}-12672\,{b}^{2}c{d}^{3}{e}^{2}+14080\,b{c}^{2}{d}^{4}e-5120\,{c}^{3}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04944, size = 416, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (630 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{3} - 1925 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 1980 \,{\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{7}{2}} - 693 \,{\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{5}{2}} + 2310 \,{\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 )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\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}\right )}}{3465 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29978, size = 706, normalized size = 2.82 \begin{align*} \frac{2 \,{\left (630 \, c^{3} e^{5} x^{5} - 2560 \, c^{3} d^{5} + 7040 \, b c^{2} d^{4} e + 3465 \, a^{2} b e^{5} - 6336 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 1848 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 4620 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 175 \,{\left (4 \, c^{3} d e^{4} - 11 \, b c^{2} e^{5}\right )} x^{4} + 20 \,{\left (40 \, c^{3} d^{2} e^{3} - 110 \, b c^{2} d e^{4} + 99 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 3 \,{\left (320 \, c^{3} d^{3} e^{2} - 880 \, b c^{2} d^{2} e^{3} + 792 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} - 231 \,{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \,{\left (640 \, c^{3} d^{4} e - 1760 \, b c^{2} d^{3} e^{2} + 1584 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 462 \,{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 1155 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 89.0048, size = 1025, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15796, size = 568, normalized size = 2.27 \begin{align*} \frac{2}{3465} \,{\left (2310 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b^{2} e^{\left (-1\right )} + 2310 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} c e^{\left (-1\right )} + 231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} b^{3} e^{\left (-2\right )} + 1386 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a b c e^{\left (-2\right )} + 396 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} b^{2} c e^{\left (-3\right )} + 396 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} a c^{2} e^{\left (-3\right )} + 55 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} b c^{2} e^{\left (-4\right )} + 10 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} c^{3} e^{\left (-5\right )} + 3465 \, \sqrt{x e + d} a^{2} b\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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