Optimal. Leaf size=286 \[ \frac{6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac{6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7} \]
[Out]
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Rubi [A] time = 0.397413, 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 \[ \frac{6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac{6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 74.3476, size = 282, normalized size = 0.99 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right )}{13 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{11}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{11 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{9 e^{7}} + \frac{6 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{7 e^{7}} + \frac{6 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{5 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.395551, size = 396, normalized size = 1.38 \[ \frac{2 (d+e x)^{3/2} \left (39 c e^2 \left (33 a^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 a b e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+143 e^3 \left (105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )-3 c^2 e \left (5 b \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )-13 a e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.012, size = 495, normalized size = 1.7 \[{\frac{6006\,{c}^{3}{x}^{6}{e}^{6}+20790\,b{c}^{2}{e}^{6}{x}^{5}-5544\,{c}^{3}d{e}^{5}{x}^{5}+24570\,{x}^{4}a{c}^{2}{e}^{6}+24570\,{b}^{2}c{e}^{6}{x}^{4}-18900\,b{c}^{2}d{e}^{5}{x}^{4}+5040\,{x}^{4}{c}^{3}{d}^{2}{e}^{4}+60060\,abc{e}^{6}{x}^{3}-21840\,{x}^{3}a{c}^{2}d{e}^{5}+10010\,{b}^{3}{e}^{6}{x}^{3}-21840\,{b}^{2}cd{e}^{5}{x}^{3}+16800\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-4480\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+38610\,{x}^{2}{a}^{2}c{e}^{6}+38610\,a{b}^{2}{e}^{6}{x}^{2}-51480\,abcd{e}^{5}{x}^{2}+18720\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-8580\,{b}^{3}d{e}^{5}{x}^{2}+18720\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-14400\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+3840\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+54054\,{a}^{2}b{e}^{6}x-30888\,x{a}^{2}cd{e}^{5}-30888\,a{b}^{2}d{e}^{5}x+41184\,abc{d}^{2}{e}^{4}x-14976\,xa{c}^{2}{d}^{3}{e}^{3}+6864\,{b}^{3}{d}^{2}{e}^{4}x-14976\,{b}^{2}c{d}^{3}{e}^{3}x+11520\,b{c}^{2}{d}^{4}{e}^{2}x-3072\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}-36036\,{a}^{2}bd{e}^{5}+20592\,{a}^{2}c{d}^{2}{e}^{4}+20592\,a{b}^{2}{d}^{2}{e}^{4}-27456\,abc{d}^{3}{e}^{3}+9984\,{c}^{2}{d}^{4}a{e}^{2}-4576\,{b}^{3}{d}^{3}{e}^{3}+9984\,{b}^{2}c{d}^{4}{e}^{2}-7680\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{45045\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.706341, size = 549, normalized size = 1.92 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} - 10395 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 12285 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 19305 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 27027 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209543, size = 691, normalized size = 2.42 \[ \frac{2 \,{\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e - 18018 \, a^{2} b d^{2} e^{5} + 15015 \, a^{3} d e^{6} + 4992 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} - 2288 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} + 10296 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 231 \,{\left (c^{3} d e^{6} + 45 \, b c^{2} e^{7}\right )} x^{6} - 63 \,{\left (4 \, c^{3} d^{2} e^{5} - 15 \, b c^{2} d e^{6} - 195 \,{\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \,{\left (b^{2} c + a c^{2}\right )} d e^{6} + 143 \,{\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} - 143 \,{\left (b^{3} + 6 \, a b c\right )} d e^{6} - 3861 \,{\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{5} e^{2} - 480 \, b c^{2} d^{4} e^{3} + 9009 \, a^{2} b e^{7} + 624 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 286 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} + 1287 \,{\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} -{\left (512 \, c^{3} d^{6} e - 1920 \, b c^{2} d^{5} e^{2} - 9009 \, a^{2} b d e^{6} - 15015 \, a^{3} e^{7} + 2496 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 1144 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 5148 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.48177, size = 539, normalized size = 1.88 \[ \frac{2 \left (\frac{c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (3 b c^{2} e - 6 c^{3} d\right )}{13 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**3*(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214056, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*sqrt(e*x + d),x, algorithm="giac")
[Out]