Integrand size = 28, antiderivative size = 427 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{9/2}}{9 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^8}-\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{15/2}}{3 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{17/2}}{17 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{19/2}}{19 e^8}+\frac {4 c^4 (d+e x)^{21/2}}{21 e^8} \]
-2/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/2)/e^8+2/9*(a*e^2-b*d*e +c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(9/2)/e^8-6/11 *(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*( e*x+d)^(11/2)/e^8+2/13*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c ^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d) ^(13/2)/e^8-2/3*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x +d)^(15/2)/e^8+6/17*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^ (17/2)/e^8-14/19*c^3*(-b*e+2*c*d)*(e*x+d)^(19/2)/e^8+4/21*c^4*(e*x+d)^(21/ 2)/e^8
Time = 0.49 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.41 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (-2 c^4 \left (2048 d^7-7168 d^6 e x+16128 d^5 e^2 x^2-29568 d^4 e^3 x^3+48048 d^3 e^4 x^4-72072 d^2 e^5 x^5+102102 d e^6 x^6-138567 e^7 x^7\right )+969 b e^4 \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )+323 c e^3 \left (286 a^3 e^3 (-2 d+7 e x)+117 a^2 b e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+36 a b^2 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^3 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-57 c^2 e^2 \left (102 a^2 e^2 \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-17 a b e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+3 b^2 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )+3 c^3 e \left (38 a e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+7 b \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )\right )}{2909907 e^8} \]
(2*(d + e*x)^(7/2)*(-2*c^4*(2048*d^7 - 7168*d^6*e*x + 16128*d^5*e^2*x^2 - 29568*d^4*e^3*x^3 + 48048*d^3*e^4*x^4 - 72072*d^2*e^5*x^5 + 102102*d*e^6*x ^6 - 138567*e^7*x^7) + 969*b*e^4*(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^3*(286*a^3*e^3*(-2*d + 7*e*x) + 117*a^2*b*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 36*a*b^2*e*(-16*d^3 + 56 *d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^3*(128*d^4 - 448*d^3*e*x + 100 8*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) - 57*c^2*e^2*(102*a^2*e^2* (16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3) - 17*a*b*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) + 3*b^2*(2 56*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)) + 3*c^3*e*(38*a*e*(-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) + 7*b*(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))))/(2909907*e^8)
Time = 0.69 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1195, 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.
| \(\displaystyle \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^{13/2} (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {(d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {(d+e x)^{5/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {7 c^3 (d+e x)^{17/2} (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^{19/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{13 e^8}+\frac {6 c^2 (d+e x)^{17/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{17 e^8}-\frac {2 c (d+e x)^{15/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}-\frac {6 (d+e x)^{11/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}+\frac {2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^8}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^8}-\frac {14 c^3 (d+e x)^{19/2} (2 c d-b e)}{19 e^8}+\frac {4 c^4 (d+e x)^{21/2}}{21 e^8}\) |
