Integrand size = 29, antiderivative size = 133 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {10 b^3 (b c-a d)^2 x}{d^5}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac {b^5 (c+d x)^3}{3 d^6}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6} \]
10*b^3*(-a*d+b*c)^2*x/d^5+1/2*(-a*d+b*c)^5/d^6/(d*x+c)^2-5*b*(-a*d+b*c)^4/ d^6/(d*x+c)-5/2*b^4*(-a*d+b*c)*(d*x+c)^2/d^6+1/3*b^5*(d*x+c)^3/d^6-10*b^2* (-a*d+b*c)^3*ln(d*x+c)/d^6
Time = 0.05 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {-3 a^5 d^5-15 a^4 b d^4 (c+2 d x)+30 a^3 b^2 c d^3 (3 c+4 d x)+30 a^2 b^3 d^2 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+15 a b^4 d \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+b^5 \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )-60 b^2 (b c-a d)^3 (c+d x)^2 \log (c+d x)}{6 d^6 (c+d x)^2} \]
(-3*a^5*d^5 - 15*a^4*b*d^4*(c + 2*d*x) + 30*a^3*b^2*c*d^3*(3*c + 4*d*x) + 30*a^2*b^3*d^2*(-5*c^3 - 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) + 15*a*b^4*d *(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + b^5*(-27*c ^5 + 6*c^4*d*x + 63*c^3*d^2*x^2 + 20*c^2*d^3*x^3 - 5*c*d^4*x^4 + 2*d^5*x^5 ) - 60*b^2*(b*c - a*d)^3*(c + d*x)^2*Log[c + d*x])/(6*d^6*(c + d*x)^2)
Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1121, 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.
| \(\displaystyle \int \frac {(a+b x)^8}{\left (x (a d+b c)+a c+b d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (-\frac {5 b^4 (c+d x) (b c-a d)}{d^5}+\frac {10 b^3 (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3}{d^5 (c+d x)}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^2}+\frac {(a d-b c)^5}{d^5 (c+d x)^3}+\frac {b^5 (c+d x)^2}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac {10 b^3 x (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac {b^5 (c+d x)^3}{3 d^6}\) |
(10*b^3*(b*c - a*d)^2*x)/d^5 + (b*c - a*d)^5/(2*d^6*(c + d*x)^2) - (5*b*(b *c - a*d)^4)/(d^6*(c + d*x)) - (5*b^4*(b*c - a*d)*(c + d*x)^2)/(2*d^6) + ( b^5*(c + d*x)^3)/(3*d^6) - (10*b^2*(b*c - a*d)^3*Log[c + d*x])/d^6
3.19.18.3.1 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 + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 2.47 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.91
| method | result | size |
| default | \(\frac {b^{3} \left (\frac {1}{3} d^{2} x^{3} b^{2}+\frac {5}{2} x^{2} a b \,d^{2}-\frac {3}{2} x^{2} b^{2} c d +10 a^{2} d^{2} x -15 a b c d x +6 b^{2} c^{2} x \right )}{d^{5}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{2 d^{6} \left (d x +c \right )^{2}}+\frac {10 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}-\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{d^{6} \left (d x +c \right )}\) | \(254\) |
