Integrand size = 20, antiderivative size = 426 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{e^9 (d+e x)}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^2}{e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^3}{3 e^9}-\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^4}{e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^9}-\frac {2 c^3 (2 c d-b e) (d+e x)^6}{3 e^9}+\frac {c^4 (d+e x)^7}{7 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^9} \]
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))*x/e^8-(a *e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)-2*(-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)^2/e^9+1/3*(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)^3/e^9-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)^4/e^9+2/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b* d))*(e*x+d)^5/e^9-2/3*c^3*(-b*e+2*c*d)*(e*x+d)^6/e^9+1/7*c^4*(e*x+d)^7/e^9 -4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)/e^9
Time = 0.20 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {c^4 \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )+35 e^4 \left (12 a^3 b d e^3-3 a^4 e^4+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+35 c e^3 \left (12 a^3 e^3 \left (-d^2+d e x+e^2 x^2\right )+18 a^2 b e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+12 a b^2 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+b^3 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+21 c^2 e^2 \left (10 a^2 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+b^2 \left (-30 d^6+150 d^5 e x+90 d^4 e^2 x^2-30 d^3 e^3 x^3+15 d^2 e^4 x^4-9 d e^5 x^5+6 e^6 x^6\right )\right )+7 c^3 e \left (6 a e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+b \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )-420 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3 (d+e x) \log (d+e x)}{105 e^9 (d+e x)} \]
(c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4* e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 3 5*e^4*(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x^2 ) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 35*c*e^3*(12*a^3*e^3 *(-d^2 + d*e*x + e^2*x^2) + 18*a^2*b*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 12*a*b^2*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^3*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 21*c^2*e^2*(10*a^2*e^2*(-3*d^4 + 9*d^3*e*x + 6 *d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b*e*(12*d^5 - 48*d^4*e*x - 30* d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^2*(-30*d^6 + 1 50*d^5*e*x + 90*d^4*e^2*x^2 - 30*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 9*d*e^5*x^ 5 + 6*e^6*x^6)) + 7*c^3*e*(6*a*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + b*(60*d^7 - 36 0*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5 *x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)*Log[d + e*x])/(105*e^9*(d + e*x))
Time = 0.92 (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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac {2 c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 c (d+e x)^3 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8}+\frac {4 (d+e x) (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^2}-\frac {4 c^3 (d+e x)^5 (2 c d-b e)}{e^8}+\frac {c^4 (d+e x)^6}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^9}+\frac {2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}-\frac {2 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {2 x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^9}-\frac {2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac {c^4 (d+e x)^7}{7 e^9}\) |
(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) )*x)/e^8 - (c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)) - (2*(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 )^2)/e^9 + ((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)^3)/(3*e^9) - (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)^4)/e^9 + (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a *e))*(d + e*x)^5)/(5*e^9) - (2*c^3*(2*c*d - b*e)*(d + e*x)^6)/(3*e^9) + (c ^4*(d + e*x)^7)/(7*e^9) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^9
3.22.52.3.1 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. \(891\) vs. \(2(418)=836\).