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^8) + (2* (c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*( d + e*x)^(9/2))/(9*e^8) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2* d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^8) + (2*(70*c ^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a *e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(13/2))/(13 *e^8) - (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(15/2))/(3*e^8) + (6*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a *e))*(d + e*x)^(17/2))/(17*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(19/2))/ (19*e^8) + (4*c^4*(d + e*x)^(21/2))/(21*e^8)
3.17.7.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.57 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {2 c^{4} x^{7}}{3}+\left (\frac {49}{19} b \,x^{6}+\frac {42}{17} a \,x^{5}\right ) c^{3}+\left (\frac {63}{17} b^{2} x^{5}+\frac {42}{13} a^{2} x^{3}+7 a b \,x^{4}\right ) c^{2}+\frac {14 \left (\frac {3}{2} x^{3} b^{3}+\frac {54}{13} a \,b^{2} x^{2}+\frac {81}{22} b \,a^{2} x +a^{3}\right ) x c}{9}+b \left (\frac {21}{11} a \,b^{2} x^{2}+\frac {7}{3} b \,a^{2} x +\frac {7}{13} x^{3} b^{3}+a^{3}\right )\right ) e^{7}-\frac {4 d \left (\frac {21 c^{4} x^{6}}{19}+\frac {63 \left (\frac {21 b x}{19}+a \right ) x^{4} c^{3}}{17}+\frac {567 \left (\frac {143}{102} b^{2} x^{2}+\frac {22}{9} a b x +a^{2}\right ) x^{2} c^{2}}{143}+\left (\frac {42}{13} x^{3} b^{3}+\frac {1134}{143} a \,b^{2} x^{2}+\frac {63}{11} b \,a^{2} x +a^{3}\right ) c +\frac {3 b^{2} \left (\frac {63}{143} b^{2} x^{2}+\frac {14}{11} a b x +a^{2}\right )}{2}\right ) e^{6}}{9}+\frac {8 d^{2} \left (\frac {154 c^{4} x^{5}}{323}+\frac {308 x^{3} \left (\frac {91 b x}{76}+a \right ) c^{3}}{221}+\frac {14 \left (\frac {33}{17} b^{2} x^{2}+3 a b x +a^{2}\right ) x \,c^{2}}{13}+b \left (\frac {14}{13} b^{2} x^{2}+\frac {28}{13} a b x +a^{2}\right ) c +\frac {\left (\frac {7 b x}{13}+a \right ) b^{3}}{3}\right ) e^{5}}{11}-\frac {32 d^{3} \left (\frac {1001 c^{4} x^{4}}{969}+\frac {42 \left (\frac {77 b x}{57}+a \right ) x^{2} c^{3}}{17}+\left (\frac {63}{17} b^{2} x^{2}+\frac {14}{3} a b x +a^{2}\right ) c^{2}+2 b^{2} \left (\frac {7 b x}{9}+a \right ) c +\frac {b^{4}}{6}\right ) e^{4}}{143}+\frac {128 d^{4} \left (\frac {154 c^{3} x^{3}}{323}+\frac {14 x \left (\frac {63 b x}{38}+a \right ) c^{2}}{17}+b \left (\frac {21 b x}{17}+a \right ) c +\frac {b^{3}}{3}\right ) c \,e^{3}}{429}-\frac {512 d^{5} \left (\frac {21 c^{2} x^{2}}{19}+\left (\frac {49 b x}{19}+a \right ) c +\frac {3 b^{2}}{2}\right ) c^{2} e^{2}}{7293}+\frac {7168 d^{6} \left (\frac {2 c x}{3}+b \right ) c^{3} e}{138567}-\frac {4096 c^{4} d^{7}}{415701}\right )}{7 e^{8}}\) | \(494\) |
| derivativedivides | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} c^{2}+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )+2 c \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )+6 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{8}}\) | \(713\) |
