Integrand size = 31, antiderivative size = 133 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=\frac {10 b^3 (b d-a e)^2 x}{e^5}+\frac {(b d-a e)^5}{2 e^6 (d+e x)^2}-\frac {5 b (b d-a e)^4}{e^6 (d+e x)}-\frac {5 b^4 (b d-a e) (d+e x)^2}{2 e^6}+\frac {b^5 (d+e x)^3}{3 e^6}-\frac {10 b^2 (b d-a e)^3 \log (d+e x)}{e^6} \]
10*b^3*(-a*e+b*d)^2*x/e^5+1/2*(-a*e+b*d)^5/e^6/(e*x+d)^2-5*b*(-a*e+b*d)^4/ e^6/(e*x+d)-5/2*b^4*(-a*e+b*d)*(e*x+d)^2/e^6+1/3*b^5*(e*x+d)^3/e^6-10*b^2* (-a*e+b*d)^3*ln(e*x+d)/e^6
Time = 0.05 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=\frac {-3 a^5 e^5-15 a^4 b e^4 (d+2 e x)+30 a^3 b^2 d e^3 (3 d+4 e x)+30 a^2 b^3 e^2 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+15 a b^4 e \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+b^5 \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )-60 b^2 (b d-a e)^3 (d+e x)^2 \log (d+e x)}{6 e^6 (d+e x)^2} \]
(-3*a^5*e^5 - 15*a^4*b*e^4*(d + 2*e*x) + 30*a^3*b^2*d*e^3*(3*d + 4*e*x) + 30*a^2*b^3*e^2*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 15*a*b^4*e *(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + b^5*(-27*d ^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5 ) - 60*b^2*(b*d - a*e)^3*(d + e*x)^2*Log[d + e*x])/(6*e^6*(d + e*x)^2)
Time = 0.34 (sec) , antiderivative size = 133, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{(d+e x)^3}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^5}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x) (b d-a e)}{e^5}+\frac {10 b^3 (b d-a e)^2}{e^5}-\frac {10 b^2 (b d-a e)^3}{e^5 (d+e x)}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)^2}+\frac {(a e-b d)^5}{e^5 (d+e x)^3}+\frac {b^5 (d+e x)^2}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 b^4 (d+e x)^2 (b d-a e)}{2 e^6}+\frac {10 b^3 x (b d-a e)^2}{e^5}-\frac {10 b^2 (b d-a e)^3 \log (d+e x)}{e^6}-\frac {5 b (b d-a e)^4}{e^6 (d+e x)}+\frac {(b d-a e)^5}{2 e^6 (d+e x)^2}+\frac {b^5 (d+e x)^3}{3 e^6}\) |
(10*b^3*(b*d - a*e)^2*x)/e^5 + (b*d - a*e)^5/(2*e^6*(d + e*x)^2) - (5*b*(b *d - a*e)^4)/(e^6*(d + e*x)) - (5*b^4*(b*d - a*e)*(d + e*x)^2)/(2*e^6) + ( b^5*(d + e*x)^3)/(3*e^6) - (10*b^2*(b*d - a*e)^3*Log[d + e*x])/e^6
3.20.14.3.1 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]
Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.91
| method | result | size |
| default | \(\frac {b^{3} \left (\frac {1}{3} b^{2} e^{2} x^{3}+\frac {5}{2} a b \,e^{2} x^{2}-\frac {3}{2} b^{2} d e \,x^{2}+10 e^{2} a^{2} x -15 a b d e x +6 b^{2} d^{2} x \right )}{e^{5}}-\frac {5 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{e^{6} \left (e x +d \right )}+\frac {10 b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}-\frac {e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}{2 e^{6} \left (e x +d \right )^{2}}\) | \(254\) |
