Integrand size = 22, antiderivative size = 286 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \] Output:
2/7*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/2)/e^7-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)^2*(e*x+d)^(9/2)/e^7+6/11*(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)^(11/2)/e^7-2/13*(-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)^(13/2)/e^7+2/5*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+ 5*b*d))*(e*x+d)^(15/2)/e^7-6/17*c^2*(-b*e+2*c*d)*(e*x+d)^(17/2)/e^7+2/19*c ^3*(e*x+d)^(19/2)/e^7
Time = 0.42 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.39 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )+1615 e^3 \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )+323 c e^2 \left (65 a^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 a b e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-19 c^2 e \left (-17 a e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+5 b \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{4849845 e^7} \] Input:
Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]
Output:
(2*(d + e*x)^(7/2)*(5*c^3*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14 784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6) + 1 615*e^3*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*e*x) + 13*a*b^2*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e ^3*x^3)) + 323*c*e^2*(65*a^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*a*b* e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^2*(128*d^4 - 44 8*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) - 19*c^2*e* (-17*a*e*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003 *e^4*x^4) + 5*b*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e^3*x ^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5))))/(4849845*e^7)
Time = 0.47 (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.
| \(\displaystyle \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{11/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{7/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {(d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{e^6}-\frac {3 c^2 (d+e x)^{15/2} (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{17/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac {6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}\) |
Input:
Int[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]
Output:
(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*( c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(9/2))/(3*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)^(11/2))/(11*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 )^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e *x)^(15/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2 *c^3*(d + e*x)^(19/2))/(19*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 = 1.07 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {7 c^{3} x^{6}}{19}+\frac {7 x^{4} \left (\frac {15 b x}{17}+a \right ) c^{2}}{5}+\left (\frac {42}{13} a b \,x^{3}+\frac {7}{5} b^{2} x^{4}+\frac {21}{11} a^{2} x^{2}\right ) c +\frac {21 a \,b^{2} x^{2}}{11}+\frac {7 a^{2} b x}{3}+a^{3}+\frac {7 b^{3} x^{3}}{13}\right ) e^{6}-\frac {2 \left (\frac {126 c^{3} x^{5}}{323}+\frac {84 \left (\frac {65 b x}{68}+a \right ) x^{3} c^{2}}{65}+\frac {14 \left (\frac {66}{65} b^{2} x^{2}+\frac {27}{13} a b x +a^{2}\right ) x c}{11}+b \left (\frac {63}{143} b^{2} x^{2}+\frac {14}{11} a b x +a^{2}\right )\right ) d \,e^{5}}{3}+\frac {8 d^{2} \left (\frac {231 c^{3} x^{4}}{323}+\frac {126 x^{2} \left (\frac {55 b x}{51}+a \right ) c^{2}}{65}+\left (\frac {126}{65} b^{2} x^{2}+\frac {42}{13} a b x +a^{2}\right ) c +b^{2} \left (\frac {7 b x}{13}+a \right )\right ) e^{4}}{33}-\frac {32 d^{3} \left (\frac {154 c^{3} x^{3}}{323}+\frac {14 \left (\frac {45 b x}{34}+a \right ) x \,c^{2}}{15}+b \left (\frac {14 b x}{15}+a \right ) c +\frac {b^{3}}{6}\right ) e^{3}}{143}+\frac {128 \left (\frac {315 c^{2} x^{2}}{323}+\left (\frac {35 b x}{17}+a \right ) c +b^{2}\right ) d^{4} c \,e^{2}}{2145}-\frac {256 d^{5} \left (\frac {14 c x}{19}+b \right ) c^{2} e}{7293}+\frac {1024 d^{6} c^{3}}{138567}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{7}}\) | \(318\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(357\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(357\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (255255 c^{3} e^{6} x^{6}+855855 x^{5} b \,c^{2} e^{6}-180180 c^{3} d \,e^{5} x^{5}+969969 x^{4} a \,c^{2} e^{6}+969969 x^{4} b^{2} c \,e^{6}-570570 x^{4} b \,c^{2} d \,e^{5}+120120 c^{3} d^{2} e^{4} x^{4}+2238390 x^{3} a b c \,e^{6}-596904 x^{3} a \,c^{2} d \,e^{5}+373065 x^{3} b^{3} e^{6}-596904 x^{3} b^{2} c d \,e^{5}+351120 x^{3} b \,c^{2} d^{2} e^{4}-73920 c^{3} d^{3} e^{3} x^{3}+1322685 x^{2} a^{2} c \,e^{6}+1322685 x^{2} a \,b^{2} e^{6}-1220940 x^{2} a b c d \,e^{5}+325584 x^{2} a \,c^{2} d^{2} e^{4}-203490 x^{2} b^{3} d \,e^{5}+325584 x^{2} b^{2} c \,d^{2} e^{4}-191520 x^{2} b \,c^{2} d^{3} e^{3}+40320 c^{3} d^{4} e^{2} x^{2}+1616615 x \,a^{2} b \,e^{6}-587860 x \,a^{2} c d \,e^{5}-587860 x a \,b^{2} d \,e^{5}+542640 x a b c \,d^{2} e^{4}-144704 x a \,c^{2} d^{3} e^{3}+90440 x \,b^{3} d^{2} e^{4}-144704 x \,b^{2} c \,d^{3} e^{3}+85120 x b \,c^{2} d^{4} e^{2}-17920 c^{3} d^{5} e x +692835 e^{6} a^{3}-461890 a^{2} b d \,e^{5}+167960 d^{2} e^{4} a^{2} c +167960 a \,b^{2} d^{2} e^{4}-155040 a b c \,d^{3} e^{3}+41344 d^{4} e^{2} a \,c^{2}-25840 b^{3} d^{3} e^{3}+41344 b^{2} c \,d^{4} e^{2}-24320 b \,c^{2} d^{5} e +5120 d^{6} c^{3}\right )}{4849845 e^{7}}\) | \(495\) |
