Integrand size = 31, antiderivative size = 130 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=-\frac {10 b^2 (b d-a e)^3 x}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b^3 (b d-a e)^2 (d+e x)^2}{e^6}-\frac {5 b^4 (b d-a e) (d+e x)^3}{3 e^6}+\frac {b^5 (d+e x)^4}{4 e^6}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6} \]
-10*b^2*(-a*e+b*d)^3*x/e^5+(-a*e+b*d)^5/e^6/(e*x+d)+5*b^3*(-a*e+b*d)^2*(e* x+d)^2/e^6-5/3*b^4*(-a*e+b*d)*(e*x+d)^3/e^6+1/4*b^5*(e*x+d)^4/e^6+5*b*(-a* e+b*d)^4*ln(e*x+d)/e^6
Time = 0.04 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=\frac {60 a^4 b d e^4-12 a^5 e^5+120 a^3 b^2 e^3 \left (-d^2+d e x+e^2 x^2\right )+60 a^2 b^3 e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+20 a b^4 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+b^5 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+60 b (b d-a e)^4 (d+e x) \log (d+e x)}{12 e^6 (d+e x)} \]
(60*a^4*b*d*e^4 - 12*a^5*e^5 + 120*a^3*b^2*e^3*(-d^2 + d*e*x + e^2*x^2) + 60*a^2*b^3*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 20*a*b^4*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^5*(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) + 60*b*(b*d - a*e)^4*(d + e*x)*Log[d + e*x])/(12*e^6*(d + e*x))
Time = 0.37 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 49, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{(d+e x)^2}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^5}{(d+e x)^2}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^2 (b d-a e)}{e^5}+\frac {10 b^3 (d+e x) (b d-a e)^2}{e^5}-\frac {10 b^2 (b d-a e)^3}{e^5}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)}+\frac {(a e-b d)^5}{e^5 (d+e x)^2}+\frac {b^5 (d+e x)^3}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 b^4 (d+e x)^3 (b d-a e)}{3 e^6}+\frac {5 b^3 (d+e x)^2 (b d-a e)^2}{e^6}-\frac {10 b^2 x (b d-a e)^3}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6}+\frac {b^5 (d+e x)^4}{4 e^6}\) |
(-10*b^2*(b*d - a*e)^3*x)/e^5 + (b*d - a*e)^5/(e^6*(d + e*x)) + (5*b^3*(b* d - a*e)^2*(d + e*x)^2)/e^6 - (5*b^4*(b*d - a*e)*(d + e*x)^3)/(3*e^6) + (b ^5*(d + e*x)^4)/(4*e^6) + (5*b*(b*d - a*e)^4*Log[d + e*x])/e^6
3.20.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(126)=252\).
Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.97
| method | result | size |
| norman | \(\frac {\frac {\left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+20 b^{2} d^{2} e^{3} a^{3}-30 b^{3} d^{3} e^{2} a^{2}+20 b^{4} d^{4} e a -5 b^{5} d^{5}\right ) x}{e^{5} d}+\frac {b^{5} x^{5}}{4 e}+\frac {5 b^{2} \left (4 a^{3} e^{3}-6 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{2}}{2 e^{4}}+\frac {5 b^{3} \left (6 e^{2} a^{2}-4 a b d e +b^{2} d^{2}\right ) x^{3}}{6 e^{3}}+\frac {5 b^{4} \left (4 a e -b d \right ) x^{4}}{12 e^{2}}}{e x +d}+\frac {5 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(256\) |
