Integrand size = 18, antiderivative size = 198 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {(3 b c+2 a d) x}{b^3 d^4}+\frac {x^2}{2 b^2 d^3}-\frac {a^6}{b^4 (b c-a d)^3 (a+b x)}-\frac {c^6}{2 d^5 (b c-a d)^2 (c+d x)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (b c-a d)^3 (c+d x)}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^4} \]
-(2*a*d+3*b*c)*x/b^3/d^4+1/2*x^2/b^2/d^3-a^6/b^4/(-a*d+b*c)^3/(b*x+a)-1/2* c^6/d^5/(-a*d+b*c)^2/(d*x+c)^2+2*c^5*(-3*a*d+2*b*c)/d^5/(-a*d+b*c)^3/(d*x+ c)-3*a^5*(-a*d+2*b*c)*ln(b*x+a)/b^4/(-a*d+b*c)^4+3*c^4*(5*a^2*d^2-6*a*b*c* d+2*b^2*c^2)*ln(d*x+c)/d^5/(-a*d+b*c)^4
Time = 0.15 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {(3 b c+2 a d) x}{b^3 d^4}+\frac {x^2}{2 b^2 d^3}-\frac {a^6}{b^4 (b c-a d)^3 (a+b x)}-\frac {c^6}{2 d^5 (b c-a d)^2 (c+d x)^2}+\frac {-4 b c^6+6 a c^5 d}{d^5 (-b c+a d)^3 (c+d x)}+\frac {3 a^5 (-2 b c+a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^4} \]
-(((3*b*c + 2*a*d)*x)/(b^3*d^4)) + x^2/(2*b^2*d^3) - a^6/(b^4*(b*c - a*d)^ 3*(a + b*x)) - c^6/(2*d^5*(b*c - a*d)^2*(c + d*x)^2) + (-4*b*c^6 + 6*a*c^5 *d)/(d^5*(-(b*c) + a*d)^3*(c + d*x)) + (3*a^5*(-2*b*c + a*d)*Log[a + b*x]) /(b^4*(b*c - a*d)^4) + (3*c^4*(2*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Log[c + d*x])/(d^5*(b*c - a*d)^4)
Time = 0.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.
| \(\displaystyle \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^6}{b^3 (a+b x)^2 (b c-a d)^3}+\frac {3 a^5 (a d-2 b c)}{b^3 (a+b x) (b c-a d)^4}+\frac {3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right )}{d^4 (c+d x) (a d-b c)^4}+\frac {-2 a d-3 b c}{b^3 d^4}+\frac {c^6}{d^4 (c+d x)^3 (a d-b c)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^4 (c+d x)^2 (a d-b c)^3}+\frac {x}{b^2 d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^6}{b^4 (a+b x) (b c-a d)^3}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac {x (2 a d+3 b c)}{b^3 d^4}-\frac {c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (c+d x) (b c-a d)^3}+\frac {x^2}{2 b^2 d^3}\) |
-(((3*b*c + 2*a*d)*x)/(b^3*d^4)) + x^2/(2*b^2*d^3) - a^6/(b^4*(b*c - a*d)^ 3*(a + b*x)) - c^6/(2*d^5*(b*c - a*d)^2*(c + d*x)^2) + (2*c^5*(2*b*c - 3*a *d))/(d^5*(b*c - a*d)^3*(c + d*x)) - (3*a^5*(2*b*c - a*d)*Log[a + b*x])/(b ^4*(b*c - a*d)^4) + (3*c^4*(2*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Log[c + d*x ])/(d^5*(b*c - a*d)^4)