| orering | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (255255 c^{3} e^{6} x^{6}+855855 x^{5} b \,c^{2} e^{6}-180180 c^{3} d \,e^{5} x^{5}+969969 x^{4} a \,c^{2} e^{6}+969969 x^{4} b^{2} c \,e^{6}-570570 x^{4} b \,c^{2} d \,e^{5}+120120 c^{3} d^{2} e^{4} x^{4}+2238390 x^{3} a b c \,e^{6}-596904 x^{3} a \,c^{2} d \,e^{5}+373065 x^{3} b^{3} e^{6}-596904 x^{3} b^{2} c d \,e^{5}+351120 x^{3} b \,c^{2} d^{2} e^{4}-73920 c^{3} d^{3} e^{3} x^{3}+1322685 x^{2} a^{2} c \,e^{6}+1322685 x^{2} a \,b^{2} e^{6}-1220940 x^{2} a b c d \,e^{5}+325584 x^{2} a \,c^{2} d^{2} e^{4}-203490 x^{2} b^{3} d \,e^{5}+325584 x^{2} b^{2} c \,d^{2} e^{4}-191520 x^{2} b \,c^{2} d^{3} e^{3}+40320 c^{3} d^{4} e^{2} x^{2}+1616615 x \,a^{2} b \,e^{6}-587860 x \,a^{2} c d \,e^{5}-587860 x a \,b^{2} d \,e^{5}+542640 x a b c \,d^{2} e^{4}-144704 x a \,c^{2} d^{3} e^{3}+90440 x \,b^{3} d^{2} e^{4}-144704 x \,b^{2} c \,d^{3} e^{3}+85120 x b \,c^{2} d^{4} e^{2}-17920 c^{3} d^{5} e x +692835 e^{6} a^{3}-461890 a^{2} b d \,e^{5}+167960 d^{2} e^{4} a^{2} c +167960 a \,b^{2} d^{2} e^{4}-155040 a b c \,d^{3} e^{3}+41344 d^{4} e^{2} a \,c^{2}-25840 b^{3} d^{3} e^{3}+41344 b^{2} c \,d^{4} e^{2}-24320 b \,c^{2} d^{5} e +5120 d^{6} c^{3}\right )}{4849845 e^{7}}\) | \(495\) |
| trager | \(\frac {2 \left (255255 c^{3} e^{9} x^{9}+855855 b \,c^{2} e^{9} x^{8}+585585 c^{3} d \,e^{8} x^{8}+969969 a \,c^{2} e^{9} x^{7}+969969 b^{2} c \,e^{9} x^{7}+1996995 b \,c^{2} d \,e^{8} x^{7}+345345 c^{3} d^{2} e^{7} x^{7}+2238390 a b c \,e^{9} x^{6}+2313003 a \,c^{2} d \,e^{8} x^{6}+373065 b^{3} e^{9} x^{6}+2313003 b^{2} c d \,e^{8} x^{6}+1206975 b \,c^{2} d^{2} e^{7} x^{6}+1155 c^{3} d^{3} e^{6} x^{6}+1322685 a^{2} c \,e^{9} x^{5}+1322685 a \,b^{2} e^{9} x^{5}+5494230 a b c d \,e^{8} x^{5}+1444779 a \,c^{2} d^{2} e^{7} x^{5}+915705 b^{3} d \,e^{8} x^{5}+1444779 b^{2} c \,d^{2} e^{7} x^{5}+5985 b \,c^{2} d^{3} e^{6} x^{5}-1260 c^{3} d^{4} e^{5} x^{5}+1616615 a^{2} b \,e^{9} x^{4}+3380195 a^{2} c d \,e^{8} x^{4}+3380195 a \,b^{2} d \,e^{8} x^{4}+3594990 a b c \,d^{2} e^{7} x^{4}+11305 a \,c^{2} d^{3} e^{6} x^{4}+599165 b^{3} d^{2} e^{7} x^{4}+11305 b^{2} c \,d^{3} e^{6} x^{4}-6650 b \,c^{2} d^{4} e^{5} x^{4}+1400 c^{3} d^{5} e^{4} x^{4}+692835 a^{3} e^{9} x^{3}+4387955 a^{2} b d \,e^{8} x^{3}+2372435 a^{2} c \,d^{2} e^{7} x^{3}+2372435 a \,b^{2} d^{2} e^{7} x^{3}+48450 a b c \,d^{3} e^{6} x^{3}-12920 a \,c^{2} d^{4} e^{5} x^{3}+8075 b^{3} d^{3} e^{6} x^{3}-12920 b^{2} c \,d^{4} e^{5} x^{3}+7600 b \,c^{2} d^{5} e^{4} x^{3}-1600 c^{3} d^{6} e^{3} x^{3}+2078505 a^{3} d \,e^{8} x^{2}+3464175 a^{2} b \,d^{2} e^{7} x^{2}+62985 a^{2} c \,d^{3} e^{6} x^{2}+62985 a \,b^{2} d^{3} e^{6} x^{2}-58140 a b c \,d^{4} e^{5} x^{2}+15504 a \,c^{2} d^{5} e^{4} x^{2}-9690 b^{3} d^{4} e^{5} x^{2}+15504 b^{2} c \,d^{5} e^{4} x^{2}-9120 b \,c^{2} d^{6} e^{3} x^{2}+1920 c^{3} d^{7} e^{2} x^{2}+2078505 a^{3} d^{2} e^{7} x +230945 a^{2} b \,d^{3} e^{6} x -83980 a^{2} c \,d^{4} e^{5} x -83980 a \,b^{2} d^{4} e^{5} x +77520 a b c \,d^{5} e^{4} x -20672 a \,c^{2} d^{6} e^{3} x +12920 b^{3} d^{5} e^{4} x -20672 b^{2} c \,d^{6} e^{3} x +12160 b \,c^{2} d^{7} e^{2} x -2560 c^{3} d^{8} e x +692835 a^{3} d^{3} e^{6}-461890 a^{2} b \,d^{4} e^{5}+167960 a^{2} c \,d^{5} e^{4}+167960 a \,b^{2} d^{5} e^{4}-155040 a b c \,d^{6} e^{3}+41344 a \,c^{2} d^{7} e^{2}-25840 b^{3} d^{6} e^{3}+41344 b^{2} c \,d^{7} e^{2}-24320 b \,c^{2} d^{8} e +5120 c^{3} d^{9}\right ) \sqrt {e x +d}}{4849845 e^{7}}\) | \(929\) |
