Integrand size = 20, antiderivative size = 430 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{2 e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^2}{2 e^9}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^3}{3 e^9}+\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^4}{2 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^5}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \log (d+e x)}{e^9} \]
-4*(-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)) *x/e^8-1/2*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^2+4*(-b*e+2*c*d)*(a*e^2-b*d*e +c*d^2)^3/e^9/(e*x+d)+1/2*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-2 0*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)^2/e^9-4/3*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 )^3/e^9+1/2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^4/e^9-4/ 5*c^3*(-b*e+2*c*d)*(e*x+d)^5/e^9+1/6*c^4*(e*x+d)^6/e^9+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))*ln(e*x+d)/e^9
Time = 0.12 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {30 e \left (-21 c^4 d^5+20 c^3 d^3 e (3 b d-2 a e)+b^3 e^4 (-3 b d+4 a e)+12 b c e^3 \left (2 b^2 d^2-3 a b d e+a^2 e^2\right )-6 c^2 d e^2 \left (10 b^2 d^2-12 a b d e+3 a^2 e^2\right )\right ) x+15 e^2 \left (15 c^4 d^4+b^4 e^4-8 c^3 d^2 e (5 b d-3 a e)-12 b^2 c e^3 (b d-a e)+6 c^2 e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) x^2+20 c e^3 (-c d+b e) \left (5 c^2 d^2+2 b^2 e^2+c e (-7 b d+6 a e)\right ) x^3+15 c^2 e^4 \left (3 c^2 d^2+3 b^2 e^2+2 c e (-3 b d+a e)\right ) x^4+6 c^3 e^5 (-3 c d+4 b e) x^5+5 c^4 e^6 x^6-\frac {15 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^2}+\frac {120 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{d+e x}+60 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2 \log (d+e x)}{30 e^9} \]
(30*e*(-21*c^4*d^5 + 20*c^3*d^3*e*(3*b*d - 2*a*e) + b^3*e^4*(-3*b*d + 4*a* e) + 12*b*c*e^3*(2*b^2*d^2 - 3*a*b*d*e + a^2*e^2) - 6*c^2*d*e^2*(10*b^2*d^ 2 - 12*a*b*d*e + 3*a^2*e^2))*x + 15*e^2*(15*c^4*d^4 + b^4*e^4 - 8*c^3*d^2* e*(5*b*d - 3*a*e) - 12*b^2*c*e^3*(b*d - a*e) + 6*c^2*e^2*(6*b^2*d^2 - 6*a* b*d*e + a^2*e^2))*x^2 + 20*c*e^3*(-(c*d) + b*e)*(5*c^2*d^2 + 2*b^2*e^2 + c *e*(-7*b*d + 6*a*e))*x^3 + 15*c^2*e^4*(3*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-3*b *d + a*e))*x^4 + 6*c^3*e^5*(-3*c*d + 4*b*e)*x^5 + 5*c^4*e^6*x^6 - (15*(c*d ^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^2 + (120*(2*c*d - b*e)*(c*d^2 + e*(-(b *d) + a*e))^3)/(d + e*x) + 60*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a* e))*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(30*e^9)
Time = 0.93 (sec) , antiderivative size = 430, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac {2 c^2 (d+e x)^3 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 c (d+e x)^2 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^2}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^3}-\frac {4 c^3 (d+e x)^4 (2 c d-b e)}{e^8}+\frac {c^4 (d+e x)^5}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^9}+\frac {c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9}-\frac {4 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 \log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}-\frac {4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^4}{2 e^9 (d+e x)^2}-\frac {4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9}\) |
(-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))*x)/e^8 - (c*d^2 - b*d*e + a*e^2)^4/(2*e^9*(d + e*x)^2) + (4*(2 *c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)) + ((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)^2)/(2*e^9) - (4*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)^3)/(3*e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^4)/(2*e^9 ) - (4*c^3*(2*c*d - b*e)*(d + e*x)^5)/(5*e^9) + (c^4*(d + e*x)^6)/(6*e^9) + (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))*Log[d + e*x])/e^9
3.22.53.3.1 Defintions of rubi rules used
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. \(889\) vs. \(2(418)=836\).