3.3.91.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.53 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {-\frac {1}{2} b d \,x^{2}+2 a d x +3 b c x}{b^{3} d^{4}}-\frac {c^{6}}{2 d^{5} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{4}}+\frac {2 c^{5} \left (3 a d -2 b c \right )}{d^{5} \left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {a^{6}}{b^{4} \left (a d -b c \right )^{3} \left (b x +a \right )}+\frac {3 a^{5} \left (a d -2 b c \right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{4}}\) | \(190\) |
| norman | \(\frac {\frac {\left (3 a^{6} d^{6}-2 a^{4} b^{2} c^{2} d^{4}-6 a^{3} b^{3} c^{3} d^{3}+20 a \,b^{5} c^{5} d -12 b^{6} c^{6}\right ) x^{2}}{d^{4} b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {x^{5}}{2 b d}-\frac {\left (3 a d +4 b c \right ) x^{4}}{2 b^{2} d^{2}}+\frac {c \left (12 a^{6} d^{6}-20 a^{4} b^{2} c^{2} d^{4}-9 a^{3} b^{3} c^{3} d^{3}+39 a^{2} b^{4} c^{4} d^{2}+8 a \,b^{5} c^{5} d -18 b^{6} c^{6}\right ) x}{2 d^{5} b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (6 a^{5} d^{5}-13 a^{3} b^{2} c^{2} d^{3}-a^{2} b^{3} c^{3} d^{2}+32 a \,b^{4} c^{4} d -18 b^{5} c^{5}\right )}{2 d^{5} b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {3 a^{5} \left (a d -2 b c \right ) \ln \left (b x +a \right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{4}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(535\) |
| risch | \(\frac {x^{2}}{2 b^{2} d^{3}}-\frac {2 a x}{d^{3} b^{3}}-\frac {3 c x}{d^{4} b^{2}}+\frac {\frac {\left (a^{6} d^{6}+6 a \,b^{5} c^{5} d -4 b^{6} c^{6}\right ) x^{2}}{b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (4 a^{6} d^{6}+12 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d -7 b^{6} c^{6}\right ) x}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b d}+\frac {a \,c^{2} \left (2 a^{5} d^{5}+11 a \,b^{4} c^{4} d -7 b^{5} c^{5}\right )}{2 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d}}{b^{3} d^{4} \left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {15 c^{4} \ln \left (d x +c \right ) a^{2}}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {18 c^{5} \ln \left (d x +c \right ) a b}{d^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {6 c^{6} \ln \left (d x +c \right ) b^{2}}{d^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {3 a^{6} \ln \left (-b x -a \right ) d}{b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {6 a^{5} \ln \left (-b x -a \right ) c}{b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(633\) |