| risch | \(\frac {b^{5} x^{3}}{3 d^{3}}+\frac {5 b^{4} x^{2} a}{2 d^{3}}-\frac {3 b^{5} x^{2} c}{2 d^{4}}+\frac {10 b^{3} a^{2} x}{d^{3}}-\frac {15 b^{4} a c x}{d^{4}}+\frac {6 b^{5} c^{2} x}{d^{5}}+\frac {\left (-5 a^{4} b \,d^{4}+20 a^{3} b^{2} c \,d^{3}-30 a^{2} b^{3} c^{2} d^{2}+20 a \,b^{4} c^{3} d -5 b^{5} c^{4}\right ) x -\frac {a^{5} d^{5}+5 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}+50 a^{2} b^{3} c^{3} d^{2}-35 a \,b^{4} c^{4} d +9 b^{5} c^{5}}{2 d}}{d^{5} \left (d x +c \right )^{2}}+\frac {10 b^{2} \ln \left (d x +c \right ) a^{3}}{d^{3}}-\frac {30 b^{3} \ln \left (d x +c \right ) a^{2} c}{d^{4}}+\frac {30 b^{4} \ln \left (d x +c \right ) a \,c^{2}}{d^{5}}-\frac {10 b^{5} \ln \left (d x +c \right ) c^{3}}{d^{6}}\) | \(279\) |
| norman | \(\frac {\frac {b^{7} x^{7}}{3 d}+\frac {b^{5} \left (46 a^{2} d^{2}-35 a b c d +10 b^{2} c^{2}\right ) x^{5}}{3 d^{3}}+\frac {b^{6} \left (19 a d -5 b c \right ) x^{6}}{6 d^{2}}-\frac {\left (198 a^{5} b^{4} d^{5}+190 a^{4} d^{4} c \,b^{5}+305 a^{3} c^{2} d^{3} b^{6}-415 a^{2} c^{3} d^{2} b^{7}+10 a \,c^{4} d \,b^{8}+90 c^{5} b^{9}\right ) x^{2}}{6 d^{6} b^{2}}-\frac {\left (120 a^{4} b^{4} d^{4}-20 a^{3} b^{5} c \,d^{3}+85 a^{2} b^{6} c^{2} d^{2}-140 a \,b^{7} c^{3} d +60 b^{8} c^{4}\right ) x^{3}}{3 d^{5} b}-\frac {\left (3 a^{5} b^{2} d^{5}+15 a^{4} b^{3} c \,d^{4}+45 a^{3} b^{4} c^{2} d^{3}+145 b^{5} c^{3} a^{2} d^{2}-230 a \,b^{6} c^{4} d +90 b^{7} c^{5}\right ) a^{2}}{6 d^{6} b^{2}}-\frac {a \left (18 a^{5} b^{3} d^{5}+90 a^{4} b^{4} c \,d^{4}+100 a^{3} b^{5} c^{2} d^{3}+5 a^{2} b^{6} c^{3} d^{2}-170 a \,b^{7} c^{4} d +90 b^{8} c^{5}\right ) x}{3 d^{6} b^{2}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {10 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(442\) |
| parallelrisch | \(\frac {-15 a^{4} b c \,d^{4}+90 a^{3} b^{2} c^{2} d^{3}-270 a^{2} b^{3} c^{3} d^{2}+270 a \,b^{4} c^{4} d -30 x \,a^{4} b \,d^{5}+15 a \,b^{4} d^{5} x^{4}+2 x^{5} b^{5} d^{5}-3 a^{5} d^{5}-90 b^{5} c^{5}-60 \ln \left (d x +c \right ) b^{5} c^{5}+60 x^{3} a^{2} b^{3} d^{5}-360 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{2} d^{3}+360 \ln \left (d x +c \right ) x a \,b^{4} c^{3} d^{2}+60 \ln \left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}-180 \ln \left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}+180 \ln \left (d x +c \right ) a \,b^{4} c^{4} d +120 x \,a^{3} b^{2} c \,d^{4}-360 x \,a^{2} b^{3} c^{2} d^{3}+360 x a \,b^{4} c^{3} d^{2}-60 x^{3} a \,b^{4} c \,d^{4}-5 x^{4} b^{5} c \,d^{4}+20 x^{3} b^{5} c^{2} d^{3}-120 x \,b^{5} c^{4} d +120 \ln \left (d x +c \right ) x \,a^{3} b^{2} c \,d^{4}-60 \ln \left (d x +c \right ) x^{2} b^{5} c^{3} d^{2}-120 \ln \left (d x +c \right ) x \,b^{5} c^{4} d +60 \ln \left (d x +c \right ) x^{2} a^{3} b^{2} d^{5}-180 \ln \left (d x +c \right ) x^{2} a^{2} b^{3} c \,d^{4}+180 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{2} d^{3}}{6 d^{6} \left (d x +c \right )^{2}}\) | \(442\) |
b^3/d^5*(1/3*d^2*x^3*b^2+5/2*x^2*a*b*d^2-3/2*x^2*b^2*c*d+10*a^2*d^2*x-15*a *b*c*d*x+6*b^2*c^2*x)-1/2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2 *b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/d^6/(d*x+c)^2+10*b^2/d^6*(a^3*d^3-3*a^ 2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(d*x+c)-5*b/d^6*(a^4*d^4-4*a^3*b*c*d^3+ 6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (127) = 254\).
