Integrand size = 16, antiderivative size = 157 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=\frac {a b}{2 (b c-a d)^3 (a+b x)^2}-\frac {b (b c+2 a d)}{(b c-a d)^4 (a+b x)}-\frac {c d}{2 (b c-a d)^3 (c+d x)^2}-\frac {d (2 b c+a d)}{(b c-a d)^4 (c+d x)}-\frac {3 b d (b c+a d) \log (a+b x)}{(b c-a d)^5}+\frac {3 b d (b c+a d) \log (c+d x)}{(b c-a d)^5} \]
1/2*a*b/(-a*d+b*c)^3/(b*x+a)^2-b*(2*a*d+b*c)/(-a*d+b*c)^4/(b*x+a)-1/2*c*d/ (-a*d+b*c)^3/(d*x+c)^2-d*(a*d+2*b*c)/(-a*d+b*c)^4/(d*x+c)-3*b*d*(a*d+b*c)* ln(b*x+a)/(-a*d+b*c)^5+3*b*d*(a*d+b*c)*ln(d*x+c)/(-a*d+b*c)^5
Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {a b (b c-a d)^2}{(a+b x)^2}-\frac {2 b (b c-a d) (b c+2 a d)}{a+b x}-\frac {c d (b c-a d)^2}{(c+d x)^2}+\frac {2 d (-b c+a d) (2 b c+a d)}{c+d x}-6 b d (b c+a d) \log (a+b x)+6 b d (b c+a d) \log (c+d x)}{2 (b c-a d)^5} \]
((a*b*(b*c - a*d)^2)/(a + b*x)^2 - (2*b*(b*c - a*d)*(b*c + 2*a*d))/(a + b* x) - (c*d*(b*c - a*d)^2)/(c + d*x)^2 + (2*d*(-(b*c) + a*d)*(2*b*c + a*d))/ (c + d*x) - 6*b*d*(b*c + a*d)*Log[a + b*x] + 6*b*d*(b*c + a*d)*Log[c + d*x ])/(2*(b*c - a*d)^5)
Time = 0.35 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {86, 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.
| \(\displaystyle \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (-\frac {3 b^2 d (a d+b c)}{(a+b x) (b c-a d)^5}+\frac {b^2 (2 a d+b c)}{(a+b x)^2 (b c-a d)^4}-\frac {a b^2}{(a+b x)^3 (b c-a d)^3}+\frac {3 b d^2 (a d+b c)}{(c+d x) (b c-a d)^5}+\frac {d^2 (a d+2 b c)}{(c+d x)^2 (b c-a d)^4}+\frac {c d^2}{(c+d x)^3 (b c-a d)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b (2 a d+b c)}{(a+b x) (b c-a d)^4}+\frac {a b}{2 (a+b x)^2 (b c-a d)^3}-\frac {d (a d+2 b c)}{(c+d x) (b c-a d)^4}-\frac {c d}{2 (c+d x)^2 (b c-a d)^3}-\frac {3 b d (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac {3 b d (a d+b c) \log (c+d x)}{(b c-a d)^5}\) |
(a*b)/(2*(b*c - a*d)^3*(a + b*x)^2) - (b*(b*c + 2*a*d))/((b*c - a*d)^4*(a + b*x)) - (c*d)/(2*(b*c - a*d)^3*(c + d*x)^2) - (d*(2*b*c + a*d))/((b*c - a*d)^4*(c + d*x)) - (3*b*d*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (3*b* d*(b*c + a*d)*Log[c + d*x])/(b*c - a*d)^5
3.4.15.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.54 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {d \left (a d +2 b c \right )}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {c d}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {3 d b \left (a d +b c \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {a b}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {b \left (2 a d +b c \right )}{\left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {3 d b \left (a d +b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}\) | \(154\) |
| norman | \(\frac {\frac {\left (a d +b c \right ) \left (-a^{2} b^{2} d^{4}-7 a \,b^{3} c \,d^{3}-b^{4} c^{2} d^{2}\right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {3 b^{2} d^{2} \left (a d +b c \right ) x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {\left (-9 a^{2} b^{3} d^{5}-18 a \,b^{4} c \,d^{4}-9 b^{5} c^{2} d^{3}\right ) x^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {c a \left (-a^{2} b^{2} d^{4}-10 a \,b^{3} c \,d^{3}-b^{4} c^{2} d^{2}\right )}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {3 b d \left (a d +b c \right ) \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {3 b d \left (a d +b c \right ) \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(541\) |