Time = 4.75 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.09
| method | result | size |
| norman | \(\frac {\frac {\left (a^{4} e^{8}-4 a^{3} b d \,e^{7}+8 a^{3} c \,d^{2} e^{6}+12 a^{2} b^{2} d^{2} e^{6}-36 a^{2} b c \,d^{3} e^{5}+24 a^{2} c^{2} d^{4} e^{4}-12 a \,b^{3} d^{3} e^{5}+48 a \,b^{2} c \,d^{4} e^{4}-60 a b \,c^{2} d^{5} e^{3}+24 a \,c^{3} d^{6} e^{2}+4 b^{4} d^{4} e^{4}-20 b^{3} c \,d^{5} e^{3}+36 b^{2} c^{2} d^{6} e^{2}-28 b \,c^{3} d^{7} e +8 c^{4} d^{8}\right ) x}{d \,e^{8}}+\frac {c^{4} x^{8}}{7 e}+\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-15 a b \,c^{2} d \,e^{3}+6 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-5 b^{3} c d \,e^{3}+9 b^{2} c^{2} d^{2} e^{2}-7 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x^{4}}{3 e^{5}}+\frac {2 \left (9 a^{2} b c \,e^{5}-6 d \,e^{4} a^{2} c^{2}+3 a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+15 a b \,c^{2} d^{2} e^{3}-6 d^{3} e^{2} c^{3} a -b^{4} d \,e^{4}+5 b^{3} c \,d^{2} e^{3}-9 b^{2} c^{2} d^{3} e^{2}+7 b \,c^{3} d^{4} e -2 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}+\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-9 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}-3 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}-15 a b \,c^{2} d^{3} e^{3}+6 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}-5 b^{3} c \,d^{3} e^{3}+9 b^{2} c^{2} d^{4} e^{2}-7 b \,c^{3} d^{5} e +2 d^{6} c^{4}\right ) x^{2}}{e^{7}}+\frac {c \left (15 a b c \,e^{3}-6 c^{2} a d \,e^{2}+5 b^{3} e^{3}-9 b^{2} d \,e^{2} c +7 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) x^{5}}{5 e^{4}}+\frac {2 c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-7 b c d e +2 c^{2} d^{2}\right ) x^{6}}{15 e^{3}}+\frac {2 c^{3} \left (7 b e -2 c d \right ) x^{7}}{21 e^{2}}}{e x +d}+\frac {4 \left (a^{3} b \,e^{7}-2 d \,e^{6} c \,a^{3}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 d^{5} e^{2} c^{3} a -b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 d^{7} c^{4}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(892\) |
| default | \(\frac {\frac {1}{3} b^{4} e^{6} x^{3}-c^{4} d^{3} e^{3} x^{4}+b^{3} c \,e^{6} x^{4}+\frac {2}{3} b \,c^{3} e^{6} x^{6}-\frac {1}{3} c^{4} d \,e^{5} x^{6}+6 a^{2} b c \,e^{6} x^{2}+4 a \,c^{3} d^{2} e^{4} x^{3}-6 a^{2} c^{2} d \,e^{5} x^{2}-8 a \,c^{3} d^{3} e^{3} x^{2}-2 a \,c^{3} d \,e^{5} x^{4}+30 b^{2} c^{2} d^{4} e^{2} x +36 a \,b^{2} c \,d^{2} e^{4} x -48 a b \,c^{2} d^{3} e^{3} x +4 a \,b^{2} c \,e^{6} x^{3}+2 a^{2} c^{2} e^{6} x^{3}-3 b^{2} c^{2} d \,e^{5} x^{4}-3 c^{4} d^{5} e \,x^{2}+4 e^{6} c \,a^{3} x +3 b \,c^{3} d^{2} e^{4} x^{4}+\frac {4}{5} a \,c^{3} e^{6} x^{5}+6 b^{3} c \,d^{2} e^{4} x^{2}-12 b^{2} c^{2} d^{3} e^{3} x^{2}+10 b \,c^{3} d^{4} e^{2} x^{2}-\frac {8}{3} b^{3} c d \,e^{5} x^{3}+6 b^{2} c^{2} d^{2} e^{4} x^{3}-\frac {16}{3} b \,c^{3} d^{3} e^{3} x^{3}+7 d^{6} c^{4} x +18 d^{2} e^{4} a^{2} c^{2} x -8 a \,b^{3} d \,e^{5} x +20 d^{4} e^{2} c^{3} a x -16 b^{3} c \,d^{3} e^{3} x -12 a \,b^{2} c d \,e^{5} x^{2}+18 a b \,c^{2} d^{2} e^{4} x^{2}+\frac {1}{7} c^{4} x^{7} e^{6}+\frac {5}{3} c^{4} d^{4} e^{2} x^{3}+2 a \,b^{3} e^{6} x^{2}+3 a b \,c^{2} e^{6} x^{4}-24 b \,c^{3} d^{5} e x -8 a b \,c^{2} d \,e^{5} x^{3}-24 a^{2} b c d \,e^{5} x -b^{4} d \,e^{5} x^{2}-\frac {8}{5} b \,c^{3} d \,e^{5} x^{5}+\frac {6}{5} b^{2} c^{2} e^{6} x^{5}+\frac {3}{5} c^{4} d^{2} e^{4} x^{5}+6 a^{2} b^{2} e^{6} x +3 b^{4} d^{2} e^{4} x}{e^{8}}-\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}}{e^{9} \left (e x +d \right )}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(979\) |