| norman | \(\frac {-\frac {e^{5} a^{5}+5 b d \,e^{4} a^{4}-30 b^{2} d^{2} e^{3} a^{3}+90 b^{3} d^{3} e^{2} a^{2}-90 b^{4} d^{4} e a +30 b^{5} d^{5}}{2 e^{6}}+\frac {b^{5} x^{5}}{3 e}-\frac {\left (5 e^{4} a^{4} b -20 e^{3} d \,a^{3} b^{2}+60 d^{2} e^{2} a^{2} b^{3}-60 d^{3} e \,b^{4} a +20 d^{4} b^{5}\right ) x}{e^{5}}+\frac {10 b^{3} \left (3 e^{2} a^{2}-3 a b d e +b^{2} d^{2}\right ) x^{3}}{3 e^{3}}+\frac {5 b^{4} \left (3 a e -b d \right ) x^{4}}{6 e^{2}}}{\left (e x +d \right )^{2}}+\frac {10 b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(254\) |
| risch | \(\frac {b^{5} x^{3}}{3 e^{3}}+\frac {5 b^{4} a \,x^{2}}{2 e^{3}}-\frac {3 b^{5} d \,x^{2}}{2 e^{4}}+\frac {10 b^{3} a^{2} x}{e^{3}}-\frac {15 b^{4} a d x}{e^{4}}+\frac {6 b^{5} d^{2} x}{e^{5}}+\frac {\left (-5 e^{4} a^{4} b +20 e^{3} d \,a^{3} b^{2}-30 d^{2} e^{2} a^{2} b^{3}+20 d^{3} e \,b^{4} a -5 d^{4} b^{5}\right ) x -\frac {e^{5} a^{5}+5 b d \,e^{4} a^{4}-30 b^{2} d^{2} e^{3} a^{3}+50 b^{3} d^{3} e^{2} a^{2}-35 b^{4} d^{4} e a +9 b^{5} d^{5}}{2 e}}{e^{5} \left (e x +d \right )^{2}}+\frac {10 b^{2} \ln \left (e x +d \right ) a^{3}}{e^{3}}-\frac {30 b^{3} \ln \left (e x +d \right ) a^{2} d}{e^{4}}+\frac {30 b^{4} \ln \left (e x +d \right ) a \,d^{2}}{e^{5}}-\frac {10 b^{5} \ln \left (e x +d \right ) d^{3}}{e^{6}}\) | \(279\) |
| parallelrisch | \(\frac {-180 \ln \left (e x +d \right ) x^{2} a^{2} b^{3} d \,e^{4}+180 \ln \left (e x +d \right ) x^{2} a \,b^{4} d^{2} e^{3}-15 b d \,e^{4} a^{4}+90 b^{2} d^{2} e^{3} a^{3}-270 b^{3} d^{3} e^{2} a^{2}+270 b^{4} d^{4} e a +15 x^{4} a \,b^{4} e^{5}-5 x^{4} b^{5} d \,e^{4}+60 x^{3} a^{2} b^{3} e^{5}+20 x^{3} b^{5} d^{2} e^{3}-30 x \,a^{4} b \,e^{5}-120 x \,b^{5} d^{4} e +60 \ln \left (e x +d \right ) x^{2} a^{3} b^{2} e^{5}-60 \ln \left (e x +d \right ) x^{2} b^{5} d^{3} e^{2}+60 \ln \left (e x +d \right ) a^{3} b^{2} d^{2} e^{3}-180 \ln \left (e x +d \right ) a^{2} b^{3} d^{3} e^{2}+180 \ln \left (e x +d \right ) a \,b^{4} d^{4} e +120 x \,a^{3} b^{2} d \,e^{4}-360 x \,a^{2} b^{3} d^{2} e^{3}+360 x a \,b^{4} d^{3} e^{2}-60 x^{3} a \,b^{4} d \,e^{4}-60 \ln \left (e x +d \right ) b^{5} d^{5}+360 \ln \left (e x +d \right ) x a \,b^{4} d^{3} e^{2}-360 \ln \left (e x +d \right ) x \,a^{2} b^{3} d^{2} e^{3}+120 \ln \left (e x +d \right ) x \,a^{3} b^{2} d \,e^{4}-90 b^{5} d^{5}-120 \ln \left (e x +d \right ) x \,b^{5} d^{4} e +2 x^{5} b^{5} e^{5}-3 e^{5} a^{5}}{6 e^{6} \left (e x +d \right )^{2}}\) | \(442\) |
b^3/e^5*(1/3*b^2*e^2*x^3+5/2*a*b*e^2*x^2-3/2*b^2*d*e*x^2+10*e^2*a^2*x-15*a *b*d*e*x+6*b^2*d^2*x)-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)/(e*x+d)+10*b^2/e^6*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2* e-b^3*d^3)*ln(e*x+d)-1/2*(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^6/(e*x+d)^2
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (127) = 254\).