| risch | \(\frac {-\frac {3 b^{2} d^{2} \left (a d +b c \right ) x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {9 b d \left (a d +b c \right )^{2} x^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {\left (a^{3} d^{3}+8 a^{2} b c \,d^{2}+8 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {a c \left (a^{2} d^{2}+10 a b c d +b^{2} c^{2}\right )}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {3 b \,d^{2} \ln \left (-b x -a \right ) a}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {3 b^{2} d \ln \left (-b x -a \right ) c}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {3 b \,d^{2} \ln \left (d x +c \right ) a}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {3 b^{2} d \ln \left (d x +c \right ) c}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(641\) |
| parallelrisch | \(\frac {9 x^{2} a \,b^{5} c^{2} d^{4}-14 x \,a^{3} b^{3} c \,d^{5}+14 x a \,b^{5} c^{3} d^{3}+6 \ln \left (b x +a \right ) x^{4} a \,b^{5} d^{6}+6 \ln \left (b x +a \right ) x^{4} b^{6} c \,d^{5}-6 \ln \left (d x +c \right ) x^{4} a \,b^{5} d^{6}-6 \ln \left (d x +c \right ) x^{4} b^{6} c \,d^{5}+12 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} d^{6}+12 \ln \left (b x +a \right ) x^{3} b^{6} c^{2} d^{4}-12 \ln \left (d x +c \right ) x^{3} a^{2} b^{4} d^{6}-12 \ln \left (d x +c \right ) x^{3} b^{6} c^{2} d^{4}+6 \ln \left (b x +a \right ) x^{2} a^{3} b^{3} d^{6}+6 \ln \left (b x +a \right ) x^{2} b^{6} c^{3} d^{3}-6 \ln \left (d x +c \right ) x^{2} a^{3} b^{3} d^{6}-6 \ln \left (d x +c \right ) x^{2} b^{6} c^{3} d^{3}+6 \ln \left (b x +a \right ) a^{3} b^{3} c^{2} d^{4}+6 \ln \left (b x +a \right ) a^{2} b^{4} c^{3} d^{3}-6 \ln \left (d x +c \right ) a^{3} b^{3} c^{2} d^{4}-6 \ln \left (d x +c \right ) a^{2} b^{4} c^{3} d^{3}-a^{4} b^{2} c \,d^{5}-9 a^{3} b^{3} c^{2} d^{4}+9 a^{2} b^{4} c^{3} d^{3}+a \,b^{5} c^{4} d^{2}-9 x^{2} a^{2} b^{4} c \,d^{5}-9 x^{2} a^{3} b^{3} d^{6}+9 x^{2} b^{6} c^{3} d^{3}-2 x \,a^{4} b^{2} d^{6}+2 x \,b^{6} c^{4} d^{2}-6 x^{3} a^{2} b^{4} d^{6}+6 x^{3} b^{6} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a \,b^{5} c \,d^{5}-24 \ln \left (d x +c \right ) x^{3} a \,b^{5} c \,d^{5}+30 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} c \,d^{5}+30 \ln \left (b x +a \right ) x^{2} a \,b^{5} c^{2} d^{4}-30 \ln \left (d x +c \right ) x^{2} a \,b^{5} c^{2} d^{4}+12 \ln \left (b x +a \right ) x \,a^{3} b^{3} c \,d^{5}+24 \ln \left (b x +a \right ) x \,a^{2} b^{4} c^{2} d^{4}+12 \ln \left (b x +a \right ) x a \,b^{5} c^{3} d^{3}-12 \ln \left (d x +c \right ) x \,a^{3} b^{3} c \,d^{5}-30 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} c \,d^{5}-24 \ln \left (d x +c \right ) x \,a^{2} b^{4} c^{2} d^{4}-12 \ln \left (d x +c \right ) x a \,b^{5} c^{3} d^{3}}{2 \left (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}\right ) \left (d x +c \right )^{2} \left (b x +a \right )^{2} b^{2} d^{2}}\) | \(829\) |
-d*(a*d+2*b*c)/(a*d-b*c)^4/(d*x+c)+1/2*c*d/(a*d-b*c)^3/(d*x+c)^2-3*d*b*(a* d+b*c)/(a*d-b*c)^5*ln(d*x+c)-1/2*a*b/(a*d-b*c)^3/(b*x+a)^2-b*(2*a*d+b*c)/( a*d-b*c)^4/(b*x+a)+3*d*b*(a*d+b*c)/(a*d-b*c)^5*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (153) = 306\).