Time = 3.21 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.07
| method | result | size |
| norman | \(\frac {-\frac {a^{4} e^{8}+4 a^{3} b d \,e^{7}-12 a^{3} c \,d^{2} e^{6}-18 a^{2} b^{2} d^{2} e^{6}+108 a^{2} b c \,d^{3} e^{5}-108 a^{2} c^{2} d^{4} e^{4}+36 a \,b^{3} d^{3} e^{5}-216 a \,b^{2} c \,d^{4} e^{4}+360 a b \,c^{2} d^{5} e^{3}-180 a \,c^{3} d^{6} e^{2}-18 b^{4} d^{4} e^{4}+120 b^{3} c \,d^{5} e^{3}-270 b^{2} c^{2} d^{6} e^{2}+252 b \,c^{3} d^{7} e -84 c^{4} d^{8}}{2 e^{9}}+\frac {c^{4} x^{8}}{6 e}+\frac {\left (18 c^{2} a^{2} e^{4}+36 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+30 c^{3} a \,d^{2} e^{2}+3 b^{4} e^{4}-20 b^{3} c d \,e^{3}+45 b^{2} c^{2} d^{2} e^{2}-42 b \,c^{3} d^{3} e +14 c^{4} d^{4}\right ) x^{4}}{6 e^{5}}+\frac {2 \left (18 a^{2} b c \,e^{5}-18 d \,e^{4} a^{2} c^{2}+6 a \,b^{3} e^{5}-36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-30 d^{3} e^{2} c^{3} a -3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-45 b^{2} c^{2} d^{3} e^{2}+42 b \,c^{3} d^{4} e -14 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}-\frac {2 \left (2 a^{3} b \,e^{7}-4 d \,e^{6} c \,a^{3}-6 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-36 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-72 a \,b^{2} c \,d^{3} e^{4}+120 a b \,c^{2} d^{4} e^{3}-60 d^{5} e^{2} c^{3} a -6 b^{4} d^{3} e^{4}+40 b^{3} c \,d^{4} e^{3}-90 b^{2} c^{2} d^{5} e^{2}+84 b \,c^{3} d^{6} e -28 d^{7} c^{4}\right ) x}{e^{8}}+\frac {c \left (60 a b c \,e^{3}-30 c^{2} a d \,e^{2}+20 b^{3} e^{3}-45 b^{2} d \,e^{2} c +42 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{5}}{15 e^{4}}+\frac {c^{2} \left (30 a c \,e^{2}+45 b^{2} e^{2}-42 b c d e +14 c^{2} d^{2}\right ) x^{6}}{30 e^{3}}+\frac {4 c^{3} \left (3 b e -c d \right ) x^{7}}{15 e^{2}}}{\left (e x +d \right )^{2}}+\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}-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 d^{4} e^{2} c^{3} a +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 d^{6} c^{4}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(890\) |
| default | \(\frac {-21 c^{4} d^{5} x +a \,c^{3} e^{5} x^{4}+\frac {1}{2} b^{4} e^{5} x^{2}-6 b^{2} c^{2} d \,e^{4} x^{3}+8 b \,c^{3} d^{2} e^{3} x^{3}-18 d \,e^{4} a^{2} c^{2} x -40 d^{3} e^{2} c^{3} a x -36 a \,b^{2} c d \,e^{4} x +72 a b \,c^{2} d^{2} e^{3} x +6 a \,b^{2} c \,e^{5} x^{2}+12 a \,c^{3} d^{2} e^{3} x^{2}-6 b^{3} c d \,e^{4} x^{2}+\frac {1}{6} c^{4} x^{6} e^{5}-3 