| parallelrisch | \(\frac {30 \ln \left (d x +c \right ) x^{2} a^{3} b^{4} c^{4} d^{4}+24 \ln \left (d x +c \right ) x^{2} a^{2} b^{5} c^{5} d^{3}-60 \ln \left (d x +c \right ) x^{2} a \,b^{6} c^{6} d^{2}-18 \ln \left (b x +a \right ) x \,a^{6} b \,c^{2} d^{6}-12 \ln \left (b x +a \right ) x \,a^{5} b^{2} c^{3} d^{5}-6 a^{6} b \,c^{3} d^{5}-13 a^{5} b^{2} c^{4} d^{4}+12 a^{4} b^{3} c^{5} d^{3}+33 a^{3} b^{4} c^{6} d^{2}-50 a^{2} b^{5} c^{7} d +18 x \,b^{7} c^{8}+6 a^{7} c^{2} d^{6}+18 a \,b^{6} c^{8}+48 x \,a^{3} b^{4} c^{5} d^{3}-31 x \,a^{2} b^{5} c^{6} d^{2}-26 x a \,b^{6} c^{7} d +6 \ln \left (b x +a \right ) x^{3} a^{6} b \,d^{8}+12 \ln \left (d x +c \right ) x^{3} b^{7} c^{6} d^{2}+24 \ln \left (d x +c \right ) x^{2} b^{7} c^{7} d +12 \ln \left (b x +a \right ) x \,a^{7} c \,d^{7}-12 \ln \left (b x +a \right ) a^{6} b \,c^{3} d^{5}+30 \ln \left (d x +c \right ) a^{3} b^{4} c^{6} d^{2}-36 \ln \left (d x +c \right ) a^{2} b^{5} c^{7} d -4 x^{5} a^{3} b^{4} c \,d^{7}+6 x^{5} a^{2} b^{5} c^{2} d^{6}-4 x^{5} a \,b^{6} c^{3} d^{5}+8 x^{4} a^{4} b^{3} c \,d^{7}-2 x^{4} a^{3} b^{4} c^{2} d^{6}-12 x^{4} a^{2} b^{5} c^{3} d^{5}+13 x^{4} a \,b^{6} c^{4} d^{4}-6 x^{2} a^{6} b c \,d^{7}-4 x^{2} a^{5} b^{2} c^{2} d^{6}-8 x^{2} a^{4} b^{3} c^{3} d^{5}+12 x^{2} a^{3} b^{4} c^{4} d^{4}+40 x^{2} a^{2} b^{5} c^{5} d^{3}-64 x^{2} a \,b^{6} c^{6} d^{2}-12 x \,a^{6} b \,c^{2} d^{6}-20 x \,a^{5} b^{2} c^{3} d^{5}+11 x \,a^{4} b^{3} c^{4} d^{4}+x^{5} a^{4} b^{3} d^{8}+x^{5} b^{7} c^{4} d^{4}-3 x^{4} a^{5} b^{2} d^{8}-4 x^{4} b^{7} c^{5} d^{3}+24 x^{2} b^{7} c^{7} d +12 x \,a^{7} c \,d^{7}+6 \ln \left (b x +a \right ) x^{2} a^{7} d^{8}+12 \ln \left (d x +c \right ) x \,b^{7} c^{8}+6 \ln \left (b x +a \right ) a^{7} c^{2} d^{6}+12 \ln \left (d x +c \right ) a \,b^{6} c^{8}+6 x^{2} a^{7} d^{8}+60 \ln \left (d x +c \right ) x \,a^{3} b^{4} c^{5} d^{3}-42 \ln \left (d x +c \right ) x \,a^{2} b^{5} c^{6} d^{2}-12 \ln \left (d x +c \right ) x a \,b^{6} c^{7} d -12 \ln \left (b x +a \right ) x^{3} a^{5} b^{2} c \,d^{7}+30 \ln \left (d x +c \right ) x^{3} a^{2} b^{5} c^{4} d^{4}-36 \ln \left (d x +c \right ) x^{3} a \,b^{6} c^{5} d^{3}-24 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} c^{2} d^{6}}{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 (d x +c \right )^{2} \left (b x +a \right ) b^{4} d^{5}}\) | \(994\) |
-1/b^3/d^4*(-1/2*b*d*x^2+2*a*d*x+3*b*c*x)-1/2/d^5*c^6/(a*d-b*c)^2/(d*x+c)^ 2+3/d^5*c^4*(5*a^2*d^2-6*a*b*c*d+2*b^2*c^2)/(a*d-b*c)^4*ln(d*x+c)+2/d^5*c^ 5*(3*a*d-2*b*c)/(a*d-b*c)^3/(d*x+c)+1/b^4*a^6/(a*d-b*c)^3/(b*x+a)+3/b^4*a^ 5*(a*d-2*b*c)/(a*d-b*c)^4*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1081 vs. \(2 (194) = 388\).
