Integrand size = 22, antiderivative size = 286 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \] Output:
2/3*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(3/2)/e^7-6/5*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)^2*(e*x+d)^(5/2)/e^7+6/7*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e* (-a*e+5*b*d))*(e*x+d)^(7/2)/e^7-2/9*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e *(-3*a*e+5*b*d))*(e*x+d)^(9/2)/e^7+6/11*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b *d))*(e*x+d)^(11/2)/e^7-6/13*c^2*(-b*e+2*c*d)*(e*x+d)^(13/2)/e^7+2/15*c^3* (e*x+d)^(15/2)/e^7
Time = 0.32 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.38 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )+143 e^3 \left (105 a^3 e^3+63 a^2 b e^2 (-2 d+3 e x)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )+39 c e^2 \left (33 a^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 a b e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-3 c^2 e \left (-13 a e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+5 b \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^7} \] Input:
Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2)^3,x]
Output:
(2*(d + e*x)^(3/2)*(c^3*(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) + 143*e^3 *(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x) + 9*a*b^2*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)) + 39*c*e^2*(33*a^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 22*a*b*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + b^2*(128*d^4 - 192*d^3*e*x + 24 0*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) - 3*c^2*e*(-13*a*e*(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) + 5*b*(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 - 69 3*e^5*x^5))))/(45045*e^7)
Time = 0.41 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{7/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{3/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^6}-\frac {3 c^2 (d+e x)^{11/2} (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{13/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac {6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac {6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}\) |
Input:
Int[Sqrt[d + e*x]*(a + b*x + c*x^2)^3,x]
Output:
(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(3/2))/(3*e^7) - (6*(2*c*d - b*e)*( c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^7) + (6*(c*d^2 - b*d*e + a* e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^ (9/2))/(9*e^7) + (6*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^ (11/2))/(11*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^7) + (2*c^ 3*(d + e*x)^(15/2))/(15*e^7)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.99 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {c^{3} x^{6}}{5}+\left (\frac {9}{13} b \,x^{5}+\frac {9}{11} a \,x^{4}\right ) c^{2}+\left (2 a b \,x^{3}+\frac {9}{11} b^{2} x^{4}+\frac {9}{7} a^{2} x^{2}\right ) c +\frac {9 a \,b^{2} x^{2}}{7}+\frac {9 a^{2} b x}{5}+a^{3}+\frac {b^{3} x^{3}}{3}\right ) e^{6}-\frac {6 d \left (\frac {2 c^{3} x^{5}}{13}+\frac {20 x^{3} \left (\frac {45 b x}{52}+a \right ) c^{2}}{33}+\left (\frac {20}{33} b^{2} x^{3}+\frac {10}{7} a b \,x^{2}+\frac {6}{7} a^{2} x \right ) c +b \left (\frac {5}{21} b^{2} x^{2}+\frac {6}{7} a b x +a^{2}\right )\right ) e^{5}}{5}+\frac {24 d^{2} \left (\frac {35 c^{3} x^{4}}{143}+\frac {10 x^{2} \left (\frac {35 b x}{39}+a \right ) c^{2}}{11}+\left (\frac {10}{11} b^{2} x^{2}+2 a b x +a^{2}\right ) c +b^{2} \left (\frac {b x}{3}+a \right )\right ) e^{4}}{35}-\frac {32 \left (\frac {70 c^{3} x^{3}}{429}+\frac {6 \left (\frac {25 b x}{26}+a \right ) x \,c^{2}}{11}+b \left (\frac {6 b x}{11}+a \right ) c +\frac {b^{3}}{6}\right ) d^{3} e^{3}}{35}+\frac {128 d^{4} \left (\frac {5 c^{2} x^{2}}{13}+\left (\frac {15 b x}{13}+a \right ) c +b^{2}\right ) c \,e^{2}}{385}-\frac {256 \left (\frac {2 c x}{5}+b \right ) d^{5} c^{2} e}{1001}+\frac {1024 d^{6} c^{3}}{15015}\right )}{3 e^{7}}\) | \(324\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\) | \(357\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\) | \(357\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 c^{3} e^{6} x^{6}+10395 x^{5} b \,c^{2} e^{6}-2772 c^{3} d \,e^{5} x^{5}+12285 x^{4} a \,c^{2} e^{6}+12285 x^{4} b^{2} c \,e^{6}-9450 x^{4} b \,c^{2} d \,e^{5}+2520 c^{3} d^{2} e^{4} x^{4}+30030 x^{3} a b c \,e^{6}-10920 x^{3} a \,c^{2} d \,e^{5}+5005 x^{3} b^{3} e^{6}-10920 x^{3} b^{2} c d \,e^{5}+8400 x^{3} b \,c^{2} d^{2} e^{4}-2240 c^{3} d^{3} e^{3} x^{3}+19305 x^{2} a^{2} c \,e^{6}+19305 x^{2} a \,b^{2} e^{6}-25740 x^{2} a b c d \,e^{5}+9360 x^{2} a \,c^{2} d^{2} e^{4}-4290 x^{2} b^{3} d \,e^{5}+9360 x^{2} b^{2} c \,d^{2} e^{4}-7200 x^{2} b \,c^{2} d^{3} e^{3}+1920 c^{3} d^{4} e^{2} x^{2}+27027 x \,a^{2} b \,e^{6}-15444 x \,a^{2} c d \,e^{5}-15444 x a \,b^{2} d \,e^{5}+20592 x a b c \,d^{2} e^{4}-7488 x a \,c^{2} d^{3} e^{3}+3432 x \,b^{3} d^{2} e^{4}-7488 x \,b^{2} c \,d^{3} e^{3}+5760 x b \,c^{2} d^{4} e^{2}-1536 c^{3} d^{5} e x +15015 e^{6} a^{3}-18018 a^{2} b d \,e^{5}+10296 d^{2} e^{4} a^{2} c +10296 a \,b^{2} d^{2} e^{4}-13728 a b c \,d^{3} e^{3}+4992 d^{4} e^{2} a \,c^{2}-2288 b^{3} d^{3} e^{3}+4992 b^{2} c \,d^{4} e^{2}-3840 b \,c^{2} d^{5} e +1024 d^{6} c^{3}\right )}{45045 e^{7}}\) | \(495\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 c^{3} e^{6} x^{6}+10395 x^{5} b \,c^{2} e^{6}-2772 c^{3} d \,e^{5} x^{5}+12285 x^{4} a \,c^{2} e^{6}+12285 x^{4} b^{2} c \,e^{6}-9450 x^{4} b \,c^{2} d \,e^{5}+2520 c^{3} d^{2} e^{4} x^{4}+30030 x^{3} a b c \,e^{6}-10920 x^{3} a \,c^{2} d \,e^{5}+5005 x^{3} b^{3} e^{6}-10920 x^{3} b^{2} c d \,e^{5}+8400 x^{3} b \,c^{2} d^{2} e^{4}-2240 c^{3} d^{3} e^{3} x^{3}+19305 x^{2} a^{2} c \,e^{6}+19305 x^{2} a \,b^{2} e^{6}-25740 x^{2} a b c d \,e^{5}+9360 x^{2} a \,c^{2} d^{2} e^{4}-4290 x^{2} b^{3} d \,e^{5}+9360 x^{2} b^{2} c \,d^{2} e^{4}-7200 x^{2} b \,c^{2} d^{3} e^{3}+1920 c^{3} d^{4} e^{2} x^{2}+27027 x \,a^{2} b \,e^{6}-15444 x \,a^{2} c d \,e^{5}-15444 x a \,b^{2} d \,e^{5}+20592 x a b c \,d^{2} e^{4}-7488 x a \,c^{2} d^{3} e^{3}+3432 x \,b^{3} d^{2} e^{4}-7488 x \,b^{2} c \,d^{3} e^{3}+5760 x b \,c^{2} d^{4} e^{2}-1536 c^{3} d^{5} e x +15015 e^{6} a^{3}-18018 a^{2} b d \,e^{5}+10296 d^{2} e^{4} a^{2} c +10296 a \,b^{2} d^{2} e^{4}-13728 a b c \,d^{3} e^{3}+4992 d^{4} e^{2} a \,c^{2}-2288 b^{3} d^{3} e^{3}+4992 b^{2} c \,d^{4} e^{2}-3840 b \,c^{2} d^{5} e +1024 d^{6} c^{3}\right )}{45045 e^{7}}\) | \(495\) |