b \,c^{3} d \,e^{4} x^{4}+18 b^{2} c^{2} d^{2} e^{3} x^{2}-20 b \,c^{3} d^{3} e^{2} x^{2}+12 a^{2} b c \,e^{5} x +\frac {4}{5} b \,c^{3} e^{5} x^{5}-\frac {3}{5} c^{4} d \,e^{4} x^{5}+\frac {3}{2} b^{2} c^{2} e^{5} x^{4}+\frac {3}{2} c^{4} d^{2} e^{3} x^{4}+\frac {4}{3} b^{3} c \,e^{5} x^{3}-\frac {10}{3} c^{4} d^{3} e^{2} x^{3}+3 a^{2} c^{2} e^{5} x^{2}+\frac {15}{2} c^{4} d^{4} e \,x^{2}+4 a \,b^{3} e^{5} x -3 b^{4} d \,e^{4} x -18 a b \,c^{2} d \,e^{4} x^{2}+24 b^{3} c \,d^{2} e^{3} x -60 b^{2} c^{2} d^{3} e^{2} x +60 b \,c^{3} d^{4} e x +4 a b \,c^{2} e^{5} x^{3}-4 a \,c^{3} d \,e^{4} x^{3}}{e^{8}}-\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} c^{3} a -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}}{e^{9} \left (e x +d \right )}-\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}}{2 e^{9} \left (e x +d \right )^{2}}+\frac {\left (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} c^{3} a +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}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(952\) |
| risch | \(\text {Expression too large to display}\) | \(1023\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1578\) |
(-1/2*(a^4*e^8+4*a^3*b*d*e^7-12*a^3*c*d^2*e^6-18*a^2*b^2*d^2*e^6+108*a^2*b *c*d^3*e^5-108*a^2*c^2*d^4*e^4+36*a*b^3*d^3*e^5-216*a*b^2*c*d^4*e^4+360*a* b*c^2*d^5*e^3-180*a*c^3*d^6*e^2-18*b^4*d^4*e^4+120*b^3*c*d^5*e^3-270*b^2*c ^2*d^6*e^2+252*b*c^3*d^7*e-84*c^4*d^8)/e^9+1/6/e*c^4*x^8+1/6*(18*a^2*c^2*e ^4+36*a*b^2*c*e^4-60*a*b*c^2*d*e^3+30*a*c^3*d^2*e^2+3*b^4*e^4-20*b^3*c*d*e ^3+45*b^2*c^2*d^2*e^2-42*b*c^3*d^3*e+14*c^4*d^4)/e^5*x^4+2/3*(18*a^2*b*c*e ^5-18*a^2*c^2*d*e^4+6*a*b^3*e^5-36*a*b^2*c*d*e^4+60*a*b*c^2*d^2*e^3-30*a*c ^3*d^3*e^2-3*b^4*d*e^4+20*b^3*c*d^2*e^3-45*b^2*c^2*d^3*e^2+42*b*c^3*d^4*e- 14*c^4*d^5)/e^6*x^3-2*(2*a^3*b*e^7-4*a^3*c*d*e^6-6*a^2*b^2*d*e^6+36*a^2*b* c*d^2*e^5-36*a^2*c^2*d^3*e^4+12*a*b^3*d^2*e^5-72*a*b^2*c*d^3*e^4+120*a*b*c ^2*d^4*e^3-60*a*c^3*d^5*e^2-6*b^4*d^3*e^4+40*b^3*c*d^4*e^3-90*b^2*c^2*d^5* e^2+84*b*c^3*d^6*e-28*c^4*d^7)/e^8*x+1/15*c*(60*a*b*c*e^3-30*a*c^2*d*e^2+2 0*b^3*e^3-45*b^2*c*d*e^2+42*b*c^2*d^2*e-14*c^3*d^3)/e^4*x^5+1/30*c^2*(30*a *c*e^2+45*b^2*e^2-42*b*c*d*e+14*c^2*d^2)/e^3*x^6+4/15*c^3*(3*b*e-c*d)/e^2* x^7)/(e*x+d)^2+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+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)/e^9*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1218 vs. \(2 (418) = 836\).
