Integrand size = 20, antiderivative size = 426 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=-\frac {c^3 (7 c d-4 b e) x}{e^8}+\frac {c^4 x^2}{2 e^7}-\frac {\left (c d^2-b d e+a e^2\right )^4}{6 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{5 e^9 (d+e x)^5}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^3}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{2 e^9 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \log (d+e x)}{e^9} \] Output:
-c^3*(-4*b*e+7*c*d)*x/e^8+1/2*c^4*x^2/e^7-1/6*(a*e^2-b*d*e+c*d^2)^4/e^9/(e *x+d)^6+4/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^5-1/2*(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^9/(e*x+d)^4+4/3*( -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^9 /(e*x+d)^3-1/2*(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^9/(e*x+d)^2+4 *c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))/e^9/(e*x+d)+2*c^2*( 14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*ln(e*x+d)/e^9
Time = 0.24 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\frac {c^4 \left (1023 d^8+5298 d^7 e x+10725 d^6 e^2 x^2+10100 d^5 e^3 x^3+3375 d^4 e^4 x^4-1170 d^3 e^5 x^5-1035 d^2 e^6 x^6-120 d e^7 x^7+15 e^8 x^8\right )-e^4 \left (5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )-2 c e^3 \left (a^3 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+3 a^2 b e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+6 a b^2 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+10 b^3 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )-3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+20 a b e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )-b^2 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+2 c^3 e \left (a d e \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-b \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 c^2 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) (d+e x)^6 \log (d+e x)}{30 e^9 (d+e x)^6} \] Input:
Integrate[(a + b*x + c*x^2)^4/(d + e*x)^7,x]
Output:
(c^4*(1023*d^8 + 5298*d^7*e*x + 10725*d^6*e^2*x^2 + 10100*d^5*e^3*x^3 + 33 75*d^4*e^4*x^4 - 1170*d^3*e^5*x^5 - 1035*d^2*e^6*x^6 - 120*d*e^7*x^7 + 15* e^8*x^8) - e^4*(5*a^4*e^4 + 4*a^3*b*e^3*(d + 6*e*x) + 3*a^2*b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b^3*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^ 3*x^3) + b^4*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4 )) - 2*c*e^3*(a^3*e^3*(d^2 + 6*d*e*x + 15*e^2*x^2) + 3*a^2*b*e^2*(d^3 + 6* d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 6*a*b^2*e*(d^4 + 6*d^3*e*x + 15*d^2 *e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 10*b^3*(d^5 + 6*d^4*e*x + 15*d^3*e ^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) - 3*c^2*e^2*(2*a^2*e^ 2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 20*a*b* e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^ 5*x^5) - b^2*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^ 3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) + 2*c^3*e*(a*d*e*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - b *(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x^3 + 4050*d^3* e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 60*c^2*(14*c^2* d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(d + e*x)^6*Log[d + e*x])/(30*e^9* (d + e*x)^6)
Time = 0.87 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)^3}+\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)}+\frac {4 c (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8 (d+e x)^2}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8 (d+e x)^4}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^5}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^6}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^7}-\frac {c^3 (7 c d-4 b e)}{e^8}+\frac {c^4 x}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{2 e^9 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac {2 c^2 \log (d+e x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^4}{6 e^9 (d+e x)^6}-\frac {c^3 x (7 c d-4 b e)}{e^8}+\frac {c^4 x^2}{2 e^7}\) |
Input:
Int[(a + b*x + c*x^2)^4/(d + e*x)^7,x]
Output:
-((c^3*(7*c*d - 4*b*e)*x)/e^8) + (c^4*x^2)/(2*e^7) - (c*d^2 - b*d*e + a*e^ 2)^4/(6*e^9*(d + e*x)^6) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(5* e^9*(d + e*x)^5) - ((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)))/(2*e^9*(d + e*x)^4) + (4*(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)))/(3*e^9*(d + e*x)^3) - (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))/(2*e^9*(d + e* x)^2) + (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(e ^9*(d + e*x)) + (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*Log[ d + e*x])/e^9
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]
Leaf count of result is larger than twice the leaf count of optimal. \(888\) vs. \(2(414)=828\).