| trager | \(\frac {2 \left (3003 c^{3} e^{7} x^{7}+10395 b \,c^{2} e^{7} x^{6}+231 c^{3} d \,e^{6} x^{6}+12285 c^{2} e^{7} a \,x^{5}+12285 b^{2} c \,e^{7} x^{5}+945 b \,c^{2} d \,e^{6} x^{5}-252 c^{3} d^{2} e^{5} x^{5}+30030 a b c \,e^{7} x^{4}+1365 a \,c^{2} d \,e^{6} x^{4}+5005 b^{3} e^{7} x^{4}+1365 b^{2} c d \,e^{6} x^{4}-1050 b \,c^{2} d^{2} e^{5} x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}+19305 a \,b^{2} e^{7} x^{3}+4290 a b c d \,e^{6} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}+715 b^{3} d \,e^{6} x^{3}-1560 b^{2} c \,d^{2} e^{5} x^{3}+1200 b \,c^{2} d^{3} e^{4} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+27027 a^{2} b \,e^{7} x^{2}+3861 a^{2} c d \,e^{6} x^{2}+3861 a \,b^{2} d \,e^{6} x^{2}-5148 a b c \,d^{2} e^{5} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}-858 b^{3} d^{2} e^{5} x^{2}+1872 b^{2} c \,d^{3} e^{4} x^{2}-1440 b \,c^{2} d^{4} e^{3} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x +9009 a^{2} b d \,e^{6} x -5148 a^{2} c \,d^{2} e^{5} x -5148 a \,b^{2} d^{2} e^{5} x +6864 a b c \,d^{3} e^{4} x -2496 a \,c^{2} d^{4} e^{3} x +1144 b^{3} d^{3} e^{4} x -2496 b^{2} c \,d^{4} e^{3} x +1920 b \,c^{2} d^{5} e^{2} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}-18018 a^{2} b \,d^{2} e^{5}+10296 a^{2} c \,d^{3} e^{4}+10296 a \,b^{2} d^{3} e^{4}-13728 a b c \,d^{4} e^{3}+4992 a \,c^{2} d^{5} e^{2}-2288 b^{3} d^{4} e^{3}+4992 b^{2} c \,d^{5} e^{2}-3840 b \,c^{2} d^{6} e +1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\) | \(637\) |
| risch | \(\frac {2 \left (3003 c^{3} e^{7} x^{7}+10395 b \,c^{2} e^{7} x^{6}+231 c^{3} d \,e^{6} x^{6}+12285 c^{2} e^{7} a \,x^{5}+12285 b^{2} c \,e^{7} x^{5}+945 b \,c^{2} d \,e^{6} x^{5}-252 c^{3} d^{2} e^{5} x^{5}+30030 a b c \,e^{7} x^{4}+1365 a \,c^{2} d \,e^{6} x^{4}+5005 b^{3} e^{7} x^{4}+1365 b^{2} c d \,e^{6} x^{4}-1050 b \,c^{2} d^{2} e^{5} x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}+19305 a \,b^{2} e^{7} x^{3}+4290 a b c d \,e^{6} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}+715 b^{3} d \,e^{6} x^{3}-1560 b^{2} c \,d^{2} e^{5} x^{3}+1200 b \,c^{2} d^{3} e^{4} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+27027 a^{2} b \,e^{7} x^{2}+3861 a^{2} c d \,e^{6} x^{2}+3861 a \,b^{2} d \,e^{6} x^{2}-5148 a b c \,d^{2} e^{5} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}-858 b^{3} d^{2} e^{5} x^{2}+1872 b^{2} c \,d^{3} e^{4} x^{2}-1440 b \,c^{2} d^{4} e^{3} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x +9009 a^{2} b d \,e^{6} x -5148 a^{2} c \,d^{2} e^{5} x -5148 a \,b^{2} d^{2} e^{5} x +6864 a b c \,d^{3} e^{4} x -2496 a \,c^{2} d^{4} e^{3} x +1144 b^{3} d^{3} e^{4} x -2496 b^{2} c \,d^{4} e^{3} x +1920 b \,c^{2} d^{5} e^{2} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}-18018 a^{2} b \,d^{2} e^{5}+10296 a^{2} c \,d^{3} e^{4}+10296 a \,b^{2} d^{3} e^{4}-13728 a b c \,d^{4} e^{3}+4992 a \,c^{2} d^{5} e^{2}-2288 b^{3} d^{4} e^{3}+4992 b^{2} c \,d^{5} e^{2}-3840 b \,c^{2} d^{6} e +1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\) | \(637\) |