Time = 0.66 (sec) , antiderivative size = 1218, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\text {Too large to display} \]
1/30*(5*c^4*e^8*x^8 + 225*c^4*d^8 - 780*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 15* a^4*e^8 + 330*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 540*(b^3*c + 3*a*b*c^2)*d^5* e^3 + 105*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 300*(a*b^3 + 3*a^2*b*c) *d^3*e^5 + 90*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 8*(c^4*d*e^7 - 3*b*c^3*e^8)* x^7 + (14*c^4*d^2*e^6 - 42*b*c^3*d*e^7 + 15*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 2*(14*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 + 15*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 20*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 5*(14*c^4*d^4*e^4 - 42*b*c^3*d^3*e^5 + 15*(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 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 20*(14*c^4*d^5*e^3 - 42*b*c^3*d^4*e^4 + 15*(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 + 3*(b ^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 15*( 69*c^4*d^6*e^2 - 200*b*c^3*d^5*e^3 + 68*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 84 *(b^3*c + 3*a*b*c^2)*d^3*e^5 + 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 16*(a*b^3 + 3*a^2*b*c)*d*e^7)*x^2 - 30*(13*c^4*d^7*e - 32*b*c^3*d^6*e^2 + 4*a^3*b*e^8 + 8*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 4*(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 + 8*(a*b^3 + 3*a^2*b*c)*d^2* e^6 - 4*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 60*(14*c^4*d^8 - 42*b*c^3*d^7*e + 15*(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 + 3*(b^ 4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 6*(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 + (14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 15*(3*...
Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (422) = 844\).
Time = 9.79 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {c^{4} x^{6}}{6 e^{3}} + x^{5} \cdot \left (\frac {4 b c^{3}}{5 e^{3}} - \frac {3 c^{4} d}{5 e^{4}}\right ) + x^{4} \left (\frac {a c^{3}}{e^{3}} + \frac {3 b^{2} c^{2}}{2 e^{3}} - \frac {3 b c^{3} d}{e^{4}} + \frac {3 c^{4} d^{2}}{2 e^{5}}\right ) + x^{3} \cdot \left (\frac {4 a b c^{2}}{e^{3}} - \frac {4 a c^{3} d}{e^{4}} + \frac {4 b^{3} c}{3 e^{3}} - \frac {6 b^{2} c^{2} d}{e^{4}} + \frac {8 b c^{3} d^{2}}{e^{5}} - \frac {10 c^{4} d^{3}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2}}{e^{3}} + \frac {6 a b^{2} c}{e^{3}} - \frac {18 a b c^{2} d}{e^{4}} + \frac {12 a c^{3} d^{2}}{e^{5}} + \frac {b^{4}}{2 e^{3}} - \frac {6 b^{3} c d}{e^{4}} + \frac {18 b^{2} c^{2} d^{2}}{e^{5}} - \frac {20 b c^{3} d^{3}}{e^{6}} + \frac {15 c^{4} d^{4}}{2 e^{7}}\right ) + x \left (\frac {12 a^{2} b c}{e^{3}} - \frac {18 a^{2} c^{2} d}{e^{4}} + \frac {4 a b^{3}}{e^{3}} - \frac {36 a b^{2} c d}{e^{4}} + \frac {72 a b c^{2} d^{2}}{e^{5}} - \frac {40 a c^{3} d^{3}}{e^{6}} - \frac {3 b^{4} d}{e^{4}} + \frac {24 b^{3} c d^{2}}{e^{5}} - \frac {60 b^{2} c^{2} d^{3}}{e^{6}} + \frac {60 b c^{3} d^{4}}{e^{7}} - \frac {21 c^{4} d^{5}}{e^{8}}\right ) + \frac {- a^{4} e^{8} - 4 a^{3} b d e^{7} + 12 a^{3} c d^{2} e^{6} + 18 a^{2} b^{2} d^{2} e^{6} - 60 a^{2} b c d^{3} e^{5} + 42 a^{2} c^{2} d^{4} e^{4} - 20 a b^{3} d^{3} e^{5} + 84 a b^{2} c d^{4} e^{4} - 108 a b c^{2} d^{5} e^{3} + 44 a c^{3} d^{6} e^{2} + 7 b^{4} d^{4} e^{4} - 36 b^{3} c d^{5} e^{3} + 66 b^{2} c^{2} d^{6} e^{2} - 52 b c^{3} d^{7} e + 15 c^{4} d^{8} + x \left (- 8 a^{3} b e^{8} + 16 a^{3} c d e^{7} + 24 a^{2} b^{2} d e^{7} - 72 a^{2} b c d^{2} e^{6} + 48 a^{2} c^{2} d^{3} e^{5} - 24 a b^{3} d^{2} e^{6} + 96 a b^{2} c d^{3} e^{5} - 120 a b c^{2} d^{4} e^{4} + 48 a c^{3} d^{5} e^{3} + 8 b^{4} d^{3} e^{5} - 40 b^{3} c d^{4} e^{4} + 72 b^{2} c^{2} d^{5} e^{3} - 56 b c^{3} d^{6} e^{2} + 16 c^{4} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} + \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{2} \cdot \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{9}} \]
c**4*x**6/(6*e**3) + x**5*(4*b*c**3/(5*e**3) - 3*c**4*d/(5*e**4)) + x**4*( a*c**3/e**3 + 3*b**2*c**2/(2*e**3) - 3*b*c**3*d/e**4 + 3*c**4*d**2/(2*e**5 )) + x**3*(4*a*b*c**2/e**3 - 4*a*c**3*d/e**4 + 4*b**3*c/(3*e**3) - 6*b**2* c**2*d/e**4 + 8*b*c**3*d**2/e**5 - 10*c**4*d**3/(3*e**6)) + x**2*(3*a**2*c **2/e**3 + 6*a*b**2*c/e**3 - 18*a*b*c**2*d/e**4 + 12*a*c**3*d**2/e**5 + b* *4/(2*e**3) - 6*b**3*c*d/e**4 + 18*b**2*c**2*d**2/e**5 - 20*b*c**3*d**3/e* *6 + 15*c**4*d**4/(2*e**7)) + x*(12*a**2*b*c/e**3 - 18*a**2*c**2*d/e**4 + 4*a*b**3/e**3 - 36*a*b**2*c*d/e**4 + 72*a*b*c**2*d**2/e**5 - 40*a*c**3*d** 3/e**6 - 3*b**4*d/e**4 + 24*b**3*c*d**2/e**5 - 60*b**2*c**2*d**3/e**6 + 60 *b*c**3*d**4/e**7 - 21*c**4*d**5/e**8) + (-a**4*e**8 - 4*a**3*b*d*e**7 + 1 2*a**3*c*d**2*e**6 + 18*a**2*b**2*d**2*e**6 - 60*a**2*b*c*d**3*e**5 + 42*a **2*c**2*d**4*e**4 - 20*a*b**3*d**3*e**5 + 84*a*b**2*c*d**4*e**4 - 108*a*b *c**2*d**5*e**3 + 44*a*c**3*d**6*e**2 + 7*b**4*d**4*e**4 - 36*b**3*c*d**5* e**3 + 66*b**2*c**2*d**6*e**2 - 52*b*c**3*d**7*e + 15*c**4*d**8 + x*(-8*a* *3*b*e**8 + 16*a**3*c*d*e**7 + 24*a**2*b**2*d*e**7 - 72*a**2*b*c*d**2*e**6 + 48*a**2*c**2*d**3*e**5 - 24*a*b**3*d**2*e**6 + 96*a*b**2*c*d**3*e**5 - 120*a*b*c**2*d**4*e**4 + 48*a*c**3*d**5*e**3 + 8*b**4*d**3*e**5 - 40*b**3* c*d**4*e**4 + 72*b**2*c**2*d**5*e**3 - 56*b*c**3*d**6*e**2 + 16*c**4*d**7* e))/(2*d**2*e**9 + 4*d*e**10*x + 2*e**11*x**2) + 2*(a*e**2 - b*d*e + c*d** 2)**2*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(d + e*...