Time = 0.27 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.13 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {2 \, b^{5} d^{5} x^{5} - 27 \, b^{5} c^{5} + 105 \, a b^{4} c^{4} d - 150 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 15 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 3 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{5} c^{3} d^{2} - 55 \, a b^{4} c^{2} d^{3} + 40 \, a^{2} b^{3} c d^{4}\right )} x^{2} + 6 \, {\left (b^{5} c^{4} d + 5 \, a b^{4} c^{3} d^{2} - 20 \, a^{2} b^{3} c^{2} d^{3} + 20 \, a^{3} b^{2} c d^{4} - 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \]
1/6*(2*b^5*d^5*x^5 - 27*b^5*c^5 + 105*a*b^4*c^4*d - 150*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 - 15*a^4*b*c*d^4 - 3*a^5*d^5 - 5*(b^5*c*d^4 - 3*a*b^4*d ^5)*x^4 + 20*(b^5*c^2*d^3 - 3*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 3*(21*b^5 *c^3*d^2 - 55*a*b^4*c^2*d^3 + 40*a^2*b^3*c*d^4)*x^2 + 6*(b^5*c^4*d + 5*a*b ^4*c^3*d^2 - 20*a^2*b^3*c^2*d^3 + 20*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x - 60*( b^5*c^5 - 3*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + (b^5*c^3*d ^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - a^3*b^2*d^5)*x^2 + 2*(b^5*c^4*d - 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 - a^3*b^2*c*d^4)*x)*log(d*x + c))/(d^ 8*x^2 + 2*c*d^7*x + c^2*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (121) = 242\).
Time = 2.74 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=\frac {b^{5} x^{3}}{3 d^{3}} + \frac {10 b^{2} \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{6}} + x^{2} \cdot \left (\frac {5 a b^{4}}{2 d^{3}} - \frac {3 b^{5} c}{2 d^{4}}\right ) + x \left (\frac {10 a^{2} b^{3}}{d^{3}} - \frac {15 a b^{4} c}{d^{4}} + \frac {6 b^{5} c^{2}}{d^{5}}\right ) + \frac {- a^{5} d^{5} - 5 a^{4} b c d^{4} + 30 a^{3} b^{2} c^{2} d^{3} - 50 a^{2} b^{3} c^{3} d^{2} + 35 a b^{4} c^{4} d - 9 b^{5} c^{5} + x \left (- 10 a^{4} b d^{5} + 40 a^{3} b^{2} c d^{4} - 60 a^{2} b^{3} c^{2} d^{3} + 40 a b^{4} c^{3} d^{2} - 10 b^{5} c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} \]
b**5*x**3/(3*d**3) + 10*b**2*(a*d - b*c)**3*log(c + d*x)/d**6 + x**2*(5*a* b**4/(2*d**3) - 3*b**5*c/(2*d**4)) + x*(10*a**2*b**3/d**3 - 15*a*b**4*c/d* *4 + 6*b**5*c**2/d**5) + (-a**5*d**5 - 5*a**4*b*c*d**4 + 30*a**3*b**2*c**2 *d**3 - 50*a**2*b**3*c**3*d**2 + 35*a*b**4*c**4*d - 9*b**5*c**5 + x*(-10*a **4*b*d**5 + 40*a**3*b**2*c*d**4 - 60*a**2*b**3*c**2*d**3 + 40*a*b**4*c**3 *d**2 - 10*b**5*c**4*d))/(2*c**2*d**6 + 4*c*d**7*x + 2*d**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 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac {2 \, b^{5} d^{2} x^{3} - 3 \, {\left (3 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{2} + 6 \, {\left (6 \, b^{5} c^{2} - 15 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x}{6 \, d^{5}} - \frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \]
-1/2*(9*b^5*c^5 - 35*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + a^5*d^5 + 10*(b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c ^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)/(d^8*x^2 + 2*c*d^7*x + c^2*d^6) + 1/6*(2*b^5*d^2*x^3 - 3*(3*b^5*c*d - 5*a*b^4*d^2)*x^2 + 6*(6*b^5*c^2 - 15* a*b^4*c*d + 10*a^2*b^3*d^2)*x)/d^5 - 10*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b ^3*c*d^2 - a^3*b^2*d^3)*log(d*x + c)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (127) = 254\).