| risch | \(\text {Expression too large to display}\) | \(1159\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1352\) |
((a^4*e^8-4*a^3*b*d*e^7+8*a^3*c*d^2*e^6+12*a^2*b^2*d^2*e^6-36*a^2*b*c*d^3* e^5+24*a^2*c^2*d^4*e^4-12*a*b^3*d^3*e^5+48*a*b^2*c*d^4*e^4-60*a*b*c^2*d^5* e^3+24*a*c^3*d^6*e^2+4*b^4*d^4*e^4-20*b^3*c*d^5*e^3+36*b^2*c^2*d^6*e^2-28* b*c^3*d^7*e+8*c^4*d^8)/d/e^8*x+1/7/e*c^4*x^8+1/3*(6*a^2*c^2*e^4+12*a*b^2*c *e^4-15*a*b*c^2*d*e^3+6*a*c^3*d^2*e^2+b^4*e^4-5*b^3*c*d*e^3+9*b^2*c^2*d^2* e^2-7*b*c^3*d^3*e+2*c^4*d^4)/e^5*x^4+2/3*(9*a^2*b*c*e^5-6*a^2*c^2*d*e^4+3* a*b^3*e^5-12*a*b^2*c*d*e^4+15*a*b*c^2*d^2*e^3-6*a*c^3*d^3*e^2-b^4*d*e^4+5* b^3*c*d^2*e^3-9*b^2*c^2*d^3*e^2+7*b*c^3*d^4*e-2*c^4*d^5)/e^6*x^3+2*(2*a^3* c*e^6+3*a^2*b^2*e^6-9*a^2*b*c*d*e^5+6*a^2*c^2*d^2*e^4-3*a*b^3*d*e^5+12*a*b ^2*c*d^2*e^4-15*a*b*c^2*d^3*e^3+6*a*c^3*d^4*e^2+b^4*d^2*e^4-5*b^3*c*d^3*e^ 3+9*b^2*c^2*d^4*e^2-7*b*c^3*d^5*e+2*c^4*d^6)/e^7*x^2+1/5*c*(15*a*b*c*e^3-6 *a*c^2*d*e^2+5*b^3*e^3-9*b^2*c*d*e^2+7*b*c^2*d^2*e-2*c^3*d^3)/e^4*x^5+2/15 *c^2*(6*a*c*e^2+9*b^2*e^2-7*b*c*d*e+2*c^2*d^2)/e^3*x^6+2/21*c^3*(7*b*e-2*c *d)/e^2*x^7)/(e*x+d)+4*(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)/e^9*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1095 vs. \(2 (418) = 836\).
Time = 0.38 (sec) , antiderivative size = 1095, normalized size of antiderivative = 2.57 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=\text {Too large to display} \]
1/105*(15*c^4*e^8*x^8 - 105*c^4*d^8 + 420*b*c^3*d^7*e + 420*a^3*b*d*e^7 - 105*a^4*e^8 - 210*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 420*(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 + 420*(a*b^3 + 3*a^2* b*c)*d^3*e^5 - 210*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 10*(2*c^4*d*e^7 - 7*b*c ^3*e^8)*x^7 + 14*(2*c^4*d^2*e^6 - 7*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)* e^8)*x^6 - 21*(2*c^4*d^3*e^5 - 7*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d *e^7 - 5*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 35*(2*c^4*d^4*e^4 - 7*b*c^3*d^3*e^ 5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 5*(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 - 70*(2*c^4*d^5*e^3 - 7*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 5*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 1 2*a*b^2*c + 6*a^2*c^2)*d*e^7 - 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 210*(2*c^4 *d^6*e^2 - 7*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 5*(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 - 3*(a*b^3 + 3 *a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 105*(7*c^4*d^7*e - 24*b *c^3*d^6*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 16*(b^3*c + 3*a*b*c^2)*d ^4*e^4 + 3*(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 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x - 420*(2*c^4*d^8 - 7*b*c^3*d^7* e - a^3*b*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 5*(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 - 3*(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 + (2*c^4*d^7*e - 7*b*c^3*d^6*e^2...
Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (423) = 846\).
Time = 1.92 (sec) , antiderivative size = 847, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {c^{4} x^{7}}{7 e^{2}} + x^{6} \cdot \left (\frac {2 b c^{3}}{3 e^{2}} - \frac {c^{4} d}{3 e^{3}}\right ) + x^{5} \cdot \left (\frac {4 a c^{3}}{5 e^{2}} + \frac {6 b^{2} c^{2}}{5 e^{2}} - \frac {8 b c^{3} d}{5 e^{3}} + \frac {3 c^{4} d^{2}}{5 e^{4}}\right ) + x^{4} \cdot \left (\frac {3 a b c^{2}}{e^{2}} - \frac {2 a c^{3} d}{e^{3}} + \frac {b^{3} c}{e^{2}} - \frac {3 b^{2} c^{2} d}{e^{3}} + \frac {3 b c^{3} d^{2}}{e^{4}} - \frac {c^{4} d^{3}}{e^{5}}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c^{2}}{e^{2}} + \frac {4 a b^{2} c}{e^{2}} - \frac {8 a b c^{2} d}{e^{3}} + \frac {4 a c^{3} d^{2}}{e^{4}} + \frac {b^{4}}{3 e^{2}} - \frac {8 b^{3} c d}{3 e^{3}} + \frac {6 b^{2} c^{2} d^{2}}{e^{4}} - \frac {16 b c^{3} d^{3}}{3 e^{5}} + \frac {5 c^{4} d^{4}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {6 a^{2} b c}{e^{2}} - \frac {6 a^{2} c^{2} d}{e^{3}} + \frac {2 a b^{3}}{e^{2}} - \frac {12 a b^{2} c d}{e^{3}} + \frac {18 a b c^{2} d^{2}}{e^{4}} - \frac {8 a c^{3} d^{3}}{e^{5}} - \frac {b^{4} d}{e^{3}} + \frac {6 b^{3} c d^{2}}{e^{4}} - \frac {12 b^{2} c^{2} d^{3}}{e^{5}} + \frac {10 b c^{3} d^{4}}{e^{6}} - \frac {3 c^{4} d^{5}}{e^{7}}\right ) + x \left (\frac {4 a^{3} c}{e^{2}} + \frac {6 a^{2} b^{2}}{e^{2}} - \frac {24 a^{2} b c d}{e^{3}} + \frac {18 a^{2} c^{2} d^{2}}{e^{4}} - \frac {8 a b^{3} d}{e^{3}} + \frac {36 a b^{2} c d^{2}}{e^{4}} - \frac {48 a b c^{2} d^{3}}{e^{5}} + \frac {20 a c^{3} d^{4}}{e^{6}} + \frac {3 b^{4} d^{2}}{e^{4}} - \frac {16 b^{3} c d^{3}}{e^{5}} + \frac {30 b^{2} c^{2} d^{4}}{e^{6}} - \frac {24 b c^{3} d^{5}}{e^{7}} + \frac {7 c^{4} d^{6}}{e^{8}}\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}}{d e^{9} + e^{10} x} + \frac {4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{9}} \]
c**4*x**7/(7*e**2) + x**6*(2*b*c**3/(3*e**2) - c**4*d/(3*e**3)) + x**5*(4* a*c**3/(5*e**2) + 6*b**2*c**2/(5*e**2) - 8*b*c**3*d/(5*e**3) + 3*c**4*d**2 /(5*e**4)) + x**4*(3*a*b*c**2/e**2 - 2*a*c**3*d/e**3 + b**3*c/e**2 - 3*b** 2*c**2*d/e**3 + 3*b*c**3*d**2/e**4 - c**4*d**3/e**5) + x**3*(2*a**2*c**2/e **2 + 4*a*b**2*c/e**2 - 8*a*b*c**2*d/e**3 + 4*a*c**3*d**2/e**4 + b**4/(3*e **2) - 8*b**3*c*d/(3*e**3) + 6*b**2*c**2*d**2/e**4 - 16*b*c**3*d**3/(3*e** 5) + 5*c**4*d**4/(3*e**6)) + x**2*(6*a**2*b*c/e**2 - 6*a**2*c**2*d/e**3 + 2*a*b**3/e**2 - 12*a*b**2*c*d/e**3 + 18*a*b*c**2*d**2/e**4 - 8*a*c**3*d**3 /e**5 - b**4*d/e**3 + 6*b**3*c*d**2/e**4 - 12*b**2*c**2*d**3/e**5 + 10*b*c **3*d**4/e**6 - 3*c**4*d**5/e**7) + x*(4*a**3*c/e**2 + 6*a**2*b**2/e**2 - 24*a**2*b*c*d/e**3 + 18*a**2*c**2*d**2/e**4 - 8*a*b**3*d/e**3 + 36*a*b**2* c*d**2/e**4 - 48*a*b*c**2*d**3/e**5 + 20*a*c**3*d**4/e**6 + 3*b**4*d**2/e* *4 - 16*b**3*c*d**3/e**5 + 30*b**2*c**2*d**4/e**6 - 24*b*c**3*d**5/e**7 + 7*c**4*d**6/e**8) + (-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)/(d*e**9 + e**10*x) + 4*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e*x)/e**9