| default | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} c^{2}+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )+2 c \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )+6 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{8}}\) | \(713\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (277134 x^{7} c^{4} e^{7}+1072071 x^{6} b \,c^{3} e^{7}-204204 x^{6} c^{4} d \,e^{6}+1027026 x^{5} a \,c^{3} e^{7}+1540539 x^{5} b^{2} c^{2} e^{7}-756756 x^{5} b \,c^{3} d \,e^{6}+144144 x^{5} c^{4} d^{2} e^{5}+2909907 x^{4} a b \,c^{2} e^{7}-684684 x^{4} a \,c^{3} d \,e^{6}+969969 x^{4} b^{3} c \,e^{7}-1027026 x^{4} b^{2} c^{2} d \,e^{6}+504504 x^{4} b \,c^{3} d^{2} e^{5}-96096 x^{4} c^{4} d^{3} e^{4}+1343034 x^{3} a^{2} c^{2} e^{7}+2686068 x^{3} a \,b^{2} c \,e^{7}-1790712 x^{3} a b \,c^{2} d \,e^{6}+421344 x^{3} a \,c^{3} d^{2} e^{5}+223839 x^{3} b^{4} e^{7}-596904 x^{3} b^{3} c d \,e^{6}+632016 x^{3} b^{2} c^{2} d^{2} e^{5}-310464 x^{3} b \,c^{3} d^{3} e^{4}+59136 x^{3} c^{4} d^{4} e^{3}+2380833 x^{2} a^{2} b c \,e^{7}-732564 x^{2} a^{2} c^{2} d \,e^{6}+793611 x^{2} a \,b^{3} e^{7}-1465128 x^{2} a \,b^{2} c d \,e^{6}+976752 x^{2} a b \,c^{2} d^{2} e^{5}-229824 x^{2} a \,c^{3} d^{3} e^{4}-122094 x^{2} b^{4} d \,e^{6}+325584 x^{2} b^{3} c \,d^{2} e^{5}-344736 x^{2} b^{2} c^{2} d^{3} e^{4}+169344 x^{2} b \,c^{3} d^{4} e^{3}-32256 x^{2} c^{4} d^{5} e^{2}+646646 x \,a^{3} c \,e^{7}+969969 x \,a^{2} b^{2} e^{7}-1058148 x \,a^{2} b c d \,e^{6}+325584 x \,a^{2} c^{2} d^{2} e^{5}-352716 x a \,b^{3} d \,e^{6}+651168 x a \,b^{2} c \,d^{2} e^{5}-434112 x a b \,c^{2} d^{3} e^{4}+102144 x a \,c^{3} d^{4} e^{3}+54264 x \,b^{4} d^{2} e^{5}-144704 x \,b^{3} c \,d^{3} e^{4}+153216 x \,b^{2} c^{2} d^{4} e^{3}-75264 x b \,c^{3} d^{5} e^{2}+14336 x \,c^{4} d^{6} e +415701 a^{3} b \,e^{7}-184756 a^{3} c d \,e^{6}-277134 a^{2} b^{2} d \,e^{6}+302328 a^{2} b c \,d^{2} e^{5}-93024 a^{2} c^{2} d^{3} e^{4}+100776 a \,b^{3} d^{2} e^{5}-186048 a \,b^{2} c \,d^{3} e^{4}+124032 a b \,c^{2} d^{4} e^{3}-29184 a \,c^{3} d^{5} e^{2}-15504 b^{4} d^{3} e^{4}+41344 b^{3} c \,d^{4} e^{3}-43776 b^{2} c^{2} d^{5} e^{2}+21504 b \,c^{3} d^{6} e -4096 c^{4} d^{7}\right )}{2909907 e^{8}}\) | \(795\) |
| trager | \(\text {Expression too large to display}\) | \(1442\) |
| risch | \(\text {Expression too large to display}\) | \(1442\) |
2/7*(e*x+d)^(7/2)*((2/3*c^4*x^7+(49/19*b*x^6+42/17*a*x^5)*c^3+(63/17*b^2*x ^5+42/13*a^2*x^3+7*a*b*x^4)*c^2+14/9*(3/2*x^3*b^3+54/13*a*b^2*x^2+81/22*b* a^2*x+a^3)*x*c+b*(21/11*a*b^2*x^2+7/3*b*a^2*x+7/13*x^3*b^3+a^3))*e^7-4/9*d *(21/19*c^4*x^6+63/17*(21/19*b*x+a)*x^4*c^3+567/143*(143/102*b^2*x^2+22/9* a*b*x+a^2)*x^2*c^2+(42/13*x^3*b^3+1134/143*a*b^2*x^2+63/11*b*a^2*x+a^3)*c+ 3/2*b^2*(63/143*b^2*x^2+14/11*a*b*x+a^2))*e^6+8/11*d^2*(154/323*c^4*x^5+30 8/221*x^3*(91/76*b*x+a)*c^3+14/13*(33/17*b^2*x^2+3*a*b*x+a^2)*x*c^2+b*(14/ 13*b^2*x^2+28/13*a*b*x+a^2)*c+1/3*(7/13*b*x+a)*b^3)*e^5-32/143*d^3*(1001/9 69*c^4*x^4+42/17*(77/57*b*x+a)*x^2*c^3+(63/17*b^2*x^2+14/3*a*b*x+a^2)*c^2+ 2*b^2*(7/9*b*x+a)*c+1/6*b^4)*e^4+128/429*d^4*(154/323*c^3*x^3+14/17*x*(63/ 38*b*x+a)*c^2+b*(21/17*b*x+a)*c+1/3*b^3)*c*e^3-512/7293*d^5*(21/19*c^2*x^2 +(49/19*b*x+a)*c+3/2*b^2)*c^2*e^2+7168/138567*d^6*(2/3*c*x+b)*c^3*e-4096/4 15701*c^4*d^7)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1113 vs. \(2 (395) = 790\).