Time = 0.88 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.09
| method | result | size |
| norman | \(\frac {-\frac {5 a^{4} e^{8}+4 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+6 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}-294 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}-441 b^{2} c^{2} d^{6} e^{2}+2058 b \,c^{3} d^{7} e -2058 c^{4} d^{8}}{30 e^{9}}+\frac {c^{4} x^{8}}{2 e}-\frac {2 \left (6 a b \,c^{2} e^{3}-12 a \,c^{3} d \,e^{2}+2 b^{3} c \,e^{3}-18 d \,e^{2} b^{2} c^{2}+84 b \,c^{3} d^{2} e -84 d^{3} c^{4}\right ) x^{5}}{e^{4}}-\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}-180 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+20 d \,e^{3} b^{3} c -270 d^{2} e^{2} b^{2} c^{2}+1260 d^{3} e b \,c^{3}-1260 d^{4} c^{4}\right ) x^{4}}{2 e^{5}}-\frac {2 \left (6 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+2 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-220 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-330 b^{2} c^{2} d^{3} e^{2}+1540 b \,c^{3} d^{4} e -1540 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}-\frac {\left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+6 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+2 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}-250 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}-375 b^{2} c^{2} d^{4} e^{2}+1750 b \,c^{3} d^{5} e -1750 d^{6} c^{4}\right ) x^{2}}{2 e^{7}}-\frac {\left (4 a^{3} b \,e^{7}+2 d \,e^{6} c \,a^{3}+3 a^{2} b^{2} d \,e^{6}+6 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+2 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-274 d^{5} e^{2} a \,c^{3}+b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-411 b^{2} c^{2} d^{5} e^{2}+1918 b \,c^{3} d^{6} e -1918 d^{7} c^{4}\right ) x}{5 e^{8}}+\frac {4 c^{3} \left (b e -c d \right ) x^{7}}{e^{2}}}{\left (e x +d \right )^{6}}+\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(889\) |
| default | \(\frac {c^{3} \left (\frac {1}{2} c e \,x^{2}+4 b e x -7 c d x \right )}{e^{8}}-\frac {12 a^{2} b c \,e^{5}-24 a^{2} c^{2} d \,e^{4}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 a \,c^{3} d^{3} e^{2}-4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{3 e^{9} \left (e x +d \right )^{3}}-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{4 e^{9} \left (e x +d \right )^{4}}+\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{9}}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} a \,c^{3}-4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {4 c \left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-9 d \,e^{2} b^{2} c +21 d^{2} e b \,c^{2}-14 d^{3} c^{3}\right )}{e^{9} \left (e x +d \right )}-\frac {6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-20 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}-140 d^{3} e b \,c^{3}+70 d^{4} c^{4}}{2 e^{9} \left (e x +d \right )^{2}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{6 e^{9} \left (e x +d \right )^{6}}\) | \(900\) |