Time = 0.20 (sec) , antiderivative size = 807, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=-\frac {c^{4} d^{8} - 4 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} + a^{4} e^{8} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6}}{e^{10} x + d e^{9}} + \frac {15 \, c^{4} e^{6} x^{7} - 35 \, {\left (c^{4} d e^{5} - 2 \, b c^{3} e^{6}\right )} x^{6} + 21 \, {\left (3 \, c^{4} d^{2} e^{4} - 8 \, b c^{3} d e^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{6}\right )} x^{5} - 105 \, {\left (c^{4} d^{3} e^{3} - 3 \, b c^{3} d^{2} e^{4} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{5} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{6}\right )} x^{4} + 35 \, {\left (5 \, c^{4} d^{4} e^{2} - 16 \, b c^{3} d^{3} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{4} - 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{6}\right )} x^{3} - 105 \, {\left (3 \, c^{4} d^{5} e - 10 \, b c^{3} d^{4} e^{2} + 4 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{3} - 6 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{5} - 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{6}\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{6} - 24 \, b c^{3} d^{5} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 16 \, {\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} - 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} x}{105 \, e^{8}} - \frac {4 \, {\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 )} \log \left (e x + d\right )}{e^{9}} \]
-(c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c ^3)*d^6*e^2 - 4*(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 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e ^6)/(e^10*x + d*e^9) + 1/105*(15*c^4*e^6*x^7 - 35*(c^4*d*e^5 - 2*b*c^3*e^6 )*x^6 + 21*(3*c^4*d^2*e^4 - 8*b*c^3*d*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*e^6)*x ^5 - 105*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4 + (3*b^2*c^2 + 2*a*c^3)*d*e^5 - (b ^3*c + 3*a*b*c^2)*e^6)*x^4 + 35*(5*c^4*d^4*e^2 - 16*b*c^3*d^3*e^3 + 6*(3*b ^2*c^2 + 2*a*c^3)*d^2*e^4 - 8*(b^3*c + 3*a*b*c^2)*d*e^5 + (b^4 + 12*a*b^2* c + 6*a^2*c^2)*e^6)*x^3 - 105*(3*c^4*d^5*e - 10*b*c^3*d^4*e^2 + 4*(3*b^2*c ^2 + 2*a*c^3)*d^3*e^3 - 6*(b^3*c + 3*a*b*c^2)*d^2*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^5 - 2*(a*b^3 + 3*a^2*b*c)*e^6)*x^2 + 105*(7*c^4*d^6 - 24* b*c^3*d^5*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 16*(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 - 8*(a*b^3 + 3*a^2*b*c)*d *e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*e^6)*x)/e^8 - 4*(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)*log(e*x + d)/e^9
Leaf count of result is larger than twice the leaf count of optimal. 1069 vs. \(2 (418) = 836\).