Time = 0.29 (sec) , antiderivative size = 1113, normalized size of antiderivative = 2.61 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/2909907*(277134*c^4*e^10*x^10 - 4096*c^4*d^10 + 21504*b*c^3*d^9*e + 4157 01*a^3*b*d^3*e^7 - 14592*(3*b^2*c^2 + 2*a*c^3)*d^8*e^2 + 41344*(b^3*c + 3* a*b*c^2)*d^7*e^3 - 15504*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^4 + 100776*( a*b^3 + 3*a^2*b*c)*d^5*e^5 - 92378*(3*a^2*b^2 + 2*a^3*c)*d^4*e^6 + 7293*(8 6*c^4*d*e^9 + 147*b*c^3*e^10)*x^9 + 3861*(94*c^4*d^2*e^8 + 637*b*c^3*d*e^9 + 133*(3*b^2*c^2 + 2*a*c^3)*e^10)*x^8 + 429*(2*c^4*d^3*e^7 + 3381*b*c^3*d ^2*e^8 + 2793*(3*b^2*c^2 + 2*a*c^3)*d*e^9 + 2261*(b^3*c + 3*a*b*c^2)*e^10) *x^7 - 231*(4*c^4*d^4*e^6 - 21*b*c^3*d^3*e^7 - 3135*(3*b^2*c^2 + 2*a*c^3)* d^2*e^8 - 10013*(b^3*c + 3*a*b*c^2)*d*e^9 - 969*(b^4 + 12*a*b^2*c + 6*a^2* c^2)*e^10)*x^6 + 63*(16*c^4*d^5*e^5 - 84*b*c^3*d^4*e^6 + 57*(3*b^2*c^2 + 2 *a*c^3)*d^3*e^7 + 22933*(b^3*c + 3*a*b*c^2)*d^2*e^8 + 8721*(b^4 + 12*a*b^2 *c + 6*a^2*c^2)*d*e^9 + 12597*(a*b^3 + 3*a^2*b*c)*e^10)*x^5 - 7*(160*c^4*d ^6*e^4 - 840*b*c^3*d^5*e^5 + 570*(3*b^2*c^2 + 2*a*c^3)*d^4*e^6 - 1615*(b^3 *c + 3*a*b*c^2)*d^3*e^7 - 51357*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^8 - 2 89731*(a*b^3 + 3*a^2*b*c)*d*e^9 - 46189*(3*a^2*b^2 + 2*a^3*c)*e^10)*x^4 + (1280*c^4*d^7*e^3 - 6720*b*c^3*d^6*e^4 + 415701*a^3*b*e^10 + 4560*(3*b^2*c ^2 + 2*a*c^3)*d^5*e^5 - 12920*(b^3*c + 3*a*b*c^2)*d^4*e^6 + 4845*(b^4 + 12 *a*b^2*c + 6*a^2*c^2)*d^3*e^7 + 1423461*(a*b^3 + 3*a^2*b*c)*d^2*e^8 + 8775 91*(3*a^2*b^2 + 2*a^3*c)*d*e^9)*x^3 - 3*(512*c^4*d^8*e^2 - 2688*b*c^3*d^7* e^3 - 415701*a^3*b*d*e^9 + 1824*(3*b^2*c^2 + 2*a*c^3)*d^6*e^4 - 5168*(b...
