Integrand size = 19, antiderivative size = 231 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac {d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^6}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{5 e^7 (d+e x)^5}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \]
-1/8*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^8+3/7*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/ (e*x+d)^7-1/2*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^6+1/5 *(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)/e^7/(e*x+d)^5-3/4*c*(b^2*e^2 -5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^4+c^2*(-b*e+2*c*d)/e^7/(e*x+d)^3-1/2*c^3 /e^7/(e*x+d)^2
Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.96 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 c e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b c^2 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]
-1/280*(b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*c*e^ 2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*c^2 *e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56* e^5*x^5) + 5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d ^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^8)
Time = 0.42 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^5}+\frac {(2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6 (d+e x)^6}+\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^7}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^4}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^9}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^8}+\frac {c^3}{e^6 (d+e x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac {d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac {c^3}{2 e^7 (d+e x)^2}\) |
-1/8*(d^3*(c*d - b*e)^3)/(e^7*(d + e*x)^8) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(7*e^7*(d + e*x)^7) - (d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e ^2))/(2*e^7*(d + e*x)^6) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e ^2))/(5*e^7*(d + e*x)^5) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(4*e^7* (d + e*x)^4) + (c^2*(2*c*d - b*e))/(e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x )^2)
3.3.57.3.1 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]
Time = 2.05 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.08
method | result | size |
risch | \(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{2} \left (b e +c d \right ) x^{5}}{e^{2}}-\frac {c \left (3 b^{2} e^{2}+5 b c d e +5 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {\left (b^{3} e^{3}+3 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e +5 c^{3} d^{3}\right ) x^{3}}{5 e^{4}}-\frac {d \left (b^{3} e^{3}+3 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e +5 c^{3} d^{3}\right ) x^{2}}{10 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+3 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e +5 c^{3} d^{3}\right ) x}{35 e^{6}}-\frac {d^{3} \left (b^{3} e^{3}+3 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e +5 c^{3} d^{3}\right )}{280 e^{7}}}{\left (e x +d \right )^{8}}\) | \(249\) |
norman | \(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {\left (e^{2} b \,c^{2}+d e \,c^{3}\right ) x^{5}}{e^{3}}-\frac {\left (3 e^{3} b^{2} c +5 d \,e^{2} b \,c^{2}+5 d^{2} e \,c^{3}\right ) x^{4}}{4 e^{4}}-\frac {\left (e^{4} b^{3}+3 b^{2} d \,e^{3} c +5 d^{2} e^{2} b \,c^{2}+5 d^{3} e \,c^{3}\right ) x^{3}}{5 e^{5}}-\frac {d \left (e^{4} b^{3}+3 b^{2} d \,e^{3} c +5 d^{2} e^{2} b \,c^{2}+5 d^{3} e \,c^{3}\right ) x^{2}}{10 e^{6}}-\frac {d^{2} \left (e^{4} b^{3}+3 b^{2} d \,e^{3} c +5 d^{2} e^{2} b \,c^{2}+5 d^{3} e \,c^{3}\right ) x}{35 e^{7}}-\frac {d^{3} \left (e^{4} b^{3}+3 b^{2} d \,e^{3} c +5 d^{2} e^{2} b \,c^{2}+5 d^{3} e \,c^{3}\right )}{280 e^{8}}}{\left (e x +d \right )^{8}}\) | \(271\) |
default | \(\frac {d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right )}{2 e^{7} \left (e x +d \right )^{6}}-\frac {c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )^{3}}-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {c^{3}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{8 e^{7} \left (e x +d \right )^{8}}\) | \(274\) |
gosper | \(-\frac {140 x^{6} c^{3} e^{6}+280 x^{5} b \,c^{2} e^{6}+280 x^{5} c^{3} d \,e^{5}+210 x^{4} b^{2} c \,e^{6}+350 x^{4} b \,c^{2} d \,e^{5}+350 x^{4} c^{3} d^{2} e^{4}+56 x^{3} b^{3} e^{6}+168 x^{3} b^{2} c d \,e^{5}+280 x^{3} b \,c^{2} d^{2} e^{4}+280 x^{3} c^{3} d^{3} e^{3}+28 x^{2} b^{3} d \,e^{5}+84 x^{2} b^{2} c \,d^{2} e^{4}+140 x^{2} b \,c^{2} d^{3} e^{3}+140 x^{2} c^{3} d^{4} e^{2}+8 x \,b^{3} d^{2} e^{4}+24 x \,b^{2} c \,d^{3} e^{3}+40 x b \,c^{2} d^{4} e^{2}+40 x \,c^{3} d^{5} e +b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}+5 b \,c^{2} d^{5} e +5 c^{3} d^{6}}{280 e^{7} \left (e x +d \right )^{8}}\) | \(285\) |