Time = 0.26 (sec) , antiderivative size = 1069, normalized size of antiderivative = 2.51 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {{\left (15 \, c^{4} - \frac {70 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {42 \, {\left (14 \, c^{4} d^{2} e^{2} - 14 \, b c^{3} d e^{3} + 3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {105 \, {\left (14 \, c^{4} d^{3} e^{3} - 21 \, b c^{3} d^{2} e^{4} + 9 \, b^{2} c^{2} d e^{5} + 6 \, a c^{3} d e^{5} - b^{3} c e^{6} - 3 \, a b c^{2} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {35 \, {\left (70 \, c^{4} d^{4} e^{4} - 140 \, b c^{3} d^{3} e^{5} + 90 \, b^{2} c^{2} d^{2} e^{6} + 60 \, 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}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {210 \, {\left (14 \, c^{4} d^{5} e^{5} - 35 \, b c^{3} d^{4} e^{6} + 30 \, b^{2} c^{2} d^{3} e^{7} + 20 \, a c^{3} d^{3} e^{7} - 10 \, b^{3} c d^{2} e^{8} - 30 \, a b c^{2} d^{2} e^{8} + b^{4} d e^{9} + 12 \, a b^{2} c d e^{9} + 6 \, a^{2} c^{2} d e^{9} - a b^{3} e^{10} - 3 \, a^{2} b c e^{10}\right )}}{{\left (e x + d\right )}^{5} e^{5}} + \frac {210 \, {\left (14 \, c^{4} d^{6} e^{6} - 42 \, b c^{3} d^{5} e^{7} + 45 \, b^{2} c^{2} d^{4} e^{8} + 30 \, a c^{3} d^{4} e^{8} - 20 \, b^{3} c d^{3} e^{9} - 60 \, a b c^{2} d^{3} e^{9} + 3 \, b^{4} d^{2} e^{10} + 36 \, a b^{2} c d^{2} e^{10} + 18 \, a^{2} c^{2} d^{2} e^{10} - 6 \, a b^{3} d e^{11} - 18 \, a^{2} b c d e^{11} + 3 \, a^{2} b^{2} e^{12} + 2 \, a^{3} c e^{12}\right )}}{{\left (e x + d\right )}^{6} e^{6}}\right )} {\left (e x + d\right )}^{7}}{105 \, e^{9}} + \frac {4 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 9 \, b^{2} c^{2} d^{5} e^{2} + 6 \, a c^{3} d^{5} e^{2} - 5 \, b^{3} c d^{4} e^{3} - 15 \, a b c^{2} d^{4} e^{3} + b^{4} d^{3} e^{4} + 12 \, a b^{2} c d^{3} e^{4} + 6 \, a^{2} c^{2} d^{3} e^{4} - 3 \, a b^{3} d^{2} e^{5} - 9 \, a^{2} b c d^{2} e^{5} + 3 \, a^{2} b^{2} d e^{6} + 2 \, a^{3} c d e^{6} - a^{3} b e^{7}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{9}} - \frac {\frac {c^{4} d^{8} e^{7}}{e x + d} - \frac {4 \, b c^{3} d^{7} e^{8}}{e x + d} + \frac {6 \, b^{2} c^{2} d^{6} e^{9}}{e x + d} + \frac {4 \, a c^{3} d^{6} e^{9}}{e x + d} - \frac {4 \, b^{3} c d^{5} e^{10}}{e x + d} - \frac {12 \, a b c^{2} d^{5} e^{10}}{e x + d} + \frac {b^{4} d^{4} e^{11}}{e x + d} + \frac {12 \, a b^{2} c d^{4} e^{11}}{e x + d} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{e x + d} - \frac {4 \, a b^{3} d^{3} e^{12}}{e x + d} - \frac {12 \, a^{2} b c d^{3} e^{12}}{e x + d} + \frac {6 \, a^{2} b^{2} d^{2} e^{13}}{e x + d} + \frac {4 \, a^{3} c d^{2} e^{13}}{e x + d} - \frac {4 \, a^{3} b d e^{14}}{e x + d} + \frac {a^{4} e^{15}}{e x + d}}{e^{16}} \]