Time = 0.27 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=-\frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} - \frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{6}} + \frac {2 \, b^{5} d^{6} x^{3} - 9 \, b^{5} c d^{5} x^{2} + 15 \, a b^{4} d^{6} x^{2} + 36 \, b^{5} c^{2} d^{4} x - 90 \, a b^{4} c d^{5} x + 60 \, a^{2} b^{3} d^{6} x}{6 \, d^{9}} \]
-10*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(abs(d*x + c))/d^6 - 1/2*(9*b^5*c^5 - 35*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 - 30*a^3* b^2*c^2*d^3 + 5*a^4*b*c*d^4 + a^5*d^5 + 10*(b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)/((d*x + c)^2*d^6) + 1/ 6*(2*b^5*d^6*x^3 - 9*b^5*c*d^5*x^2 + 15*a*b^4*d^6*x^2 + 36*b^5*c^2*d^4*x - 90*a*b^4*c*d^5*x + 60*a^2*b^3*d^6*x)/d^9
Time = 9.84 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx=x^2\,\left (\frac {5\,a\,b^4}{2\,d^3}-\frac {3\,b^5\,c}{2\,d^4}\right )-\frac {\frac {a^5\,d^5+5\,a^4\,b\,c\,d^4-30\,a^3\,b^2\,c^2\,d^3+50\,a^2\,b^3\,c^3\,d^2-35\,a\,b^4\,c^4\,d+9\,b^5\,c^5}{2\,d}+x\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{c^2\,d^5+2\,c\,d^6\,x+d^7\,x^2}-x\,\left (\frac {3\,c\,\left (\frac {5\,a\,b^4}{d^3}-\frac {3\,b^5\,c}{d^4}\right )}{d}-\frac {10\,a^2\,b^3}{d^3}+\frac {3\,b^5\,c^2}{d^5}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )}{d^6}+\frac {b^5\,x^3}{3\,d^3} \]
x^2*((5*a*b^4)/(2*d^3) - (3*b^5*c)/(2*d^4)) - ((a^5*d^5 + 9*b^5*c^5 + 50*a ^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 35*a*b^4*c^4*d + 5*a^4*b*c*d^4)/(2*d ) + x*(5*b^5*c^4 + 5*a^4*b*d^4 - 20*a^3*b^2*c*d^3 + 30*a^2*b^3*c^2*d^2 - 2 0*a*b^4*c^3*d))/(c^2*d^5 + d^7*x^2 + 2*c*d^6*x) - x*((3*c*((5*a*b^4)/d^3 - (3*b^5*c)/d^4))/d - (10*a^2*b^3)/d^3 + (3*b^5*c^2)/d^5) - (log(c + d*x)*( 10*b^5*c^3 - 10*a^3*b^2*d^3 + 30*a^2*b^3*c*d^2 - 30*a*b^4*c^2*d))/d^6 + (b ^5*x^3)/(3*d^3)