Time = 0.28 (sec) , antiderivative size = 1081, normalized size of antiderivative = 5.46 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=\frac {7 \, a b^{6} c^{8} - 18 \, a^{2} b^{5} c^{7} d + 11 \, a^{3} b^{4} c^{6} d^{2} - 2 \, a^{6} b c^{3} d^{5} + 2 \, a^{7} c^{2} d^{6} + {\left (b^{7} c^{4} d^{4} - 4 \, a b^{6} c^{3} d^{5} + 6 \, a^{2} b^{5} c^{2} d^{6} - 4 \, a^{3} b^{4} c d^{7} + a^{4} b^{3} d^{8}\right )} x^{5} - {\left (4 \, b^{7} c^{5} d^{3} - 13 \, a b^{6} c^{4} d^{4} + 12 \, a^{2} b^{5} c^{3} d^{5} + 2 \, a^{3} b^{4} c^{2} d^{6} - 8 \, a^{4} b^{3} c d^{7} + 3 \, a^{5} b^{2} d^{8}\right )} x^{4} - {\left (11 \, b^{7} c^{6} d^{2} - 32 \, a b^{6} c^{5} d^{3} + 22 \, a^{2} b^{5} c^{4} d^{4} + 12 \, a^{3} b^{4} c^{3} d^{5} - 13 \, a^{4} b^{3} c^{2} d^{6} - 4 \, a^{5} b^{2} c d^{7} + 4 \, a^{6} b d^{8}\right )} x^{3} + {\left (2 \, b^{7} c^{7} d - 11 \, a b^{6} c^{6} d^{2} + 28 \, a^{2} b^{5} c^{5} d^{3} - 34 \, a^{3} b^{4} c^{4} d^{4} + 6 \, a^{4} b^{3} c^{3} d^{5} + 17 \, a^{5} b^{2} c^{2} d^{6} - 10 \, a^{6} b c d^{7} + 2 \, a^{7} d^{8}\right )} x^{2} + {\left (7 \, b^{7} c^{8} - 16 \, a b^{6} c^{7} d + 11 \, a^{2} b^{5} c^{6} d^{2} - 8 \, a^{3} b^{4} c^{5} d^{3} + 10 \, a^{5} b^{2} c^{3} d^{5} - 8 \, a^{6} b c^{2} d^{6} + 4 \, a^{7} c d^{7}\right )} x - 6 \, {\left (2 \, a^{6} b c^{3} d^{5} - a^{7} c^{2} d^{6} + {\left (2 \, a^{5} b^{2} c d^{7} - a^{6} b d^{8}\right )} x^{3} + {\left (4 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + {\left (2 \, a^{5} b^{2} c^{3} d^{5} + 3 \, a^{6} b c^{2} d^{6} - 2 \, a^{7} c d^{7}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (2 \, a b^{6} c^{8} - 6 \, a^{2} b^{5} c^{7} d + 5 \, a^{3} b^{4} c^{6} d^{2} + {\left (2 \, b^{7} c^{6} d^{2} - 6 \, a b^{6} c^{5} d^{3} + 5 \, a^{2} b^{5} c^{4} d^{4}\right )} x^{3} + {\left (4 \, b^{7} c^{7} d - 10 \, a b^{6} c^{6} d^{2} + 4 \, a^{2} b^{5} c^{5} d^{3} + 5 \, a^{3} b^{4} c^{4} d^{4}\right )} x^{2} + {\left (2 \, b^{7} c^{8} - 2 \, a b^{6} c^{7} d - 7 \, a^{2} b^{5} c^{6} d^{2} + 10 \, a^{3} b^{4} c^{5} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{8} c^{6} d^{5} - 4 \, a^{2} b^{7} c^{5} d^{6} + 6 \, a^{3} b^{6} c^{4} d^{7} - 4 \, a^{4} b^{5} c^{3} d^{8} + a^{5} b^{4} c^{2} d^{9} + {\left (b^{9} c^{4} d^{7} - 4 \, a b^{8} c^{3} d^{8} + 6 \, a^{2} b^{7} c^{2} d^{9} - 4 \, a^{3} b^{6} c d^{10} + a^{4} b^{5} d^{11}\right )} x^{3} + {\left (2 \, b^{9} c^{5} d^{6} - 7 \, a b^{8} c^{4} d^{7} + 8 \, a^{2} b^{7} c^{3} d^{8} - 2 \, a^{3} b^{6} c^{2} d^{9} - 2 \, a^{4} b^{5} c d^{10} + a^{5} b^{4} d^{11}\right )} x^{2} + {\left (b^{9} c^{6} d^{5} - 2 \, a b^{8} c^{5} d^{6} - 2 \, a^{2} b^{7} c^{4} d^{7} + 8 \, a^{3} b^{6} c^{3} d^{8} - 7 \, a^{4} b^{5} c^{2} d^{9} + 2 \, a^{5} b^{4} c d^{10}\right )} x\right )}} \]