| risch | \(\frac {c^{4} x^{2}}{2 e^{7}}+\frac {4 c^{3} b x}{e^{7}}-\frac {7 c^{4} d x}{e^{8}}+\frac {\left (-12 c^{2} e^{7} a b +24 a \,c^{3} d \,e^{6}-4 b^{3} c \,e^{7}+36 b^{2} c^{2} d \,e^{6}-84 b \,c^{3} d^{2} e^{5}+56 c^{4} d^{3} e^{4}\right ) x^{5}-\frac {e^{3} \left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}-180 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+20 d \,e^{3} b^{3} c -270 d^{2} e^{2} b^{2} c^{2}+700 d^{3} e b \,c^{3}-490 d^{4} c^{4}\right ) x^{4}}{2}-\frac {2 e^{2} \left (6 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+2 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-220 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-330 b^{2} c^{2} d^{3} e^{2}+910 b \,c^{3} d^{4} e -658 c^{4} d^{5}\right ) x^{3}}{3}-\frac {e \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+6 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+2 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}-250 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}-375 b^{2} c^{2} d^{4} e^{2}+1078 b \,c^{3} d^{5} e -798 d^{6} c^{4}\right ) x^{2}}{2}+\left (-\frac {4}{5} a^{3} b \,e^{7}-\frac {2}{5} d \,e^{6} c \,a^{3}-\frac {3}{5} a^{2} b^{2} d \,e^{6}-\frac {6}{5} a^{2} b c \,d^{2} e^{5}-\frac {6}{5} d^{3} e^{4} a^{2} c^{2}-\frac {2}{5} a \,b^{3} d^{2} e^{5}-\frac {12}{5} a \,b^{2} c \,d^{3} e^{4}-12 a b \,c^{2} d^{4} e^{3}+\frac {274}{5} d^{5} e^{2} a \,c^{3}-\frac {1}{5} b^{4} d^{3} e^{4}-4 b^{3} c \,d^{4} e^{3}+\frac {411}{5} b^{2} c^{2} d^{5} e^{2}-\frac {1218}{5} b \,c^{3} d^{6} e +\frac {918}{5} d^{7} c^{4}\right ) x -\frac {5 a^{4} e^{8}+4 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+6 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}-294 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}-441 b^{2} c^{2} d^{6} e^{2}+1338 b \,c^{3} d^{7} e -1023 c^{4} d^{8}}{30 e}}{e^{8} \left (e x +d \right )^{6}}+\frac {4 c^{3} \ln \left (e x +d \right ) a}{e^{7}}+\frac {6 c^{2} \ln \left (e x +d \right ) b^{2}}{e^{7}}-\frac {28 c^{3} \ln \left (e x +d \right ) b d}{e^{8}}+\frac {28 c^{4} \ln \left (e x +d \right ) d^{2}}{e^{9}}\) | \(910\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1519\) |
Input:
int((c*x^2+b*x+a)^4/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
(-1/30*(5*a^4*e^8+4*a^3*b*d*e^7+2*a^3*c*d^2*e^6+3*a^2*b^2*d^2*e^6+6*a^2*b* c*d^3*e^5+6*a^2*c^2*d^4*e^4+2*a*b^3*d^3*e^5+12*a*b^2*c*d^4*e^4+60*a*b*c^2* d^5*e^3-294*a*c^3*d^6*e^2+b^4*d^4*e^4+20*b^3*c*d^5*e^3-441*b^2*c^2*d^6*e^2 +2058*b*c^3*d^7*e-2058*c^4*d^8)/e^9+1/2*c^4/e*x^8-2*(6*a*b*c^2*e^3-12*a*c^ 3*d*e^2+2*b^3*c*e^3-18*b^2*c^2*d*e^2+84*b*c^3*d^2*e-84*c^4*d^3)/e^4*x^5-1/ 2*(6*a^2*c^2*e^4+12*a*b^2*c*e^4+60*a*b*c^2*d*e^3-180*a*c^3*d^2*e^2+b^4*e^4 +20*b^3*c*d*e^3-270*b^2*c^2*d^2*e^2+1260*b*c^3*d^3*e-1260*c^4*d^4)/e^5*x^4 -2/3*(6*a^2*b*c*e^5+6*a^2*c^2*d*e^4+2*a*b^3*e^5+12*a*b^2*c*d*e^4+60*a*b*c^ 