| default | \(\frac {b^{2} \left (\frac {1}{4} b^{3} x^{4} e^{3}+\frac {5}{3} x^{3} a \,b^{2} e^{3}-\frac {2}{3} x^{3} b^{3} d \,e^{2}+5 x^{2} a^{2} b \,e^{3}-5 x^{2} a \,b^{2} d \,e^{2}+\frac {3}{2} x^{2} b^{3} d^{2} e +10 a^{3} e^{3} x -20 a^{2} b d \,e^{2} x +15 a \,b^{2} d^{2} e x -4 b^{3} d^{3} x \right )}{e^{5}}-\frac {e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}{e^{6} \left (e x +d \right )}+\frac {5 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(259\) |
| risch | \(\frac {b^{5} x^{4}}{4 e^{2}}+\frac {5 b^{4} x^{3} a}{3 e^{2}}-\frac {2 b^{5} x^{3} d}{3 e^{3}}+\frac {5 b^{3} x^{2} a^{2}}{e^{2}}-\frac {5 b^{4} x^{2} a d}{e^{3}}+\frac {3 b^{5} x^{2} d^{2}}{2 e^{4}}+\frac {10 b^{2} a^{3} x}{e^{2}}-\frac {20 b^{3} a^{2} d x}{e^{3}}+\frac {15 b^{4} a \,d^{2} x}{e^{4}}-\frac {4 b^{5} d^{3} x}{e^{5}}-\frac {a^{5}}{e \left (e x +d \right )}+\frac {5 b d \,a^{4}}{e^{2} \left (e x +d \right )}-\frac {10 b^{2} d^{2} a^{3}}{e^{3} \left (e x +d \right )}+\frac {10 b^{3} d^{3} a^{2}}{e^{4} \left (e x +d \right )}-\frac {5 b^{4} d^{4} a}{e^{5} \left (e x +d \right )}+\frac {b^{5} d^{5}}{e^{6} \left (e x +d \right )}+\frac {5 b \ln \left (e x +d \right ) a^{4}}{e^{2}}-\frac {20 b^{2} \ln \left (e x +d \right ) d \,a^{3}}{e^{3}}+\frac {30 b^{3} \ln \left (e x +d \right ) d^{2} a^{2}}{e^{4}}-\frac {20 b^{4} \ln \left (e x +d \right ) d^{3} a}{e^{5}}+\frac {5 b^{5} \ln \left (e x +d \right ) d^{4}}{e^{6}}\) | \(326\) |
| parallelrisch | \(\frac {60 b d \,e^{4} a^{4}-240 b^{2} d^{2} e^{3} a^{3}+360 b^{3} d^{3} e^{2} a^{2}-240 b^{4} d^{4} e a +20 x^{4} a \,b^{4} e^{5}-5 x^{4} b^{5} d \,e^{4}+60 x^{3} a^{2} b^{3} e^{5}+10 x^{3} b^{5} d^{2} e^{3}+120 x^{2} a^{3} b^{2} e^{5}-30 x^{2} b^{5} d^{3} e^{2}-240 \ln \left (e x +d \right ) a^{3} b^{2} d^{2} e^{3}+360 \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{2}-240 \ln \left (e x +d \right ) a \,b^{4} d^{4} e -40 x^{3} a \,b^{4} d \,e^{4}-180 x^{2} a^{2} b^{3} d \,e^{4}+120 x^{2} a \,b^{4} d^{2} e^{3}+60 \ln \left (e x +d \right ) a^{4} b d \,e^{4}+60 \ln \left (e x +d \right ) b^{5} d^{5}-240 \ln \left (e x +d \right ) x a \,b^{4} d^{3} e^{2}+360 \ln \left (e x +d \right ) x \,a^{2} b^{3} d^{2} e^{3}-240 \ln \left (e x +d \right ) x \,a^{3} b^{2} d \,e^{4}+60 b^{5} d^{5}+60 \ln \left (e x +d \right ) x \,b^{5} d^{4} e +60 \ln \left (e x +d \right ) x \,a^{4} b \,e^{5}+3 x^{5} b^{5} e^{5}-12 e^{5} a^{5}}{12 e^{6} \left (e x +d \right )}\) | \(389\) |
((a^5*e^5-5*a^4*b*d*e^4+20*a^3*b^2*d^2*e^3-30*a^2*b^3*d^3*e^2+20*a*b^4*d^4 *e-5*b^5*d^5)/e^5/d*x+1/4*b^5/e*x^5+5/2*b^2*(4*a^3*e^3-6*a^2*b*d*e^2+4*a*b ^2*d^2*e-b^3*d^3)/e^4*x^2+5/6*b^3*(6*a^2*e^2-4*a*b*d*e+b^2*d^2)/e^3*x^3+5/ 12*b^4*(4*a*e-b*d)/e^2*x^4)/(e*x+d)+5*b/e^6*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b ^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (126) = 252\).