Time = 0.23 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {15 \, c^{4} d^{8} - 52 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} - a^{4} e^{8} + 22 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 36 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 7 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 20 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 8 \, {\left (2 \, c^{4} d^{7} e - 7 \, b c^{3} d^{6} e^{2} - a^{3} b e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{2 \, {\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac {5 \, c^{4} e^{5} x^{6} - 6 \, {\left (3 \, c^{4} d e^{4} - 4 \, b c^{3} e^{5}\right )} x^{5} + 15 \, {\left (3 \, c^{4} d^{2} e^{3} - 6 \, b c^{3} d e^{4} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{5}\right )} x^{4} - 20 \, {\left (5 \, c^{4} d^{3} e^{2} - 12 \, b c^{3} d^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{4} - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{5}\right )} x^{3} + 15 \, {\left (15 \, c^{4} d^{4} e - 40 \, b c^{3} d^{3} e^{2} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{3} - 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{5}\right )} x^{2} - 30 \, {\left (21 \, c^{4} d^{5} - 60 \, b c^{3} d^{4} e + 20 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 24 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} x}{30 \, e^{8}} + \frac {2 \, {\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 )} \log \left (e x + d\right )}{e^{9}} \]
1/2*(15*c^4*d^8 - 52*b*c^3*d^7*e - 4*a^3*b*d*e^7 - a^4*e^8 + 22*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 36*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 7*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 20*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 6*(3*a^2*b^2 + 2* a^3*c)*d^2*e^6 + 8*(2*c^4*d^7*e - 7*b*c^3*d^6*e^2 - a^3*b*e^8 + 3*(3*b^2*c ^2 + 2*a*c^3)*d^5*e^3 - 5*(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 - 3*(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^11*x^2 + 2*d*e^10*x + d^2*e^9) + 1/30*(5*c^4*e^5*x^6 - 6*( 3*c^4*d*e^4 - 4*b*c^3*e^5)*x^5 + 15*(3*c^4*d^2*e^3 - 6*b*c^3*d*e^4 + (3*b^ 2*c^2 + 2*a*c^3)*e^5)*x^4 - 20*(5*c^4*d^3*e^2 - 12*b*c^3*d^2*e^3 + 3*(3*b^ 2*c^2 + 2*a*c^3)*d*e^4 - 2*(b^3*c + 3*a*b*c^2)*e^5)*x^3 + 15*(15*c^4*d^4*e - 40*b*c^3*d^3*e^2 + 12*(3*b^2*c^2 + 2*a*c^3)*d^2*e^3 - 12*(b^3*c + 3*a*b *c^2)*d*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^5)*x^2 - 30*(21*c^4*d^5 - 6 0*b*c^3*d^4*e + 20*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 24*(b^3*c + 3*a*b*c^2)* d^2*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - 4*(a*b^3 + 3*a^2*b*c)*e ^5)*x)/e^8 + 2*(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)*log(e*x + d)/e^9
Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (418) = 836\).