Time = 0.24 (sec) , antiderivative size = 891, normalized size of antiderivative = 5.68 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a b^{3} c^{4} + 9 \, a^{2} b^{2} c^{3} d - 9 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 9 \, {\left (b^{4} c^{3} d + a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} + 7 \, a b^{3} c^{3} d - 7 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x + 6 \, {\left (a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} + {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{3} + {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{3} d + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 6 \, {\left (a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} + {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{3} + {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{3} d + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \]
-1/2*(a*b^3*c^4 + 9*a^2*b^2*c^3*d - 9*a^3*b*c^2*d^2 - a^4*c*d^3 + 6*(b^4*c ^2*d^2 - a^2*b^2*d^4)*x^3 + 9*(b^4*c^3*d + a*b^3*c^2*d^2 - a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + 2*(b^4*c^4 + 7*a*b^3*c^3*d - 7*a^3*b*c*d^3 - a^4*d^4)*x + 6*(a^2*b^2*c^3*d + a^3*b*c^2*d^2 + (b^4*c*d^3 + a*b^3*d^4)*x^4 + 2*(b^4* c^2*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^3 + (b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + 2*(a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x)*log(b*x + a) - 6*(a^2*b^2*c^3*d + a^3*b*c^2*d^2 + (b^4*c*d ^3 + a*b^3*d^4)*x^4 + 2*(b^4*c^2*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^3 + (b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + 2*(a*b^3 *c^3*d + 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x)*log(d*x + c))/(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^ 4 - a^7*c^2*d^5 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10 *a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 4*a *b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a ^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^ 4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*x^ 2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (141) = 282\).
Time = 1.58 (sec) , antiderivative size = 1052, normalized size of antiderivative = 6.70 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=- \frac {3 b d \left (a d + b c\right ) \log {\left (x + \frac {- \frac {3 a^{6} b d^{7} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {18 a^{5} b^{2} c d^{6} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {45 a^{4} b^{3} c^{2} d^{5} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {60 a^{3} b^{4} c^{3} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {45 a^{2} b^{5} c^{4} d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a^{2} b d^{3} + \frac {18 a b^{6} c^{5} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 6 a b^{2} c d^{2} - \frac {3 b^{7} c^{6} d \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 b^{3} c^{2} d}{6 a b^{2} d^{3} + 6 b^{3} c d^{2}} \right )}}{\left (a d - b c\right )^{5}} + \frac {3 b d \left (a d + b c\right ) \log {\left (x + \frac {\frac {3 a^{6} b d^{7} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {18 a^{5} b^{2} c d^{6} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {45 