Time = 0.29 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.87 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=\frac {3 \, b^{5} e^{5} x^{5} + 12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e + 120 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} - 5 \, {\left (b^{5} d e^{4} - 4 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 4 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 4 \, a^{3} b^{2} e^{5}\right )} x^{2} - 12 \, {\left (4 \, b^{5} d^{4} e - 15 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4}\right )} x + 60 \, {\left (b^{5} d^{5} - 4 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} - 4 \, a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \]
1/12*(3*b^5*e^5*x^5 + 12*b^5*d^5 - 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 12*a^5*e^5 - 5*(b^5*d*e^4 - 4*a*b^4 *e^5)*x^4 + 10*(b^5*d^2*e^3 - 4*a*b^4*d*e^4 + 6*a^2*b^3*e^5)*x^3 - 30*(b^5 *d^3*e^2 - 4*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 4*a^3*b^2*e^5)*x^2 - 12*(4* b^5*d^4*e - 15*a*b^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 - 10*a^3*b^2*d*e^4)*x + 60*(b^5*d^5 - 4*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 - 4*a^3*b^2*d^2*e^3 + a^4* b*d*e^4 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e ^4 + a^4*b*e^5)*x)*log(e*x + d))/(e^7*x + d*e^6)
Time = 0.46 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=\frac {b^{5} x^{4}}{4 e^{2}} + \frac {5 b \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{6}} + x^{3} \cdot \left (\frac {5 a b^{4}}{3 e^{2}} - \frac {2 b^{5} d}{3 e^{3}}\right ) + x^{2} \cdot \left (\frac {5 a^{2} b^{3}}{e^{2}} - \frac {5 a b^{4} d}{e^{3}} + \frac {3 b^{5} d^{2}}{2 e^{4}}\right ) + x \left (\frac {10 a^{3} b^{2}}{e^{2}} - \frac {20 a^{2} b^{3} d}{e^{3}} + \frac {15 a b^{4} d^{2}}{e^{4}} - \frac {4 b^{5} d^{3}}{e^{5}}\right ) + \frac {- a^{5} e^{5} + 5 a^{4} b d e^{4} - 10 a^{3} b^{2} d^{2} e^{3} + 10 a^{2} b^{3} d^{3} e^{2} - 5 a b^{4} d^{4} e + b^{5} d^{5}}{d e^{6} + e^{7} x} \]
b**5*x**4/(4*e**2) + 5*b*(a*e - b*d)**4*log(d + e*x)/e**6 + x**3*(5*a*b**4 /(3*e**2) - 2*b**5*d/(3*e**3)) + x**2*(5*a**2*b**3/e**2 - 5*a*b**4*d/e**3 + 3*b**5*d**2/(2*e**4)) + x*(10*a**3*b**2/e**2 - 20*a**2*b**3*d/e**3 + 15* a*b**4*d**2/e**4 - 4*b**5*d**3/e**5) + (-a**5*e**5 + 5*a**4*b*d*e**4 - 10* a**3*b**2*d**2*e**3 + 10*a**2*b**3*d**3*e**2 - 5*a*b**4*d**4*e + b**5*d**5 )/(d*e**6 + e**7*x)
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (126) = 252\).