Time = 0.27 (sec) , antiderivative size = 958, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\frac {2 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 45 \, b^{2} c^{2} d^{4} e^{2} + 30 \, a c^{3} d^{4} e^{2} - 20 \, b^{3} c d^{3} e^{3} - 60 \, a b c^{2} d^{3} e^{3} + 3 \, b^{4} d^{2} e^{4} + 36 \, a b^{2} c d^{2} e^{4} + 18 \, a^{2} c^{2} d^{2} e^{4} - 6 \, a b^{3} d e^{5} - 18 \, a^{2} b c d e^{5} + 3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} + \frac {15 \, c^{4} d^{8} - 52 \, b c^{3} d^{7} e + 66 \, b^{2} c^{2} d^{6} e^{2} + 44 \, a c^{3} d^{6} e^{2} - 36 \, b^{3} c d^{5} e^{3} - 108 \, a b c^{2} d^{5} e^{3} + 7 \, b^{4} d^{4} e^{4} + 84 \, a b^{2} c d^{4} e^{4} + 42 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a b^{3} d^{3} e^{5} - 60 \, a^{2} b c d^{3} e^{5} + 18 \, a^{2} b^{2} d^{2} e^{6} + 12 \, a^{3} c d^{2} e^{6} - 4 \, a^{3} b d e^{7} - a^{4} e^{8} + 8 \, {\left (2 \, c^{4} d^{7} e - 7 \, b c^{3} d^{6} e^{2} + 9 \, b^{2} c^{2} d^{5} e^{3} + 6 \, a c^{3} d^{5} e^{3} - 5 \, b^{3} c d^{4} e^{4} - 15 \, 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} - 3 \, a b^{3} d^{2} e^{6} - 9 \, a^{2} b c d^{2} e^{6} + 3 \, a^{2} b^{2} d e^{7} + 2 \, a^{3} c d e^{7} - a^{3} b e^{8}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{9}} + \frac {5 \, c^{4} e^{15} x^{6} - 18 \, c^{4} d e^{14} x^{5} + 24 \, b c^{3} e^{15} x^{5} + 45 \, c^{4} d^{2} e^{13} x^{4} - 90 \, b c^{3} d e^{14} x^{4} + 45 \, b^{2} c^{2} e^{15} x^{4} + 30 \, a c^{3} e^{15} x^{4} - 100 \, c^{4} d^{3} e^{12} x^{3} + 240 \, b c^{3} d^{2} e^{13} x^{3} - 180 \, b^{2} c^{2} d e^{14} x^{3} - 120 \, a c^{3} d e^{14} x^{3} + 40 \, b^{3} c e^{15} x^{3} + 120 \, a b c^{2} e^{15} x^{3} + 225 \, c^{4} d^{4} e^{11} x^{2} - 600 \, b c^{3} d^{3} e^{12} x^{2} + 540 \, b^{2} c^{2} d^{2} e^{13} x^{2} + 360 \, a c^{3} d^{2} e^{13} x^{2} - 180 \, b^{3} c d e^{14} x^{2} - 540 \, a b c^{2} d e^{14} x^{2} + 15 \, b^{4} e^{15} x^{2} + 180 \, a b^{2} c e^{15} x^{2} + 90 \, a^{2} c^{2} e^{15} x^{2} - 630 \, c^{4} d^{5} e^{10} x + 1800 \, b c^{3} d^{4} e^{11} x - 1800 \, b^{2} c^{2} d^{3} e^{12} x - 1200 \, a c^{3} d^{3} e^{12} x + 720 \, b^{3} c d^{2} e^{13} x + 2160 \, a b c^{2} d^{2} e^{13} x - 90 \, b^{4} d e^{14} x - 1080 \, a b^{2} c d e^{14} x - 540 \, a^{2} c^{2} d e^{14} x + 120 \, a b^{3} e^{15} x + 360 \, a^{2} b c e^{15} x}{30 \, e^{18}} \]
2*(14*c^4*d^6 - 42*b*c^3*d^5*e + 45*b^2*c^2*d^4*e^2 + 30*a*c^3*d^4*e^2 - 2 0*b^3*c*d^3*e^3 - 60*a*b*c^2*d^3*e^3 + 3*b^4*d^2*e^4 + 36*a*b^2*c*d^2*e^4 + 18*a^2*c^2*d^2*e^4 - 6*a*b^3*d*e^5 - 18*a^2*b*c*d*e^5 + 3*a^2*b^2*e^6 + 2*a^3*c*e^6)*log(abs(e*x + d))/e^9 + 1/2*(15*c^4*d^8 - 52*b*c^3*d^7*e + 66 *b^2*c^2*d^6*e^2 + 44*a*c^3*d^6*e^2 - 36*b^3*c*d^5*e^3 - 108*a*b*c^2*d^5*e ^3 + 7*b^4*d^4*e^4 + 84*a*b^2*c*d^4*e^4 + 42*a^2*c^2*d^4*e^4 - 20*a*b^3*d^ 