a^{4} b^{3} c^{2} d^{5} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {60 a^{3} b^{4} c^{3} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {45 a^{2} b^{5} c^{4} d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a^{2} b d^{3} - \frac {18 a b^{6} c^{5} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 6 a b^{2} c d^{2} + \frac {3 b^{7} c^{6} d \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 b^{3} c^{2} d}{6 a b^{2} d^{3} + 6 b^{3} c d^{2}} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{3} c d^{2} - 10 a^{2} b c^{2} d - a b^{2} c^{3} + x^{3} \left (- 6 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x^{2} \left (- 9 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 9 b^{3} c^{2} d\right ) + x \left (- 2 a^{3} d^{3} - 16 a^{2} b c d^{2} - 16 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{6} c^{2} d^{4} - 8 a^{5} b c^{3} d^{3} + 12 a^{4} b^{2} c^{4} d^{2} - 8 a^{3} b^{3} c^{5} d + 2 a^{2} b^{4} c^{6} + x^{4} \cdot \left (2 a^{4} b^{2} d^{6} - 8 a^{3} b^{3} c d^{5} + 12 a^{2} b^{4} c^{2} d^{4} - 8 a b^{5} c^{3} d^{3} + 2 b^{6} c^{4} d^{2}\right ) + x^{3} \cdot \left (4 a^{5} b d^{6} - 12 a^{4} b^{2} c d^{5} + 8 a^{3} b^{3} c^{2} d^{4} + 8 a^{2} b^{4} c^{3} d^{3} - 12 a b^{5} c^{4} d^{2} + 4 b^{6} c^{5} d\right ) + x^{2} \cdot \left (2 a^{6} d^{6} - 18 a^{4} b^{2} c^{2} d^{4} + 32 a^{3} b^{3} c^{3} d^{3} - 18 a^{2} b^{4} c^{4} d^{2} + 2 b^{6} c^{6}\right ) + x \left (4 a^{6} c d^{5} - 12 a^{5} b c^{2} d^{4} + 8 a^{4} b^{2} c^{3} d^{3} + 8 a^{3} b^{3} c^{4} d^{2} - 12 a^{2} b^{4} c^{5} d + 4 a b^{5} c^{6}\right )} \]
-3*b*d*(a*d + b*c)*log(x + (-3*a**6*b*d**7*(a*d + b*c)/(a*d - b*c)**5 + 18 *a**5*b**2*c*d**6*(a*d + b*c)/(a*d - b*c)**5 - 45*a**4*b**3*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 + 60*a**3*b**4*c**3*d**4*(a*d + b*c)/(a*d - b*c)**5 - 45*a**2*b**5*c**4*d**3*(a*d + b*c)/(a*d - b*c)**5 + 3*a**2*b*d**3 + 18* a*b**6*c**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 6*a*b**2*c*d**2 - 3*b**7*c** 6*d*(a*d + b*c)/(a*d - b*c)**5 + 3*b**3*c**2*d)/(6*a*b**2*d**3 + 6*b**3*c* d**2))/(a*d - b*c)**5 + 3*b*d*(a*d + b*c)*log(x + (3*a**6*b*d**7*(a*d + b* c)/(a*d - b*c)**5 - 18*a**5*b**2*c*d**6*(a*d + b*c)/(a*d - b*c)**5 + 45*a* *4*b**3*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 - 60*a**3*b**4*c**3*d**4*(a*d + b*c)/(a*d - b*c)**5 + 45*a**2*b**5*c**4*d**3*(a*d + b*c)/(a*d - b*c)**5 + 3*a**2*b*d**3 - 18*a*b**6*c**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 6*a*b* *2*c*d**2 + 3*b**7*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 3*b**3*c**2*d)/(6*a *b**2*d**3 + 6*b**3*c*d**2))/(a*d - b*c)**5 + (-a**3*c*d**2 - 10*a**2*b*c* *2*d - a*b**2*c**3 + x**3*(-6*a*b**2*d**3 - 6*b**3*c*d**2) + x**2*(-9*a**2 *b*d**3 - 18*a*b**2*c*d**2 - 9*b**3*c**2*d) + x*(-2*a**3*d**3 - 16*a**2*b* c*d**2 - 16*a*b**2*c**2*d - 2*b**3*c**3))/(2*a**6*c**2*d**4 - 8*a**5*b*c** 3*d**3 + 12*a**4*b**2*c**4*d**2 - 8*a**3*b**3*c**5*d + 2*a**2*b**4*c**6 + x**4*(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b**4*c**2*d**4 - 8*a *b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**3*(4*a**5*b*d**6 - 12*a**4*b**2*c *d**5 + 8*a**3*b**3*c**2*d**4 + 8*a**2*b**4*c**3*d**3 - 12*a*b**5*c**4*...
