Optimal. Leaf size=286 \[ \frac{6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac{6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7} \]
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Rubi [A] time = 0.129942, antiderivative size = 286, 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 = {698} \[ \frac{6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac{6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}{e^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^6}+\frac{3 \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 ) (d+e x)^{5/2}}{e^6}+\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)^{7/2}}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{11/2}}{e^6}+\frac{c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac{6 \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)^{7/2}}{7 e^7}-\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)^{9/2}}{9 e^7}+\frac{6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}\\ \end{align*}
Mathematica [A] time = 0.387385, size = 396, normalized size = 1.38 \[ \frac{2 (d+e x)^{3/2} \left (39 c e^2 \left (33 a^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 a b e \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+b^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )+143 e^3 \left (63 a^2 b e^2 (3 e x-2 d)+105 a^3 e^3+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )\right )-3 c^2 e \left (5 b \left (480 d^3 e^2 x^2-560 d^2 e^3 x^3-384 d^4 e x+256 d^5+630 d e^4 x^4-693 e^5 x^5\right )-13 a e \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )+c^3 \left (1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-1536 d^5 e x+1024 d^6-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 495, normalized size = 1.7 \begin{align*}{\frac{6006\,{c}^{3}{x}^{6}{e}^{6}+20790\,b{c}^{2}{e}^{6}{x}^{5}-5544\,{c}^{3}d{e}^{5}{x}^{5}+24570\,a{c}^{2}{e}^{6}{x}^{4}+24570\,{b}^{2}c{e}^{6}{x}^{4}-18900\,b{c}^{2}d{e}^{5}{x}^{4}+5040\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+60060\,abc{e}^{6}{x}^{3}-21840\,a{c}^{2}d{e}^{5}{x}^{3}+10010\,{b}^{3}{e}^{6}{x}^{3}-21840\,{b}^{2}cd{e}^{5}{x}^{3}+16800\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-4480\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+38610\,{a}^{2}c{e}^{6}{x}^{2}+38610\,a{b}^{2}{e}^{6}{x}^{2}-51480\,abcd{e}^{5}{x}^{2}+18720\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-8580\,{b}^{3}d{e}^{5}{x}^{2}+18720\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-14400\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+54054\,{a}^{2}b{e}^{6}x-30888\,{a}^{2}cd{e}^{5}x-30888\,a{b}^{2}d{e}^{5}x+41184\,abc{d}^{2}{e}^{4}x-14976\,a{c}^{2}{d}^{3}{e}^{3}x+6864\,{b}^{3}{d}^{2}{e}^{4}x-14976\,{b}^{2}c{d}^{3}{e}^{3}x+11520\,b{c}^{2}{d}^{4}{e}^{2}x-3072\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}-36036\,{a}^{2}bd{e}^{5}+20592\,{a}^{2}c{d}^{2}{e}^{4}+20592\,a{b}^{2}{d}^{2}{e}^{4}-27456\,abc{d}^{3}{e}^{3}+9984\,a{c}^{2}{d}^{4}{e}^{2}-4576\,{b}^{3}{d}^{3}{e}^{3}+9984\,{b}^{2}c{d}^{4}{e}^{2}-7680\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{45045\,{e}^{7}} \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 = 0.999574, size = 549, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} - 10395 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 12285 \,{\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{11}{2}} - 5005 \,{\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{9}{2}} + 19305 \,{\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{7}{2}} - 27027 \,{\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 )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00285, size = 1170, normalized size = 4.09 \begin{align*} \frac{2 \,{\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e - 18018 \, a^{2} b d^{2} e^{5} + 15015 \, a^{3} d e^{6} + 4992 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} - 2288 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} + 10296 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 231 \,{\left (c^{3} d e^{6} + 45 \, b c^{2} e^{7}\right )} x^{6} - 63 \,{\left (4 \, c^{3} d^{2} e^{5} - 15 \, b c^{2} d e^{6} - 195 \,{\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \,{\left (b^{2} c + a c^{2}\right )} d e^{6} + 143 \,{\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} - 143 \,{\left (b^{3} + 6 \, a b c\right )} d e^{6} - 3861 \,{\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{5} e^{2} - 480 \, b c^{2} d^{4} e^{3} + 9009 \, a^{2} b e^{7} + 624 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 286 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} + 1287 \,{\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} -{\left (512 \, c^{3} d^{6} e - 1920 \, b c^{2} d^{5} e^{2} - 9009 \, a^{2} b d e^{6} - 15015 \, a^{3} e^{7} + 2496 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 1144 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 5148 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.26919, size = 539, normalized size = 1.88 \begin{align*} \frac{2 \left (\frac{c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (3 b c^{2} e - 6 c^{3} d\right )}{13 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12241, size = 752, normalized size = 2.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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