Time = 0.19 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=\frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{e^{7} x + d e^{6}} + \frac {3 \, b^{5} e^{3} x^{4} - 4 \, {\left (2 \, b^{5} d e^{2} - 5 \, a b^{4} e^{3}\right )} x^{3} + 6 \, {\left (3 \, b^{5} d^{2} e - 10 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{2} - 12 \, {\left (4 \, b^{5} d^{3} - 15 \, a b^{4} d^{2} e + 20 \, a^{2} b^{3} d e^{2} - 10 \, a^{3} b^{2} e^{3}\right )} x}{12 \, e^{5}} + \frac {5 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]
(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4 *b*d*e^4 - a^5*e^5)/(e^7*x + d*e^6) + 1/12*(3*b^5*e^3*x^4 - 4*(2*b^5*d*e^2 - 5*a*b^4*e^3)*x^3 + 6*(3*b^5*d^2*e - 10*a*b^4*d*e^2 + 10*a^2*b^3*e^3)*x^ 2 - 12*(4*b^5*d^3 - 15*a*b^4*d^2*e + 20*a^2*b^3*d*e^2 - 10*a^3*b^2*e^3)*x) /e^5 + 5*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*log(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (126) = 252\).
Time = 0.26 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.61 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=\frac {{\left (3 \, b^{5} - \frac {20 \, {\left (b^{5} d e - a b^{4} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {60 \, {\left (b^{5} d^{2} e^{2} - 2 \, a b^{4} d e^{3} + a^{2} b^{3} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}}\right )} {\left (e x + d\right )}^{4}}{12 \, e^{6}} - \frac {5 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{6}} + \frac {\frac {b^{5} d^{5} e^{4}}{e x + d} - \frac {5 \, a b^{4} d^{4} e^{5}}{e x + d} + \frac {10 \, a^{2} b^{3} d^{3} e^{6}}{e x + d} - \frac {10 \, a^{3} b^{2} d^{2} e^{7}}{e x + d} + \frac {5 \, a^{4} b d e^{8}}{e x + d} - \frac {a^{5} e^{9}}{e x + d}}{e^{10}} \]
1/12*(3*b^5 - 20*(b^5*d*e - a*b^4*e^2)/((e*x + d)*e) + 60*(b^5*d^2*e^2 - 2 *a*b^4*d*e^3 + a^2*b^3*e^4)/((e*x + d)^2*e^2) - 120*(b^5*d^3*e^3 - 3*a*b^4 *d^2*e^4 + 3*a^2*b^3*d*e^5 - a^3*b^2*e^6)/((e*x + d)^3*e^3))*(e*x + d)^4/e ^6 - 5*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^ 4*b*e^4)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^6 + (b^5*d^5*e^4/(e*x + d) - 5*a*b^4*d^4*e^5/(e*x + d) + 10*a^2*b^3*d^3*e^6/(e*x + d) - 10*a^3*b^2 *d^2*e^7/(e*x + d) + 5*a^4*b*d*e^8/(e*x + d) - a^5*e^9/(e*x + d))/e^10
Time = 10.71 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx=x^3\,\left (\frac {5\,a\,b^4}{3\,e^2}-\frac {2\,b^5\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e}-\frac {5\,a^2\,b^3}{e^2}+\frac {b^5\,d^2}{2\,e^4}\right )+x\,\left (\frac {10\,a^3\,b^2}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e}-\frac {10\,a^2\,b^3}{e^2}+\frac {b^5\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )}{e^6}-\frac {a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {b^5\,x^4}{4\,e^2} \]
x^3*((5*a*b^4)/(3*e^2) - (2*b^5*d)/(3*e^3)) - x^2*((d*((5*a*b^4)/e^2 - (2* b^5*d)/e^3))/e - (5*a^2*b^3)/e^2 + (b^5*d^2)/(2*e^4)) + x*((10*a^3*b^2)/e^ 2 + (2*d*((2*d*((5*a*b^4)/e^2 - (2*b^5*d)/e^3))/e - (10*a^2*b^3)/e^2 + (b^ 5*d^2)/e^4))/e - (d^2*((5*a*b^4)/e^2 - (2*b^5*d)/e^3))/e^2) + (log(d + e*x )*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a* b^4*d^3*e))/e^6 - (a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2 *e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)/(e*(d*e^5 + e^6*x)) + (b^5*x^4)/(4*e ^2)