| risch | \(\frac {2 \left (255255 c^{3} e^{9} x^{9}+855855 b \,c^{2} e^{9} x^{8}+585585 c^{3} d \,e^{8} x^{8}+969969 a \,c^{2} e^{9} x^{7}+969969 b^{2} c \,e^{9} x^{7}+1996995 b \,c^{2} d \,e^{8} x^{7}+345345 c^{3} d^{2} e^{7} x^{7}+2238390 a b c \,e^{9} x^{6}+2313003 a \,c^{2} d \,e^{8} x^{6}+373065 b^{3} e^{9} x^{6}+2313003 b^{2} c d \,e^{8} x^{6}+1206975 b \,c^{2} d^{2} e^{7} x^{6}+1155 c^{3} d^{3} e^{6} x^{6}+1322685 a^{2} c \,e^{9} x^{5}+1322685 a \,b^{2} e^{9} x^{5}+5494230 a b c d \,e^{8} x^{5}+1444779 a \,c^{2} d^{2} e^{7} x^{5}+915705 b^{3} d \,e^{8} x^{5}+1444779 b^{2} c \,d^{2} e^{7} x^{5}+5985 b \,c^{2} d^{3} e^{6} x^{5}-1260 c^{3} d^{4} e^{5} x^{5}+1616615 a^{2} b \,e^{9} x^{4}+3380195 a^{2} c d \,e^{8} x^{4}+3380195 a \,b^{2} d \,e^{8} x^{4}+3594990 a b c \,d^{2} e^{7} x^{4}+11305 a \,c^{2} d^{3} e^{6} x^{4}+599165 b^{3} d^{2} e^{7} x^{4}+11305 b^{2} c \,d^{3} e^{6} x^{4}-6650 b \,c^{2} d^{4} e^{5} x^{4}+1400 c^{3} d^{5} e^{4} x^{4}+692835 a^{3} e^{9} x^{3}+4387955 a^{2} b d \,e^{8} x^{3}+2372435 a^{2} c \,d^{2} e^{7} x^{3}+2372435 a \,b^{2} d^{2} e^{7} x^{3}+48450 a b c \,d^{3} e^{6} x^{3}-12920 a \,c^{2} d^{4} e^{5} x^{3}+8075 b^{3} d^{3} e^{6} x^{3}-12920 b^{2} c \,d^{4} e^{5} x^{3}+7600 b \,c^{2} d^{5} e^{4} x^{3}-1600 c^{3} d^{6} e^{3} x^{3}+2078505 a^{3} d \,e^{8} x^{2}+3464175 a^{2} b \,d^{2} e^{7} x^{2}+62985 a^{2} c \,d^{3} e^{6} x^{2}+62985 a \,b^{2} d^{3} e^{6} x^{2}-58140 a b c \,d^{4} e^{5} x^{2}+15504 a \,c^{2} d^{5} e^{4} x^{2}-9690 b^{3} d^{4} e^{5} x^{2}+15504 b^{2} c \,d^{5} e^{4} x^{2}-9120 b \,c^{2} d^{6} e^{3} x^{2}+1920 c^{3} d^{7} e^{2} x^{2}+2078505 a^{3} d^{2} e^{7} x +230945 a^{2} b \,d^{3} e^{6} x -83980 a^{2} c \,d^{4} e^{5} x -83980 a \,b^{2} d^{4} e^{5} x +77520 a b c \,d^{5} e^{4} x -20672 a \,c^{2} d^{6} e^{3} x +12920 b^{3} d^{5} e^{4} x -20672 b^{2} c \,d^{6} e^{3} x +12160 b \,c^{2} d^{7} e^{2} x -2560 c^{3} d^{8} e x +692835 a^{3} d^{3} e^{6}-461890 a^{2} b \,d^{4} e^{5}+167960 a^{2} c \,d^{5} e^{4}+167960 a \,b^{2} d^{5} e^{4}-155040 a b c \,d^{6} e^{3}+41344 a \,c^{2} d^{7} e^{2}-25840 b^{3} d^{6} e^{3}+41344 b^{2} c \,d^{7} e^{2}-24320 b \,c^{2} d^{8} e +5120 c^{3} d^{9}\right ) \sqrt {e x +d}}{4849845 e^{7}}\) | \(929\) |
Input:
int((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
2/7*((7/19*c^3*x^6+7/5*x^4*(15/17*b*x+a)*c^2+(42/13*a*b*x^3+7/5*b^2*x^4+21 /11*a^2*x^2)*c+21/11*a*b^2*x^2+7/3*a^2*b*x+a^3+7/13*b^3*x^3)*e^6-2/3*(126/ 323*c^3*x^5+84/65*(65/68*b*x+a)*x^3*c^2+14/11*(66/65*b^2*x^2+27/13*a*b*x+a ^2)*x*c+b*(63/143*b^2*x^2+14/11*a*b*x+a^2))*d*e^5+8/33*d^2*(231/323*c^3*x^ 4+126/65*x^2*(55/51*b*x+a)*c^2+(126/65*b^2*x^2+42/13*a*b*x+a^2)*c+b^2*(7/1 3*b*x+a))*e^4-32/143*d^3*(154/323*c^3*x^3+14/15*(45/34*b*x+a)*x*c^2+b*(14/ 15*b*x+a)*c+1/6*b^3)*e^3+128/2145*(315/323*c^2*x^2+(35/17*b*x+a)*c+b^2)*d^ 4*c*e^2-256/7293*d^5*(14/19*c*x+b)*c^2*e+1024/138567*d^6*c^3)*(e*x+d)^(7/2 )/e^7
Leaf count of result is larger than twice the leaf count of optimal. 727 vs. \(2 (258) = 516\).