2*d^2*e^3-220*a*c^3*d^3*e^2+b^4*d*e^4+20*b^3*c*d^2*e^3-330*b^2*c^2*d^3*e^2 +1540*b*c^3*d^4*e-1540*c^4*d^5)/e^6*x^3-1/2*(2*a^3*c*e^6+3*a^2*b^2*e^6+6*a ^2*b*c*d*e^5+6*a^2*c^2*d^2*e^4+2*a*b^3*d*e^5+12*a*b^2*c*d^2*e^4+60*a*b*c^2 *d^3*e^3-250*a*c^3*d^4*e^2+b^4*d^2*e^4+20*b^3*c*d^3*e^3-375*b^2*c^2*d^4*e^ 2+1750*b*c^3*d^5*e-1750*c^4*d^6)/e^7*x^2-1/5*(4*a^3*b*e^7+2*a^3*c*d*e^6+3* a^2*b^2*d*e^6+6*a^2*b*c*d^2*e^5+6*a^2*c^2*d^3*e^4+2*a*b^3*d^2*e^5+12*a*b^2 *c*d^3*e^4+60*a*b*c^2*d^4*e^3-274*a*c^3*d^5*e^2+b^4*d^3*e^4+20*b^3*c*d^4*e ^3-411*b^2*c^2*d^5*e^2+1918*b*c^3*d^6*e-1918*c^4*d^7)/e^8*x+4*c^3*(b*e-c*d )/e^2*x^7)/(e*x+d)^6+2*c^2/e^9*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2) *ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1191 vs. \(2 (414) = 828\).
Time = 0.08 (sec) , antiderivative size = 1191, normalized size of antiderivative = 2.80 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d)^7,x, algorithm="fricas")
Output:
1/30*(15*c^4*e^8*x^8 + 1023*c^4*d^8 - 1338*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 5 *a^4*e^8 + 147*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^5* e^3 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 2*(a*b^3 + 3*a^2*b*c)*d^3*e ^5 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 120*(c^4*d*e^7 - b*c^3*e^8)*x^7 - 45* (23*c^4*d^2*e^6 - 16*b*c^3*d*e^7)*x^6 - 30*(39*c^4*d^3*e^5 + 24*b*c^3*d^2* e^6 - 12*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 4*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 15 *(225*c^4*d^4*e^4 - 540*b*c^3*d^3*e^5 + 90*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 20*(b^3*c + 3*a*b*c^2)*d*e^7 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 20*(505*c^4*d^5*e^3 - 820*b*c^3*d^4*e^4 + 110*(3*b^2*c^2 + 2*a*c^3)*d^3*e^ 5 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^6 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 2*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 15*(715*c^4*d^6*e^2 - 1030*b*c^3*d^5*e^ 3 + 125*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^5 - ( b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 2*(a*b^3 + 3*a^2*b*c)*d*e^7 - (3*a ^2*b^2 + 2*a^3*c)*e^8)*x^2 + 6*(883*c^4*d^7*e - 1198*b*c^3*d^6*e^2 - 4*a^3 *b*e^8 + 137*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 20*(b^3*c + 3*a*b*c^2)*d^4*e^ 4 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 2*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - (3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 60*(14*c^4*d^8 - 14*b*c^3*d^7*e + (3*b ^2*c^2 + 2*a*c^3)*d^6*e^2 + (14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 6*(14*c^4*d^3*e^5 - 14*b*c^3*d^2*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7)*x^5 + 15*(14*c^4*d^4*e^4 - 14*b*c^3*d^3*e^5 + (3*b^2*c^...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=\text {Timed out} \] Input:
integrate((c*x**2+b*x+a)**4/(e*x+d)**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (414) = 828\).