parallelrisch | \(\frac {-140 c^{3} x^{6} e^{7}-280 b \,c^{2} e^{7} x^{5}-280 c^{3} d \,e^{6} x^{5}-210 b^{2} c \,e^{7} x^{4}-350 b \,c^{2} d \,e^{6} x^{4}-350 c^{3} d^{2} e^{5} x^{4}-56 b^{3} e^{7} x^{3}-168 b^{2} c d \,e^{6} x^{3}-280 b \,c^{2} d^{2} e^{5} x^{3}-280 c^{3} d^{3} e^{4} x^{3}-28 b^{3} d \,e^{6} x^{2}-84 b^{2} c \,d^{2} e^{5} x^{2}-140 b \,c^{2} d^{3} e^{4} x^{2}-140 c^{3} d^{4} e^{3} x^{2}-8 b^{3} d^{2} e^{5} x -24 b^{2} c \,d^{3} e^{4} x -40 b \,c^{2} d^{4} e^{3} x -40 c^{3} d^{5} e^{2} x -b^{3} d^{3} e^{4}-3 b^{2} c \,d^{4} e^{3}-5 b \,c^{2} d^{5} e^{2}-5 c^{3} d^{6} e}{280 e^{8} \left (e x +d \right )^{8}}\) | \(291\) |
(-1/2*c^3*x^6/e-c^2/e^2*(b*e+c*d)*x^5-1/4*c/e^3*(3*b^2*e^2+5*b*c*d*e+5*c^2 *d^2)*x^4-1/5/e^4*(b^3*e^3+3*b^2*c*d*e^2+5*b*c^2*d^2*e+5*c^3*d^3)*x^3-1/10 *d/e^5*(b^3*e^3+3*b^2*c*d*e^2+5*b*c^2*d^2*e+5*c^3*d^3)*x^2-1/35*d^2/e^6*(b ^3*e^3+3*b^2*c*d*e^2+5*b*c^2*d^2*e+5*c^3*d^3)*x-1/280*d^3/e^7*(b^3*e^3+3*b ^2*c*d*e^2+5*b*c^2*d^2*e+5*c^3*d^3))/(e*x+d)^8
Time = 0.27 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.49 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/280*(140*c^3*e^6*x^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + b^ 3*d^3*e^3 + 280*(c^3*d*e^5 + b*c^2*e^6)*x^5 + 70*(5*c^3*d^2*e^4 + 5*b*c^2* d*e^5 + 3*b^2*c*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + 3*b^2*c*d *e^5 + b^3*e^6)*x^3 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^ 4 + b^3*d*e^5)*x^2 + 8*(5*c^3*d^5*e + 5*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x ^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^ 8*e^7)
Timed out. \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.49 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/280*(140*c^3*e^6*x^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + b^ 3*d^3*e^3 + 280*(c^3*d*e^5 + b*c^2*e^6)*x^5 + 70*(5*c^3*d^2*e^4 + 5*b*c^2* d*e^5 + 3*b^2*c*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + 3*b^2*c*d *e^5 + b^3*e^6)*x^3 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^ 4 + b^3*d*e^5)*x^2 + 8*(5*c^3*d^5*e + 5*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x ^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^ 8*e^7)
Time = 0.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.23 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 280 \, b c^{2} e^{6} x^{5} + 350 \, c^{3} d^{2} e^{4} x^{4} + 350 \, b c^{2} d e^{5} x^{4} + 210 \, b^{2} c e^{6} x^{4} + 280 \, c^{3} d^{3} e^{3} x^{3} + 280 \, b c^{2} d^{2} e^{4} x^{3} + 168 \, b^{2} c d e^{5} x^{3} + 56 \, b^{3} e^{6} x^{3} + 140 \, c^{3} d^{4} e^{2} x^{2} + 140 \, b c^{2} d^{3} e^{3} x^{2} + 84 \, b^{2} c d^{2} e^{4} x^{2} + 28 \, b^{3} d e^{5} x^{2} + 40 \, c^{3} d^{5} e x + 40 \, b c^{2} d^{4} e^{2} x + 24 \, b^{2} c d^{3} e^{3} x + 8 \, b^{3} d^{2} e^{4} x + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3}}{280 \, {\left (e x + d\right )}^{8} e^{7}} \]
-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 280*b*c^2*e^6*x^5 + 350*c^3* d^2*e^4*x^4 + 350*b*c^2*d*e^5*x^4 + 210*b^2*c*e^6*x^4 + 280*c^3*d^3*e^3*x^ 3 + 280*b*c^2*d^2*e^4*x^3 + 168*b^2*c*d*e^5*x^3 + 56*b^3*e^6*x^3 + 140*c^3 *d^4*e^2*x^2 + 140*b*c^2*d^3*e^3*x^2 + 84*b^2*c*d^2*e^4*x^2 + 28*b^3*d*e^5 *x^2 + 40*c^3*d^5*e*x + 40*b*c^2*d^4*e^2*x + 24*b^2*c*d^3*e^3*x + 8*b^3*d^ 2*e^4*x + 5*c^3*d^6 + 5*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + b^3*d^3*e^3)/((e*x + d)^8*e^7)
Time = 9.60 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.41 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {\frac {d^3\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{280\,e^7}+\frac {x^3\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{5\,e^4}+\frac {c^3\,x^6}{2\,e}+\frac {c^2\,x^5\,\left (b\,e+c\,d\right )}{e^2}+\frac {c\,x^4\,\left (3\,b^2\,e^2+5\,b\,c\,d\,e+5\,c^2\,d^2\right )}{4\,e^3}+\frac {d\,x^2\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{10\,e^5}+\frac {d^2\,x\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{35\,e^6}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
-((d^3*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(280*e^7) + (x^3*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(5*e^4) + (c^3 *x^6)/(2*e) + (c^2*x^5*(b*e + c*d))/e^2 + (c*x^4*(3*b^2*e^2 + 5*c^2*d^2 + 5*b*c*d*e))/(4*e^3) + (d*x^2*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2* c*d*e^2))/(10*e^5) + (d^2*x*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2*c *d*e^2))/(35*e^6))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5* e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)