Time = 0.08 (sec) , antiderivative size = 727, normalized size of antiderivative = 2.54 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e - 461890* a^2*b*d^4*e^5 + 692835*a^3*d^3*e^6 + 41344*(b^2*c + a*c^2)*d^7*e^2 - 25840 *(b^3 + 6*a*b*c)*d^6*e^3 + 167960*(a*b^2 + a^2*c)*d^5*e^4 + 45045*(13*c^3* d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323* (b^2*c + a*c^2)*e^9)*x^7 + 231*(5*c^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013 *(b^2*c + a*c^2)*d*e^8 + 1615*(b^3 + 6*a*b*c)*e^9)*x^6 - 63*(20*c^3*d^4*e^ 5 - 95*b*c^2*d^3*e^6 - 22933*(b^2*c + a*c^2)*d^2*e^7 - 14535*(b^3 + 6*a*b* c)*d*e^8 - 20995*(a*b^2 + a^2*c)*e^9)*x^5 + 35*(40*c^3*d^5*e^4 - 190*b*c^2 *d^4*e^5 + 46189*a^2*b*e^9 + 323*(b^2*c + a*c^2)*d^3*e^6 + 17119*(b^3 + 6* a*b*c)*d^2*e^7 + 96577*(a*b^2 + a^2*c)*d*e^8)*x^4 - 5*(320*c^3*d^6*e^3 - 1 520*b*c^2*d^5*e^4 - 877591*a^2*b*d*e^8 - 138567*a^3*e^9 + 2584*(b^2*c + a* c^2)*d^4*e^5 - 1615*(b^3 + 6*a*b*c)*d^3*e^6 - 474487*(a*b^2 + a^2*c)*d^2*e ^7)*x^3 + 3*(640*c^3*d^7*e^2 - 3040*b*c^2*d^6*e^3 + 1154725*a^2*b*d^2*e^7 + 692835*a^3*d*e^8 + 5168*(b^2*c + a*c^2)*d^5*e^4 - 3230*(b^3 + 6*a*b*c)*d ^4*e^5 + 20995*(a*b^2 + a^2*c)*d^3*e^6)*x^2 - (2560*c^3*d^8*e - 12160*b*c^ 2*d^7*e^2 - 230945*a^2*b*d^3*e^6 - 2078505*a^3*d^2*e^7 + 20672*(b^2*c + a* c^2)*d^6*e^3 - 12920*(b^3 + 6*a*b*c)*d^5*e^4 + 83980*(a*b^2 + a^2*c)*d^4*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 = 1.55 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.21 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{6}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{7 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**3,x)
Output:
Piecewise((2*(c**3*(d + e*x)**(19/2)/(19*e**6) + (d + e*x)**(17/2)*(3*b*c* *2*e - 6*c**3*d)/(17*e**6) + (d + e*x)**(15/2)*(3*a*c**2*e**2 + 3*b**2*c*e **2 - 15*b*c**2*d*e + 15*c**3*d**2)/(15*e**6) + (d + e*x)**(13/2)*(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)/(13*e**6) + (d + e*x)**(11/2)*(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)/(11*e**6) + (d + e*x)**(9/2)*(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)/(9*e**6) + (d + e*x)**(7/2)*(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)/(7*e**6))/e, Ne(e, 0)), (d**(5/2)*(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 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (255255 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{3} - 855855 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 373065 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{4849845 \, e^{7}} \] Input:
integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(15/2) - 373065*(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)^(13/2) + 1322685*(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)^(11/2) - 1616615*(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)^(9/2) + 692835*(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)^(7/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 2883 vs. \(2 (258) = 516\).