Time = 0.07 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d)^7,x, algorithm="maxima")
Output:
1/30*(1023*c^4*d^8 - 1338*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 5*a^4*e^8 + 147*(3 *b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^5*e^3 - (b^4 + 12*a *b^2*c + 6*a^2*c^2)*d^4*e^4 - 2*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 120*(14*c^4*d^3*e^5 - 21*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 15*(490*c^4*d^4*e^4 - 70 0*b*c^3*d^3*e^5 + 90*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 20*(b^3*c + 3*a*b*c^2 )*d*e^7 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 20*(658*c^4*d^5*e^3 - 910*b*c^3*d^4*e^4 + 110*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(b^3*c + 3*a*b* c^2)*d^2*e^6 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 2*(a*b^3 + 3*a^2*b*c )*e^8)*x^3 + 15*(798*c^4*d^6*e^2 - 1078*b*c^3*d^5*e^3 + 125*(3*b^2*c^2 + 2 *a*c^3)*d^4*e^4 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^5 - (b^4 + 12*a*b^2*c + 6*a ^2*c^2)*d^2*e^6 - 2*(a*b^3 + 3*a^2*b*c)*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*e^8) *x^2 + 6*(918*c^4*d^7*e - 1218*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 137*(3*b^2*c^ 2 + 2*a*c^3)*d^5*e^3 - 20*(b^3*c + 3*a*b*c^2)*d^4*e^4 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 2*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - (3*a^2*b^2 + 2*a^3* c)*d*e^7)*x)/(e^15*x^6 + 6*d*e^14*x^5 + 15*d^2*e^13*x^4 + 20*d^3*e^12*x^3 + 15*d^4*e^11*x^2 + 6*d^5*e^10*x + d^6*e^9) + 1/2*(c^4*e*x^2 - 2*(7*c^4*d - 4*b*c^3*e)*x)/e^8 + 2*(14*c^4*d^2 - 14*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3) *e^2)*log(e*x + d)/e^9
Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (414) = 828\).
Time = 0.35 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d)^7,x, algorithm="giac")
Output:
2*(14*c^4*d^2 - 14*b*c^3*d*e + 3*b^2*c^2*e^2 + 2*a*c^3*e^2)*log(abs(e*x + d))/e^9 + 1/2*(c^4*e^7*x^2 - 14*c^4*d*e^6*x + 8*b*c^3*e^7*x)/e^14 + 1/30*( 1023*c^4*d^8 - 1338*b*c^3*d^7*e + 441*b^2*c^2*d^6*e^2 + 294*a*c^3*d^6*e^2 - 20*b^3*c*d^5*e^3 - 60*a*b*c^2*d^5*e^3 - b^4*d^4*e^4 - 12*a*b^2*c*d^4*e^4 - 6*a^2*c^2*d^4*e^4 - 2*a*b^3*d^3*e^5 - 6*a^2*b*c*d^3*e^5 - 3*a^2*b^2*d^2 *e^6 - 2*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 - 5*a^4*e^8 + 120*(14*c^4*d^3*e^5 - 21*b*c^3*d^2*e^6 + 9*b^2*c^2*d*e^7 + 6*a*c^3*d*e^7 - b^3*c*e^8 - 3*a*b*c^ 2*e^8)*x^5 + 15*(490*c^4*d^4*e^4 - 700*b*c^3*d^3*e^5 + 270*b^2*c^2*d^2*e^6 + 180*a*c^3*d^2*e^6 - 20*b^3*c*d*e^7 - 60*a*b*c^2*d*e^7 - b^4*e^8 - 12*a* b^2*c*e^8 - 6*a^2*c^2*e^8)*x^4 + 20*(658*c^4*d^5*e^3 - 910*b*c^3*d^4*e^4 + 330*b^2*c^2*d^3*e^5 + 220*a*c^3*d^3*e^5 - 20*b^3*c*d^2*e^6 - 60*a*b*c^2*d ^2*e^6 - b^4*d*e^7 - 12*a*b^2*c*d*e^7 - 6*a^2*c^2*d*e^7 - 2*a*b^3*e^8 - 6* a^2*b*c*e^8)*x^3 + 15*(798*c^4*d^6*e^2 - 1078*b*c^3*d^5*e^3 + 375*b^2*c^2* d^4*e^4 + 250*a*c^3*d^4*e^4 - 20*b^3*c*d^3*e^5 - 60*a*b*c^2*d^3*e^5 - b^4* d^2*e^6 - 12*a*b^2*c*d^2*e^6 - 6*a^2*c^2*d^2*e^6 - 2*a*b^3*d*e^7 - 6*a^2*b *c*d*e^7 - 3*a^2*b^2*e^8 - 2*a^3*c*e^8)*x^2 + 6*(918*c^4*d^7*e - 1218*b*c^ 3*d^6*e^2 + 411*b^2*c^2*d^5*e^3 + 274*a*c^3*d^5*e^3 - 20*b^3*c*d^4*e^4 - 6 0*a*b*c^2*d^4*e^4 - b^4*d^3*e^5 - 12*a*b^2*c*d^3*e^5 - 6*a^2*c^2*d^3*e^5 - 2*a*b^3*d^2*e^6 - 6*a^2*b*c*d^2*e^6 - 3*a^2*b^2*d*e^7 - 2*a^3*c*d*e^7 - 4 *a^3*b*e^8)*x)/((e*x + d)^6*e^9)
Time = 0.12 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx=x\,\left (\frac {4\,b\,c^3}{e^7}-\frac {7\,c^4\,d}{e^8}\right )-\frac {x^3\,\left (4\,a^2\,b\,c\,e^7+4\,a^2\,c^2\,d\,e^6+\frac {4\,a\,b^3\,e^7}{3}+8\,a\,b^2\,c\,d\,e^6+40\,a\,b\,c^2\,d^2\,e^5-\frac {440\,a\,c^3\,d^3\,e^4}{3}+\frac {2\,b^4\,d\,e^6}{3}+\frac {40\,b^3\,c\,d^2\,e^5}{3}-220\,b^2\,c^2\,d^3\,e^4+\frac {1820\,b\,c^3\,d^4\,e^3}{3}-\frac {1316\,c^4\,d^5\,e^2}{3}\right )+x\,\left (\frac {4\,a^3\,b\,e^7}{5}+\frac {2\,a^3\,c\,d\,e^6}{5}+\frac {3\,a^2\,b^2\,d\,e^6}{5}+\frac {6\,a^2\,b\,c\,d^2\,e^5}{5}+\frac {6\,a^2\,c^2\,d^3\,e^4}{5}+\frac {2\,a\,b^3\,d^2\,e^5}{5}+\frac {12\,a\,b^2\,c\,d^3\,e^4}{5}+12\,a\,b\,c^2\,d^4\,e^3-\frac {274\,a\,c^3\,d^5\,e^2}{5}+\frac {b^4\,d^3\,e^4}{5}+4\,b^3\,c\,d^4\,e^3-\frac {411\,b^2\,c^2\,d^5\,e^2}{5}+\frac {1218\,b\,c^3\,d^6\,e}{5}-\frac {918\,c^4\,d^7}{5}\right )+x^4\,\left (3\,a^2\,c^2\,e^7+6\,a\,b^2\,c\,e^7+30\,a\,b\,c^2\,d\,e^6-90\,a\,c^3\,d^2\,e^5+\frac {b^4\,e^7}{2}+10\,b^3\,c\,d\,e^6-135\,b^2\,c^2\,d^2\,e^5+350\,b\,c^3\,d^3\,e^4-245\,c^4\,d^4\,e^3\right )+x^5\,\left (4\,b^3\,c\,e^7-36\,b^2\,c^2\,d\,e^6+84\,b\,c^3\,d^2\,e^5+12\,a\,b\,c^2\,e^7-56\,c^4\,d^3\,e^4-24\,a\,c^3\,d\,e^6\right )+\frac {5\,a^4\,e^8+4\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+6\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+2\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+60\,a\,b\,c^2\,d^5\,e^3-294\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+20\,b^3\,c\,d^5\,e^3-441\,b^2\,c^2\,d^6\,e^2+1338\,b\,c^3\,d^7\,e-1023\,c^4\,d^8}{30\,e}+x^2\,\left (a^3\,c\,e^7+\frac {3\,a^2\,b^2\,e^7}{2}+3\,a^2\,b\,c\,d\,e^6+3\,a^2\,c^2\,d^2\,e^5+a\,b^3\,d\,e^6+6\,a\,b^2\,c\,d^2\,e^5+30\,a\,b\,c^2\,d^3\,e^4-125\,a\,c^3\,d^4\,e^3+\frac {b^4\,d^2\,e^5}{2}+10\,b^3\,c\,d^3\,e^4-\frac {375\,b^2\,c^2\,d^4\,e^3}{2}+539\,b\,c^3\,d^5\,e^2-399\,c^4\,d^6\,e\right )}{d^6\,e^8+6\,d^5\,e^9\,x+15\,d^4\,e^{10}\,x^2+20\,d^3\,e^{11}\,x^3+15\,d^2\,e^{12}\,x^4+6\,d\,e^{13}\,x^5+e^{14}\,x^6}+\frac {\ln \left (d+e\,x\right )\,\left (6\,b^2\,c^2\,e^2-28\,b\,c^3\,d\,e+28\,c^4\,d^2+4\,a\,c^3\,e^2\right )}{e^9}+\frac {c^4\,x^2}{2\,e^7} \] Input:
int((a + b*x + c*x^2)^4/(d + e*x)^7,x)
Output:
x*((4*b*c^3)/e^7 - (7*c^4*d)/e^8) - (x^3*((4*a*b^3*e^7)/3 + (2*b^4*d*e^6)/ 3 - (1316*c^4*d^5*e^2)/3 - (440*a*c^3*d^3*e^4)/3 + 4*a^2*c^2*d*e^6 + (1820 *b*c^3*d^4*e^3)/3 + (40*b^3*c*d^2*e^5)/3 - 220*b^2*c^2*d^3*e^4 + 4*a^2*b*c *e^7 + 8*a*b^2*c*d*e^6 + 40*a*b*c^2*d^2*e^5) + x*((4*a^3*b*e^7)/5 - (918*c ^4*d^7)/5 + (b^4*d^3*e^4)/5 + (2*a*b^3*d^2*e^5)/5 + (3*a^2*b^2*d*e^6)/5 - (274*a*c^3*d^5*e^2)/5 + 4*b^3*c*d^4*e^3 + (6*a^2*c^2*d^3*e^4)/5 - (411*b^2 *c^2*d^5*e^2)/5 + (2*a^3*c*d*e^6)/5 + (1218*b*c^3*d^6*e)/5 + 12*a*b*c^2*d^ 4*e^3 + (12*a*b^2*c*d^3*e^4)/5 + (6*a^2*b*c*d^2*e^5)/5) + x^4*((b^4*e^7)/2 + 3*a^2*c^2*e^7 - 245*c^4*d^4*e^3 - 90*a*c^3*d^2*e^5 + 350*b*c^3*d^3*e^4 - 135*b^2*c^2*d^2*e^5 + 6*a*b^2*c*e^7 + 10*b^3*c*d*e^6 + 30*a*b*c^2*d*e^6) + x^5*(4*b^3*c*e^7 - 56*c^4*d^3*e^4 + 84*b*c^3*d^2*e^5 - 36*b^2*c^2*d*e^6 + 12*a*b*c^2*e^7 - 24*a*c^3*d*e^6) + (5*a^4*e^8 - 1023*c^4*d^8 + b^4*d^4* e^4 + 2*a*b^3*d^3*e^5 - 294*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 + 20*b^3*c*d^5 *e^3 + 3*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 - 441*b^2*c^2*d^6*e^2 + 4*a^3 *b*d*e^7 + 1338*b*c^3*d^7*e + 60*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 + 6* a^2*b*c*d^3*e^5)/(30*e) + x^2*(a^3*c*e^7 - 399*c^4*d^6*e + (3*a^2*b^2*e^7) /2 + (b^4*d^2*e^5)/2 - 125*a*c^3*d^4*e^3 + 539*b*c^3*d^5*e^2 + 10*b^3*c*d^ 3*e^4 + 3*a^2*c^2*d^2*e^5 - (375*b^2*c^2*d^4*e^3)/2 + a*b^3*d*e^6 + 3*a^2* b*c*d*e^6 + 30*a*b*c^2*d^3*e^4 + 6*a*b^2*c*d^2*e^5))/(d^6*e^8 + e^14*x^6 + 6*d^5*e^9*x + 6*d*e^13*x^5 + 15*d^4*e^10*x^2 + 20*d^3*e^11*x^3 + 15*d^...