3*e^5 - 60*a^2*b*c*d^3*e^5 + 18*a^2*b^2*d^2*e^6 + 12*a^3*c*d^2*e^6 - 4*a^3 *b*d*e^7 - a^4*e^8 + 8*(2*c^4*d^7*e - 7*b*c^3*d^6*e^2 + 9*b^2*c^2*d^5*e^3 + 6*a*c^3*d^5*e^3 - 5*b^3*c*d^4*e^4 - 15*a*b*c^2*d^4*e^4 + b^4*d^3*e^5 + 1 2*a*b^2*c*d^3*e^5 + 6*a^2*c^2*d^3*e^5 - 3*a*b^3*d^2*e^6 - 9*a^2*b*c*d^2*e^ 6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7 - a^3*b*e^8)*x)/((e*x + d)^2*e^9) + 1/ 30*(5*c^4*e^15*x^6 - 18*c^4*d*e^14*x^5 + 24*b*c^3*e^15*x^5 + 45*c^4*d^2*e^ 13*x^4 - 90*b*c^3*d*e^14*x^4 + 45*b^2*c^2*e^15*x^4 + 30*a*c^3*e^15*x^4 - 1 00*c^4*d^3*e^12*x^3 + 240*b*c^3*d^2*e^13*x^3 - 180*b^2*c^2*d*e^14*x^3 - 12 0*a*c^3*d*e^14*x^3 + 40*b^3*c*e^15*x^3 + 120*a*b*c^2*e^15*x^3 + 225*c^4*d^ 4*e^11*x^2 - 600*b*c^3*d^3*e^12*x^2 + 540*b^2*c^2*d^2*e^13*x^2 + 360*a*c^3 *d^2*e^13*x^2 - 180*b^3*c*d*e^14*x^2 - 540*a*b*c^2*d*e^14*x^2 + 15*b^4*e^1 5*x^2 + 180*a*b^2*c*e^15*x^2 + 90*a^2*c^2*e^15*x^2 - 630*c^4*d^5*e^10*x + 1800*b*c^3*d^4*e^11*x - 1800*b^2*c^2*d^3*e^12*x - 1200*a*c^3*d^3*e^12*x + 720*b^3*c*d^2*e^13*x + 2160*a*b*c^2*d^2*e^13*x - 90*b^4*d*e^14*x - 1080...
Time = 9.93 (sec) , antiderivative size = 1444, normalized size of antiderivative = 3.36 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^3} \, dx=\text {Too large to display} \]
(x*(8*c^4*d^7 - 4*a^3*b*e^7 + 4*b^4*d^3*e^4 - 12*a*b^3*d^2*e^5 + 12*a^2*b^ 2*d*e^6 + 24*a*c^3*d^5*e^2 - 20*b^3*c*d^4*e^3 + 24*a^2*c^2*d^3*e^4 + 36*b^ 2*c^2*d^5*e^2 + 8*a^3*c*d*e^6 - 28*b*c^3*d^6*e - 60*a*b*c^2*d^4*e^3 + 48*a *b^2*c*d^3*e^4 - 36*a^2*b*c*d^2*e^5) + (15*c^4*d^8 - a^4*e^8 + 7*b^4*d^4*e ^4 - 20*a*b^3*d^3*e^5 + 44*a*c^3*d^6*e^2 + 12*a^3*c*d^2*e^6 - 36*b^3*c*d^5 *e^3 + 18*a^2*b^2*d^2*e^6 + 42*a^2*c^2*d^4*e^4 + 66*b^2*c^2*d^6*e^2 - 4*a^ 3*b*d*e^7 - 52*b*c^3*d^7*e - 108*a*b*c^2*d^5*e^3 + 84*a*b^2*c*d^4*e^4 - 60 *a^2*b*c*d^3*e^5)/(2*e))/(d^2*e^8 + e^10*x^2 + 2*d*e^9*x) + x^5*((4*b*c^3) /(5*e^3) - (3*c^4*d)/(5*e^4)) - x^3*((c^4*d^3)/(3*e^6) + (d^2*((4*b*c^3)/e ^3 - (3*c^4*d)/e^4))/e^2 - (d*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - ( 4*a*c^3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e - (4*b*c*(3*a*c + b^2))/(3* e^3)) + x*((d^3*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^ 2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e^3 - (3*d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/ e^3 + (3*d^2*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c ^2)/e^3 + (3*c^4*d^2)/e^5))/e^2 + (3*d*((c^4*d^3)/e^6 + (3*d^2*((4*b*c^3)/ e^3 - (3*c^4*d)/e^4))/e^2 - (3*d*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e - (4*b*c*(3*a*c + b^2))/ e^3))/e - (d^3*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e^3))/e + (3*d^2*((c^4*d^3 )/e^6 + (3*d^2*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e^2 - (3*d*((3*d*((4*b*c^3 )/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5...