Time = 2.06 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.02 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{4} \left (d + e x\right )^{\frac {21}{2}}}{21 e^{7}} + \frac {\left (d + e x\right )^{\frac {19}{2}} \cdot \left (7 b c^{3} e - 14 c^{4} d\right )}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 42 b c^{3} d e + 42 c^{4} d^{2}\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (15 a b c^{2} e^{3} - 30 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 45 b^{2} c^{2} d e^{2} + 105 b c^{3} d^{2} e - 70 c^{4} d^{3}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (9 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 90 a b c^{2} d^{2} e^{3} - 60 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 30 b^{3} c d^{2} e^{3} - 90 b^{2} c^{2} d^{3} e^{2} + 105 b c^{3} d^{4} e - 42 c^{4} d^{5}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a^{3} c e^{6} + 3 a^{2} b^{2} e^{6} - 18 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 6 a b^{3} d e^{5} + 36 a b^{2} c d^{2} e^{4} - 60 a b c^{2} d^{3} e^{3} + 30 a c^{3} d^{4} e^{2} + 3 b^{4} d^{2} e^{4} - 20 b^{3} c d^{3} e^{3} + 45 b^{2} c^{2} d^{4} e^{2} - 42 b c^{3} d^{5} e + 14 c^{4} d^{6}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} + 9 a^{2} b c d^{2} e^{5} - 6 a^{2} c^{2} d^{3} e^{4} + 3 a b^{3} d^{2} e^{5} - 12 a b^{2} c d^{3} e^{4} + 15 a b c^{2} d^{4} e^{3} - 6 a c^{3} d^{5} e^{2} - b^{4} d^{3} e^{4} + 5 b^{3} c d^{4} e^{3} - 9 b^{2} c^{2} d^{5} e^{2} + 7 b c^{3} d^{6} e - 2 c^{4} d^{7}\right )}{7 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{4}}{4} & \text {otherwise} \end {cases} \]
Piecewise((2*(2*c**4*(d + e*x)**(21/2)/(21*e**7) + (d + e*x)**(19/2)*(7*b* c**3*e - 14*c**4*d)/(19*e**7) + (d + e*x)**(17/2)*(6*a*c**3*e**2 + 9*b**2* c**2*e**2 - 42*b*c**3*d*e + 42*c**4*d**2)/(17*e**7) + (d + e*x)**(15/2)*(1 5*a*b*c**2*e**3 - 30*a*c**3*d*e**2 + 5*b**3*c*e**3 - 45*b**2*c**2*d*e**2 + 105*b*c**3*d**2*e - 70*c**4*d**3)/(15*e**7) + (d + e*x)**(13/2)*(6*a**2*c **2*e**4 + 12*a*b**2*c*e**4 - 60*a*b*c**2*d*e**3 + 60*a*c**3*d**2*e**2 + b **4*e**4 - 20*b**3*c*d*e**3 + 90*b**2*c**2*d**2*e**2 - 140*b*c**3*d**3*e + 70*c**4*d**4)/(13*e**7) + (d + e*x)**(11/2)*(9*a**2*b*c*e**5 - 18*a**2*c* *2*d*e**4 + 3*a*b**3*e**5 - 36*a*b**2*c*d*e**4 + 90*a*b*c**2*d**2*e**3 - 6 0*a*c**3*d**3*e**2 - 3*b**4*d*e**4 + 30*b**3*c*d**2*e**3 - 90*b**2*c**2*d* *3*e**2 + 105*b*c**3*d**4*e - 42*c**4*d**5)/(11*e**7) + (d + e*x)**(9/2)*( 2*a**3*c*e**6 + 3*a**2*b**2*e**6 - 18*a**2*b*c*d*e**5 + 18*a**2*c**2*d**2* e**4 - 6*a*b**3*d*e**5 + 36*a*b**2*c*d**2*e**4 - 60*a*b*c**2*d**3*e**3 + 3 0*a*c**3*d**4*e**2 + 3*b**4*d**2*e**4 - 20*b**3*c*d**3*e**3 + 45*b**2*c**2 *d**4*e**2 - 42*b*c**3*d**5*e + 14*c**4*d**6)/(9*e**7) + (d + e*x)**(7/2)* (a**3*b*e**7 - 2*a**3*c*d*e**6 - 3*a**2*b**2*d*e**6 + 9*a**2*b*c*d**2*e**5 - 6*a**2*c**2*d**3*e**4 + 3*a*b**3*d**2*e**5 - 12*a*b**2*c*d**3*e**4 + 15 *a*b*c**2*d**4*e**3 - 6*a*c**3*d**5*e**2 - b**4*d**3*e**4 + 5*b**3*c*d**4* e**3 - 9*b**2*c**2*d**5*e**2 + 7*b*c**3*d**6*e - 2*c**4*d**7)/(7*e**7))/e, Ne(e, 0)), (d**(5/2)*(a + b*x + c*x**2)**4/4, True))
Time = 0.20 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.51 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (277134 \, {\left (e x + d\right )}^{\frac {21}{2}} c^{4} - 1072071 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 513513 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 969969 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 223839 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 793611 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 323323 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 415701 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{2909907 \, e^{8}} \]