Time = 0.21 (sec) , antiderivative size = 1472, normalized size of antiderivative = 3.46 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^4/(e*x+d)^7,x)
Output:
(120*log(d + e*x)*a*c**3*d**7*e**2 + 720*log(d + e*x)*a*c**3*d**6*e**3*x + 1800*log(d + e*x)*a*c**3*d**5*e**4*x**2 + 2400*log(d + e*x)*a*c**3*d**4*e **5*x**3 + 1800*log(d + e*x)*a*c**3*d**3*e**6*x**4 + 720*log(d + e*x)*a*c* *3*d**2*e**7*x**5 + 120*log(d + e*x)*a*c**3*d*e**8*x**6 + 180*log(d + e*x) *b**2*c**2*d**7*e**2 + 1080*log(d + e*x)*b**2*c**2*d**6*e**3*x + 2700*log( d + e*x)*b**2*c**2*d**5*e**4*x**2 + 3600*log(d + e*x)*b**2*c**2*d**4*e**5* x**3 + 2700*log(d + e*x)*b**2*c**2*d**3*e**6*x**4 + 1080*log(d + e*x)*b**2 *c**2*d**2*e**7*x**5 + 180*log(d + e*x)*b**2*c**2*d*e**8*x**6 - 840*log(d + e*x)*b*c**3*d**8*e - 5040*log(d + e*x)*b*c**3*d**7*e**2*x - 12600*log(d + e*x)*b*c**3*d**6*e**3*x**2 - 16800*log(d + e*x)*b*c**3*d**5*e**4*x**3 - 12600*log(d + e*x)*b*c**3*d**4*e**5*x**4 - 5040*log(d + e*x)*b*c**3*d**3*e **6*x**5 - 840*log(d + e*x)*b*c**3*d**2*e**7*x**6 + 840*log(d + e*x)*c**4* d**9 + 5040*log(d + e*x)*c**4*d**8*e*x + 12600*log(d + e*x)*c**4*d**7*e**2 *x**2 + 16800*log(d + e*x)*c**4*d**6*e**3*x**3 + 12600*log(d + e*x)*c**4*d **5*e**4*x**4 + 5040*log(d + e*x)*c**4*d**4*e**5*x**5 + 840*log(d + e*x)*c **4*d**3*e**6*x**6 - 5*a**4*d*e**8 - 4*a**3*b*d**2*e**7 - 24*a**3*b*d*e**8 *x - 2*a**3*c*d**3*e**6 - 12*a**3*c*d**2*e**7*x - 30*a**3*c*d*e**8*x**2 - 3*a**2*b**2*d**3*e**6 - 18*a**2*b**2*d**2*e**7*x - 45*a**2*b**2*d*e**8*x** 2 - 6*a**2*b*c*d**4*e**5 - 36*a**2*b*c*d**3*e**6*x - 90*a**2*b*c*d**2*e**7 *x**2 - 120*a**2*b*c*d*e**8*x**3 - 6*a**2*c**2*d**5*e**4 - 36*a**2*c**2...