Time = 0.41 (sec) , antiderivative size = 2883, normalized size of antiderivative = 10.08 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
2/4849845*(4849845*sqrt(e*x + d)*a^3*d^3 + 4849845*((e*x + d)^(3/2) - 3*sq rt(e*x + d)*d)*a^3*d^2 + 4849845*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2 *b*d^3/e + 969969*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*d + 969969*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sq rt(e*x + d)*d^2)*a*b^2*d^3/e^2 + 969969*(3*(e*x + d)^(5/2) - 10*(e*x + d)^ (3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*c*d^3/e^2 + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b*d^2/e + 138567*(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^3 + 138567*(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)*b^3*d^3/e^3 + 831402*(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^3/e^3 + 1247103*(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*d^2/e^2 + 1247103*(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*d^2/e^2 + 1247103*(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*b*d/e + 46189*( 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^3/e^4 + 46189*(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^3/e^4 + 46189*(35*(e*x...
Time = 5.76 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.04 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{15\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{13\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^7} \] Input:
int((d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x)
Output:
((d + e*x)^(11/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 36*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))/(1 1*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(17/2))/(17*e^7) + ((d + e*x)^(15/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^ 2*c*e^2 - 30*b*c^2*d*e))/(15*e^7) + (2*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b* d*e)^3)/(7*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(13/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(13*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(a* e^2 + c*d^2 - b*d*e)^2)/(3*e^7)
Time = 0.18 (sec) , antiderivative size = 927, normalized size of antiderivative = 3.24 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:
int((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x)
Output:
(2*sqrt(d + e*x)*(692835*a**3*d**3*e**6 + 2078505*a**3*d**2*e**7*x + 20785 05*a**3*d*e**8*x**2 + 692835*a**3*e**9*x**3 - 461890*a**2*b*d**4*e**5 + 23 0945*a**2*b*d**3*e**6*x + 3464175*a**2*b*d**2*e**7*x**2 + 4387955*a**2*b*d *e**8*x**3 + 1616615*a**2*b*e**9*x**4 + 167960*a**2*c*d**5*e**4 - 83980*a* *2*c*d**4*e**5*x + 62985*a**2*c*d**3*e**6*x**2 + 2372435*a**2*c*d**2*e**7* x**3 + 3380195*a**2*c*d*e**8*x**4 + 1322685*a**2*c*e**9*x**5 + 167960*a*b* *2*d**5*e**4 - 83980*a*b**2*d**4*e**5*x + 62985*a*b**2*d**3*e**6*x**2 + 23 72435*a*b**2*d**2*e**7*x**3 + 3380195*a*b**2*d*e**8*x**4 + 1322685*a*b**2* e**9*x**5 - 155040*a*b*c*d**6*e**3 + 77520*a*b*c*d**5*e**4*x - 58140*a*b*c *d**4*e**5*x**2 + 48450*a*b*c*d**3*e**6*x**3 + 3594990*a*b*c*d**2*e**7*x** 4 + 5494230*a*b*c*d*e**8*x**5 + 2238390*a*b*c*e**9*x**6 + 41344*a*c**2*d** 7*e**2 - 20672*a*c**2*d**6*e**3*x + 15504*a*c**2*d**5*e**4*x**2 - 12920*a* c**2*d**4*e**5*x**3 + 11305*a*c**2*d**3*e**6*x**4 + 1444779*a*c**2*d**2*e* *7*x**5 + 2313003*a*c**2*d*e**8*x**6 + 969969*a*c**2*e**9*x**7 - 25840*b** 3*d**6*e**3 + 12920*b**3*d**5*e**4*x - 9690*b**3*d**4*e**5*x**2 + 8075*b** 3*d**3*e**6*x**3 + 599165*b**3*d**2*e**7*x**4 + 915705*b**3*d*e**8*x**5 + 373065*b**3*e**9*x**6 + 41344*b**2*c*d**7*e**2 - 20672*b**2*c*d**6*e**3*x + 15504*b**2*c*d**5*e**4*x**2 - 12920*b**2*c*d**4*e**5*x**3 + 11305*b**2*c *d**3*e**6*x**4 + 1444779*b**2*c*d**2*e**7*x**5 + 2313003*b**2*c*d*e**8*x* *6 + 969969*b**2*c*e**9*x**7 - 24320*b*c**2*d**8*e + 12160*b*c**2*d**7*...