1/105*(15*c^4 - 70*(2*c^4*d*e - b*c^3*e^2)/((e*x + d)*e) + 42*(14*c^4*d^2* e^2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*e^4 + 2*a*c^3*e^4)/((e*x + d)^2*e^2) - 10 5*(14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e^5 - b ^3*c*e^6 - 3*a*b*c^2*e^6)/((e*x + d)^3*e^3) + 35*(70*c^4*d^4*e^4 - 140*b*c ^3*d^3*e^5 + 90*b^2*c^2*d^2*e^6 + 60*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)/((e*x + d)^4*e^4) - 210*(14*c^4*d^5*e^5 - 35*b*c^3*d^4*e^6 + 30*b^2*c^2*d^3*e^7 + 20*a*c^3* d^3*e^7 - 10*b^3*c*d^2*e^8 - 30*a*b*c^2*d^2*e^8 + b^4*d*e^9 + 12*a*b^2*c*d *e^9 + 6*a^2*c^2*d*e^9 - a*b^3*e^10 - 3*a^2*b*c*e^10)/((e*x + d)^5*e^5) + 210*(14*c^4*d^6*e^6 - 42*b*c^3*d^5*e^7 + 45*b^2*c^2*d^4*e^8 + 30*a*c^3*d^4 *e^8 - 20*b^3*c*d^3*e^9 - 60*a*b*c^2*d^3*e^9 + 3*b^4*d^2*e^10 + 36*a*b^2*c *d^2*e^10 + 18*a^2*c^2*d^2*e^10 - 6*a*b^3*d*e^11 - 18*a^2*b*c*d*e^11 + 3*a ^2*b^2*e^12 + 2*a^3*c*e^12)/((e*x + d)^6*e^6))*(e*x + d)^7/e^9 + 4*(2*c^4* d^7 - 7*b*c^3*d^6*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^ 3 - 15*a*b*c^2*d^4*e^3 + b^4*d^3*e^4 + 12*a*b^2*c*d^3*e^4 + 6*a^2*c^2*d^3* e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*e^6 + 2*a^3*c*d*e^ 6 - a^3*b*e^7)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^9 - (c^4*d^8*e^7/( e*x + d) - 4*b*c^3*d^7*e^8/(e*x + d) + 6*b^2*c^2*d^6*e^9/(e*x + d) + 4*a*c ^3*d^6*e^9/(e*x + d) - 4*b^3*c*d^5*e^10/(e*x + d) - 12*a*b*c^2*d^5*e^10/(e *x + d) + b^4*d^4*e^11/(e*x + d) + 12*a*b^2*c*d^4*e^11/(e*x + d) + 6*a^...
Time = 9.88 (sec) , antiderivative size = 1679, normalized size of antiderivative = 3.94 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx=\text {Too large to display} \]
x*((4*a^3*c + 6*a^2*b^2)/e^2 + (2*d*((2*d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/ e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2 )/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/ e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/ e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/e))/e + (d^2*((2*d *((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c ^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a* c + b^2))/e^2))/e^2 - (4*a*b*(3*a*c + b^2))/e^2))/e - (d^2*((b^4 + 6*a^2*c ^2 + 12*a*b^2*c)/e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4* a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c^3) /e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/e ))/e^2) + x^6*((2*b*c^3)/(3*e^2) - (c^4*d)/(3*e^3)) - x^2*((d*((b^4 + 6*a^ 2*c^2 + 12*a*b^2*c)/e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c ^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/ e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2) )/e))/e + (d^2*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3) )/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/(2*e^2) - (2*a*b*(3*a*c + b^2))/e^2...