2/2909907*(277134*(e*x + d)^(21/2)*c^4 - 1072071*(2*c^4*d - b*c^3*e)*(e*x + d)^(19/2) + 513513*(14*c^4*d^2 - 14*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^ 2)*(e*x + d)^(17/2) - 969969*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(15/2) + 223839*(70*c ^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(13/2) - 793611*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 1 0*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a* b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(11/2) + 323323*(14*c^4*d^6 - 42*b*c^3*d^5 *e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3 *(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3 *a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(9/2) - 415701*(2*c^4*d^7 - 7*b*c^3*d^6 *e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d ^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^ 2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*(e*x + d)^(7/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 4281 vs. \(2 (395) = 790\).
Time = 0.32 (sec) , antiderivative size = 4281, normalized size of antiderivative = 10.03 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/14549535*(14549535*sqrt(e*x + d)*a^3*b*d^3 + 14549535*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*b*d^2 + 14549535*((e*x + d)^(3/2) - 3*sqrt(e*x + d )*d)*a^2*b^2*d^3/e + 9699690*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*c*d ^3/e + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d )*d^2)*a^3*b*d + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sq rt(e*x + d)*d^2)*a*b^3*d^3/e^2 + 8729721*(3*(e*x + d)^(5/2) - 10*(e*x + d) ^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b*c*d^3/e^2 + 8729721*(3*(e*x + d)^(5 /2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2*d^2/e + 5819814 *(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*c*d ^2/e + 415701*(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*b + 415701*(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^4*d^3/e^3 + 4988412*(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*c*d^3/e^3 + 2494206*(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^2*d^3/e^3 + 3741309*(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^3*d^2/e^2 + 11223927*(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*c*d^2/e^2 + 3741309*(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^2*d/e + 249420...
Time = 10.94 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.04 \[ \int (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{17/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{17\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{13\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{7\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{11\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{9/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{9\,e^8}+\frac {2\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{15/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{3\,e^8} \]
((d + e*x)^(17/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d *e))/(17*e^8) + (4*c^4*(d + e*x)^(21/2))/(21*e^8) - ((28*c^4*d - 14*b*c^3* e)*(d + e*x)^(19/2))/(19*e^8) + ((d + e*x)^(13/2)*(2*b^4*e^4 + 140*c^4*d^4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e ^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*e^3 - 120*a*b*c^2*d*e^3))/(13*e^8) + (2* (b*e - 2*c*d)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(7*e^8) + (6*(b*e - 2*c*d)*(d + e*x)^(11/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^ 3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d^2*e^2 - 14*b*c^2*d^3*e - 10*a*b*c*d*e^3)) /(11*e^8) + (2*(d + e*x)^(9/2)*(a*e^2 + c*d^2 - b*d*e)^2*(3*b^2*e^2 + 14*c ^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(9*e^8) + (2*c*(b*e - 2*c*d)*(d + e*x)^( 15/2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(3*e^8)