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (153) = 306\).
Time = 0.21 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.99 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {3 \, {\left (b^{2} c d + a b d^{2}\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {3 \, {\left (b^{2} c d + a b d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + a^{3} c d^{2} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} + 8 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{2 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
-3*(b^2*c*d + a*b*d^2)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3* c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) + 3*(b^2*c*d + a*b *d^2)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3* b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 1/2*(a*b^2*c^3 + 10*a^2*b*c^2*d + a^3*c*d^2 + 6*(b^3*c*d^2 + a*b^2*d^3)*x^3 + 9*(b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 2*(b^3*c^3 + 8*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*x )/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a ^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^ 3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^ 3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (153) = 306\).
Time = 0.27 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.47 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {3 \, {\left (b^{3} c d + a b^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} + \frac {3 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {6 \, b^{3} c d^{2} x^{3} + 6 \, a b^{2} d^{3} x^{3} + 9 \, b^{3} c^{2} d x^{2} + 18 \, a b^{2} c d^{2} x^{2} + 9 \, a^{2} b d^{3} x^{2} + 2 \, b^{3} c^{3} x + 16 \, a b^{2} c^{2} d x + 16 \, a^{2} b c d^{2} x + 2 \, a^{3} d^{3} x + a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + a^{3} c d^{2}}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]
-3*(b^3*c*d + a*b^2*d^2)*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a ^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) + 3*(b^ 2*c*d^2 + a*b*d^3)*log(abs(d*x + c))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 + 10*a^2 *b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6) - 1/2*(6*b^3* c*d^2*x^3 + 6*a*b^2*d^3*x^3 + 9*b^3*c^2*d*x^2 + 18*a*b^2*c*d^2*x^2 + 9*a^2 *b*d^3*x^2 + 2*b^3*c^3*x + 16*a*b^2*c^2*d*x + 16*a^2*b*c*d^2*x + 2*a^3*d^3 *x + a*b^2*c^3 + 10*a^2*b*c^2*d + a^3*c*d^2)/((b^4*c^4 - 4*a*b^3*c^3*d + 6 *a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(b*d*x^2 + b*c*x + a*d*x + a*c )^2)
Time = 0.65 (sec) , antiderivative size = 495, normalized size of antiderivative = 3.15 \[ \int \frac {x}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6\,b\,d\,\mathrm {atanh}\left (\frac {\left (a\,d+b\,c+2\,b\,d\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^5}-\frac {\frac {a^3\,c\,d^2+10\,a^2\,b\,c^2\,d+a\,b^2\,c^3}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {9\,x^2\,\left (a^2\,b\,d^3+2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {3\,b^2\,d^2\,x^3\,\left (a\,d+b\,c\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4} \]
(6*b*d*atanh(((a*d + b*c + 2*b*d*x)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(a*d - b*c)^5)*(a*d + b*c))/(a*d - b*c) ^5 - ((a*b^2*c^3 + a^3*c*d^2 + 10*a^2*b*c^2*d)/(2*(a^4*d^4 + b^4*c^4 + 6*a ^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (9*x^2*(a^2*b*d^3 + b^3 *c^2*d + 2*a*b^2*c*d^2))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b ^3*c^3*d - 4*a^3*b*c*d^3)) + (x*(a*d + b*c)*(a^2*d^2 + b^2*c^2 + 7*a*b*c*d ))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3) + (3*b^2*d^2*x^3*(a*d + b*c))/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4* a*b^3*c^3*d - 4*a^3*b*c*d^3))/(x*(2*a*b*c^2 + 2*a^2*c*d) + x^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*a*b*d^2 + 2*b^2*c*d) + a^2*c^2 + b^2*d^2*x^4 )