Time = 0.28 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.13 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=\frac {2 \, b^{5} e^{5} x^{5} - 27 \, b^{5} d^{5} + 105 \, a b^{4} d^{4} e - 150 \, a^{2} b^{3} d^{3} e^{2} + 90 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} - 3 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{5} d^{3} e^{2} - 55 \, a b^{4} d^{2} e^{3} + 40 \, a^{2} b^{3} d e^{4}\right )} x^{2} + 6 \, {\left (b^{5} d^{4} e + 5 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - 3 \, a b^{4} d^{4} e + 3 \, a^{2} b^{3} d^{3} e^{2} - a^{3} b^{2} d^{2} e^{3} + {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{5} d^{4} e - 3 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3} - a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]
1/6*(2*b^5*e^5*x^5 - 27*b^5*d^5 + 105*a*b^4*d^4*e - 150*a^2*b^3*d^3*e^2 + 90*a^3*b^2*d^2*e^3 - 15*a^4*b*d*e^4 - 3*a^5*e^5 - 5*(b^5*d*e^4 - 3*a*b^4*e ^5)*x^4 + 20*(b^5*d^2*e^3 - 3*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 3*(21*b^5 *d^3*e^2 - 55*a*b^4*d^2*e^3 + 40*a^2*b^3*d*e^4)*x^2 + 6*(b^5*d^4*e + 5*a*b ^4*d^3*e^2 - 20*a^2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x - 60*( b^5*d^5 - 3*a*b^4*d^4*e + 3*a^2*b^3*d^3*e^2 - a^3*b^2*d^2*e^3 + (b^5*d^3*e ^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 2*(b^5*d^4*e - 3*a*b^4*d^3*e^2 + 3*a^2*b^3*d^2*e^3 - a^3*b^2*d*e^4)*x)*log(e*x + d))/(e^ 8*x^2 + 2*d*e^7*x + d^2*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (121) = 242\).
Time = 0.84 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=\frac {b^{5} x^{3}}{3 e^{3}} + \frac {10 b^{2} \left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{6}} + x^{2} \cdot \left (\frac {5 a b^{4}}{2 e^{3}} - \frac {3 b^{5} d}{2 e^{4}}\right ) + x \left (\frac {10 a^{2} b^{3}}{e^{3}} - \frac {15 a b^{4} d}{e^{4}} + \frac {6 b^{5} d^{2}}{e^{5}}\right ) + \frac {- a^{5} e^{5} - 5 a^{4} b d e^{4} + 30 a^{3} b^{2} d^{2} e^{3} - 50 a^{2} b^{3} d^{3} e^{2} + 35 a b^{4} d^{4} e - 9 b^{5} d^{5} + x \left (- 10 a^{4} b e^{5} + 40 a^{3} b^{2} d e^{4} - 60 a^{2} b^{3} d^{2} e^{3} + 40 a b^{4} d^{3} e^{2} - 10 b^{5} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} \]
b**5*x**3/(3*e**3) + 10*b**2*(a*e - b*d)**3*log(d + e*x)/e**6 + x**2*(5*a* b**4/(2*e**3) - 3*b**5*d/(2*e**4)) + x*(10*a**2*b**3/e**3 - 15*a*b**4*d/e* *4 + 6*b**5*d**2/e**5) + (-a**5*e**5 - 5*a**4*b*d*e**4 + 30*a**3*b**2*d**2 *e**3 - 50*a**2*b**3*d**3*e**2 + 35*a*b**4*d**4*e - 9*b**5*d**5 + x*(-10*a **4*b*e**5 + 40*a**3*b**2*d*e**4 - 60*a**2*b**3*d**2*e**3 + 40*a*b**4*d**3 *e**2 - 10*b**5*d**4*e))/(2*d**2*e**6 + 4*d*e**7*x + 2*e**8*x**2)
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (127) = 254\).
Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=-\frac {9 \, b^{5} d^{5} - 35 \, a b^{4} d^{4} e + 50 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} + a^{5} e^{5} + 10 \, {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {2 \, b^{5} e^{2} x^{3} - 3 \, {\left (3 \, b^{5} d e - 5 \, a b^{4} e^{2}\right )} x^{2} + 6 \, {\left (6 \, b^{5} d^{2} - 15 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x}{6 \, e^{5}} - \frac {10 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \]
-1/2*(9*b^5*d^5 - 35*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 + a^5*e^5 + 10*(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)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6) + 1/6*(2*b^5*e^2*x^3 - 3*(3*b^5*d*e - 5*a*b^4*e^2)*x^2 + 6*(6*b^5*d^2 - 15* a*b^4*d*e + 10*a^2*b^3*e^2)*x)/e^5 - 10*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b ^3*d*e^2 - a^3*b^2*e^3)*log(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (127) = 254\).
Time = 0.25 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=-\frac {10 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} - \frac {9 \, b^{5} d^{5} - 35 \, a b^{4} d^{4} e + 50 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} + a^{5} e^{5} + 10 \, {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{6}} + \frac {2 \, b^{5} e^{6} x^{3} - 9 \, b^{5} d e^{5} x^{2} + 15 \, a b^{4} e^{6} x^{2} + 36 \, b^{5} d^{2} e^{4} x - 90 \, a b^{4} d e^{5} x + 60 \, a^{2} b^{3} e^{6} x}{6 \, e^{9}} \]
-10*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*log(abs(e*x + d))/e^6 - 1/2*(9*b^5*d^5 - 35*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 - 30*a^3* b^2*d^2*e^3 + 5*a^4*b*d*e^4 + a^5*e^5 + 10*(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)/((e*x + d)^2*e^6) + 1/ 6*(2*b^5*e^6*x^3 - 9*b^5*d*e^5*x^2 + 15*a*b^4*e^6*x^2 + 36*b^5*d^2*e^4*x - 90*a*b^4*d*e^5*x + 60*a^2*b^3*e^6*x)/e^9
Time = 0.09 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx=x^2\,\left (\frac {5\,a\,b^4}{2\,e^3}-\frac {3\,b^5\,d}{2\,e^4}\right )-\frac {\frac {a^5\,e^5+5\,a^4\,b\,d\,e^4-30\,a^3\,b^2\,d^2\,e^3+50\,a^2\,b^3\,d^3\,e^2-35\,a\,b^4\,d^4\,e+9\,b^5\,d^5}{2\,e}+x\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )}{d^2\,e^5+2\,d\,e^6\,x+e^7\,x^2}-x\,\left (\frac {3\,d\,\left (\frac {5\,a\,b^4}{e^3}-\frac {3\,b^5\,d}{e^4}\right )}{e}-\frac {10\,a^2\,b^3}{e^3}+\frac {3\,b^5\,d^2}{e^5}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )}{e^6}+\frac {b^5\,x^3}{3\,e^3} \]
x^2*((5*a*b^4)/(2*e^3) - (3*b^5*d)/(2*e^4)) - ((a^5*e^5 + 9*b^5*d^5 + 50*a ^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 - 35*a*b^4*d^4*e + 5*a^4*b*d*e^4)/(2*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 - 2 0*a*b^4*d^3*e))/(d^2*e^5 + e^7*x^2 + 2*d*e^6*x) - x*((3*d*((5*a*b^4)/e^3 - (3*b^5*d)/e^4))/e - (10*a^2*b^3)/e^3 + (3*b^5*d^2)/e^5) - (log(d + e*x)*( 10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e))/e^6 + (b ^5*x^3)/(3*e^3)