Input:
int((e*x+d)^(1/2)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
2/3*(e*x+d)^(3/2)*((1/5*c^3*x^6+(9/13*b*x^5+9/11*a*x^4)*c^2+(2*a*b*x^3+9/1 1*b^2*x^4+9/7*a^2*x^2)*c+9/7*a*b^2*x^2+9/5*a^2*b*x+a^3+1/3*b^3*x^3)*e^6-6/ 5*d*(2/13*c^3*x^5+20/33*x^3*(45/52*b*x+a)*c^2+(20/33*b^2*x^3+10/7*a*b*x^2+ 6/7*a^2*x)*c+b*(5/21*b^2*x^2+6/7*a*b*x+a^2))*e^5+24/35*d^2*(35/143*c^3*x^4 +10/11*x^2*(35/39*b*x+a)*c^2+(10/11*b^2*x^2+2*a*b*x+a^2)*c+b^2*(1/3*b*x+a) )*e^4-32/35*(70/429*c^3*x^3+6/11*(25/26*b*x+a)*x*c^2+b*(6/11*b*x+a)*c+1/6* b^3)*d^3*e^3+128/385*d^4*(5/13*c^2*x^2+(15/13*b*x+a)*c+b^2)*c*e^2-256/1001 *(2/5*c*x+b)*d^5*c^2*e+1024/15015*d^6*c^3)/e^7
Time = 0.07 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.79 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e - 18018 \, a^{2} b d^{2} e^{5} + 15015 \, a^{3} d e^{6} + 4992 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} - 2288 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} + 10296 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 231 \, {\left (c^{3} d e^{6} + 45 \, b c^{2} e^{7}\right )} x^{6} - 63 \, {\left (4 \, c^{3} d^{2} e^{5} - 15 \, b c^{2} d e^{6} - 195 \, {\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \, {\left (b^{2} c + a c^{2}\right )} d e^{6} + 143 \, {\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} - 143 \, {\left (b^{3} + 6 \, a b c\right )} d e^{6} - 3861 \, {\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{5} e^{2} - 480 \, b c^{2} d^{4} e^{3} + 9009 \, a^{2} b e^{7} + 624 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 286 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} + 1287 \, {\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{6} e - 1920 \, b c^{2} d^{5} e^{2} - 9009 \, a^{2} b d e^{6} - 15015 \, a^{3} e^{7} + 2496 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 1144 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 5148 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \] Input:
integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
2/45045*(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*(b^2*c + a*c^2)*d^5*e^2 - 2288*(b^3 + 6*a *b*c)*d^4*e^3 + 10296*(a*b^2 + a^2*c)*d^3*e^4 + 231*(c^3*d*e^6 + 45*b*c^2* e^7)*x^6 - 63*(4*c^3*d^2*e^5 - 15*b*c^2*d*e^6 - 195*(b^2*c + a*c^2)*e^7)*x ^5 + 35*(8*c^3*d^3*e^4 - 30*b*c^2*d^2*e^5 + 39*(b^2*c + a*c^2)*d*e^6 + 143 *(b^3 + 6*a*b*c)*e^7)*x^4 - 5*(64*c^3*d^4*e^3 - 240*b*c^2*d^3*e^4 + 312*(b ^2*c + a*c^2)*d^2*e^5 - 143*(b^3 + 6*a*b*c)*d*e^6 - 3861*(a*b^2 + a^2*c)*e ^7)*x^3 + 3*(128*c^3*d^5*e^2 - 480*b*c^2*d^4*e^3 + 9009*a^2*b*e^7 + 624*(b ^2*c + a*c^2)*d^3*e^4 - 286*(b^3 + 6*a*b*c)*d^2*e^5 + 1287*(a*b^2 + a^2*c) *d*e^6)*x^2 - (512*c^3*d^6*e - 1920*b*c^2*d^5*e^2 - 9009*a^2*b*d*e^6 - 150 15*a^3*e^7 + 2496*(b^2*c + a*c^2)*d^4*e^3 - 1144*(b^3 + 6*a*b*c)*d^3*e^4 + 5148*(a*b^2 + a^2*c)*d^2*e^5)*x)*sqrt(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (284) = 568\).
Time = 1.31 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.21 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**(1/2)*(c*x**2+b*x+a)**3,x)
Output:
Piecewise((2*(c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(3*b*c* *2*e - 6*c**3*d)/(13*e**6) + (d + e*x)**(11/2)*(3*a*c**2*e**2 + 3*b**2*c*e **2 - 15*b*c**2*d*e + 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(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)/(9*e**6) + (d + e*x)**(7/2)*(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)/(7*e**6) + (d + e*x)**(5/2)*(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)/(5*e**6) + (d + e*x)**(3/2)*(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 )/(3*e**6))/e, Ne(e, 0)), (sqrt(d)*(a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6 /2 + c**3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3*(3*a**2*c + 3*a*b**2)/3), True))
Time = 0.04 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.42 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 10395 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \] Input:
integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
2/45045*(3003*(e*x + d)^(15/2)*c^3 - 10395*(2*c^3*d - b*c^2*e)*(e*x + d)^( 13/2) + 12285*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(1 1/2) - 5005*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*(e*x + d)^(9/2) + 19305*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*( b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(7/2) - 27027*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a* c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d )^(5/2) + 15015*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^ 2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4 )*(e*x + d)^(3/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (258) = 516\).
Time = 0.38 (sec) , antiderivative size = 1177, normalized size of antiderivative = 4.12 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
2/45045*(45045*sqrt(e*x + d)*a^3*d + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3 + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*b*d/e + 9009* (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^2*d/ e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^ 2)*a^2*c*d/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt( e*x + d)*d^2)*a^2*b/e + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 3 5*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3*d/e^3 + 7722*(5*(e*x + d )^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d) *d^3)*a*b*c*d/e^3 + 3861*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e *x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b^2/e^2 + 3861*(5*(e*x + d)^(7 /2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3 )*a^2*c/e^2 + 429*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*c*d/e ^4 + 429*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2) *d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*c^2*d/e^4 + 143* (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 42 0*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^3/e^3 + 858*(35*(e*x + d) ^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^( 3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b*c/e^3 + 195*(63*(e*x + d)^(11/2) - 3 85*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d...
Time = 5.75 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.04 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{11\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{3\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{9\,e^7}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{5\,e^7} \] Input:
int((d + e*x)^(1/2)*(a + b*x + c*x^2)^3,x)
Output:
((d + e*x)^(7/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 3 6*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(7* e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e* x)^(13/2))/(13*e^7) + ((d + e*x)^(11/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2* c*e^2 - 30*b*c^2*d*e))/(11*e^7) + (2*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d* e)^3)/(3*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(b^2*e^2 + 10*c^2*d^2 + 6 *a*c*e^2 - 10*b*c*d*e))/(9*e^7) + (6*(b*e - 2*c*d)*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(5*e^7)
Time = 0.22 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.22 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \sqrt {e x +d}\, \left (3003 c^{3} e^{7} x^{7}+10395 b \,c^{2} e^{7} x^{6}+231 c^{3} d \,e^{6} x^{6}+12285 a \,c^{2} e^{7} x^{5}+12285 b^{2} c \,e^{7} x^{5}+945 b \,c^{2} d \,e^{6} x^{5}-252 c^{3} d^{2} e^{5} x^{5}+30030 a b c \,e^{7} x^{4}+1365 a \,c^{2} d \,e^{6} x^{4}+5005 b^{3} e^{7} x^{4}+1365 b^{2} c d \,e^{6} x^{4}-1050 b \,c^{2} d^{2} e^{5} x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}+19305 a \,b^{2} e^{7} x^{3}+4290 a b c d \,e^{6} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}+715 b^{3} d \,e^{6} x^{3}-1560 b^{2} c \,d^{2} e^{5} x^{3}+1200 b \,c^{2} d^{3} e^{4} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+27027 a^{2} b \,e^{7} x^{2}+3861 a^{2} c d \,e^{6} x^{2}+3861 a \,b^{2} d \,e^{6} x^{2}-5148 a b c \,d^{2} e^{5} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}-858 b^{3} d^{2} e^{5} x^{2}+1872 b^{2} c \,d^{3} e^{4} x^{2}-1440 b \,c^{2} d^{4} e^{3} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x +9009 a^{2} b d \,e^{6} x -5148 a^{2} c \,d^{2} e^{5} x -5148 a \,b^{2} d^{2} e^{5} x +6864 a b c \,d^{3} e^{4} x -2496 a \,c^{2} d^{4} e^{3} x +1144 b^{3} d^{3} e^{4} x -2496 b^{2} c \,d^{4} e^{3} x +1920 b \,c^{2} d^{5} e^{2} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}-18018 a^{2} b \,d^{2} e^{5}+10296 a^{2} c \,d^{3} e^{4}+10296 a \,b^{2} d^{3} e^{4}-13728 a b c \,d^{4} e^{3}+4992 a \,c^{2} d^{5} e^{2}-2288 b^{3} d^{4} e^{3}+4992 b^{2} c \,d^{5} e^{2}-3840 b \,c^{2} d^{6} e +1024 c^{3} d^{7}\right )}{45045 e^{7}} \] Input:
int((e*x+d)^(1/2)*(c*x^2+b*x+a)^3,x)
Output:
(2*sqrt(d + e*x)*(15015*a**3*d*e**6 + 15015*a**3*e**7*x - 18018*a**2*b*d** 2*e**5 + 9009*a**2*b*d*e**6*x + 27027*a**2*b*e**7*x**2 + 10296*a**2*c*d**3 *e**4 - 5148*a**2*c*d**2*e**5*x + 3861*a**2*c*d*e**6*x**2 + 19305*a**2*c*e **7*x**3 + 10296*a*b**2*d**3*e**4 - 5148*a*b**2*d**2*e**5*x + 3861*a*b**2* d*e**6*x**2 + 19305*a*b**2*e**7*x**3 - 13728*a*b*c*d**4*e**3 + 6864*a*b*c* d**3*e**4*x - 5148*a*b*c*d**2*e**5*x**2 + 4290*a*b*c*d*e**6*x**3 + 30030*a *b*c*e**7*x**4 + 4992*a*c**2*d**5*e**2 - 2496*a*c**2*d**4*e**3*x + 1872*a* c**2*d**3*e**4*x**2 - 1560*a*c**2*d**2*e**5*x**3 + 1365*a*c**2*d*e**6*x**4 + 12285*a*c**2*e**7*x**5 - 2288*b**3*d**4*e**3 + 1144*b**3*d**3*e**4*x - 858*b**3*d**2*e**5*x**2 + 715*b**3*d*e**6*x**3 + 5005*b**3*e**7*x**4 + 499 2*b**2*c*d**5*e**2 - 2496*b**2*c*d**4*e**3*x + 1872*b**2*c*d**3*e**4*x**2 - 1560*b**2*c*d**2*e**5*x**3 + 1365*b**2*c*d*e**6*x**4 + 12285*b**2*c*e**7 *x**5 - 3840*b*c**2*d**6*e + 1920*b*c**2*d**5*e**2*x - 1440*b*c**2*d**4*e* *3*x**2 + 1200*b*c**2*d**3*e**4*x**3 - 1050*b*c**2*d**2*e**5*x**4 + 945*b* c**2*d*e**6*x**5 + 10395*b*c**2*e**7*x**6 + 1024*c**3*d**7 - 512*c**3*d**6 *e*x + 384*c**3*d**5*e**2*x**2 - 320*c**3*d**4*e**3*x**3 + 280*c**3*d**3*e **4*x**4 - 252*c**3*d**2*e**5*x**5 + 231*c**3*d*e**6*x**6 + 3003*c**3*e**7 *x**7))/(45045*e**7)