1/2*(7*a*b^6*c^8 - 18*a^2*b^5*c^7*d + 11*a^3*b^4*c^6*d^2 - 2*a^6*b*c^3*d^5 + 2*a^7*c^2*d^6 + (b^7*c^4*d^4 - 4*a*b^6*c^3*d^5 + 6*a^2*b^5*c^2*d^6 - 4* a^3*b^4*c*d^7 + a^4*b^3*d^8)*x^5 - (4*b^7*c^5*d^3 - 13*a*b^6*c^4*d^4 + 12* a^2*b^5*c^3*d^5 + 2*a^3*b^4*c^2*d^6 - 8*a^4*b^3*c*d^7 + 3*a^5*b^2*d^8)*x^4 - (11*b^7*c^6*d^2 - 32*a*b^6*c^5*d^3 + 22*a^2*b^5*c^4*d^4 + 12*a^3*b^4*c^ 3*d^5 - 13*a^4*b^3*c^2*d^6 - 4*a^5*b^2*c*d^7 + 4*a^6*b*d^8)*x^3 + (2*b^7*c ^7*d - 11*a*b^6*c^6*d^2 + 28*a^2*b^5*c^5*d^3 - 34*a^3*b^4*c^4*d^4 + 6*a^4* b^3*c^3*d^5 + 17*a^5*b^2*c^2*d^6 - 10*a^6*b*c*d^7 + 2*a^7*d^8)*x^2 + (7*b^ 7*c^8 - 16*a*b^6*c^7*d + 11*a^2*b^5*c^6*d^2 - 8*a^3*b^4*c^5*d^3 + 10*a^5*b ^2*c^3*d^5 - 8*a^6*b*c^2*d^6 + 4*a^7*c*d^7)*x - 6*(2*a^6*b*c^3*d^5 - a^7*c ^2*d^6 + (2*a^5*b^2*c*d^7 - a^6*b*d^8)*x^3 + (4*a^5*b^2*c^2*d^6 - a^7*d^8) *x^2 + (2*a^5*b^2*c^3*d^5 + 3*a^6*b*c^2*d^6 - 2*a^7*c*d^7)*x)*log(b*x + a) + 6*(2*a*b^6*c^8 - 6*a^2*b^5*c^7*d + 5*a^3*b^4*c^6*d^2 + (2*b^7*c^6*d^2 - 6*a*b^6*c^5*d^3 + 5*a^2*b^5*c^4*d^4)*x^3 + (4*b^7*c^7*d - 10*a*b^6*c^6*d^ 2 + 4*a^2*b^5*c^5*d^3 + 5*a^3*b^4*c^4*d^4)*x^2 + (2*b^7*c^8 - 2*a*b^6*c^7* d - 7*a^2*b^5*c^6*d^2 + 10*a^3*b^4*c^5*d^3)*x)*log(d*x + c))/(a*b^8*c^6*d^ 5 - 4*a^2*b^7*c^5*d^6 + 6*a^3*b^6*c^4*d^7 - 4*a^4*b^5*c^3*d^8 + a^5*b^4*c^ 2*d^9 + (b^9*c^4*d^7 - 4*a*b^8*c^3*d^8 + 6*a^2*b^7*c^2*d^9 - 4*a^3*b^6*c*d ^10 + a^4*b^5*d^11)*x^3 + (2*b^9*c^5*d^6 - 7*a*b^8*c^4*d^7 + 8*a^2*b^7*c^3 *d^8 - 2*a^3*b^6*c^2*d^9 - 2*a^4*b^5*c*d^10 + a^5*b^4*d^11)*x^2 + (b^9*...
Timed out. \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (194) = 388\).
Time = 0.24 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.76 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {3 \, {\left (2 \, a^{5} b c - a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}} + \frac {3 \, {\left (2 \, b^{2} c^{6} - 6 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}} + \frac {7 \, a b^{5} c^{7} - 11 \, a^{2} b^{4} c^{6} d - 2 \, a^{6} c^{2} d^{5} + 2 \, {\left (4 \, b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} - a^{6} d^{7}\right )} x^{2} + {\left (7 \, b^{6} c^{7} - 3 \, a b^{5} c^{6} d - 12 \, a^{2} b^{4} c^{5} d^{2} - 4 \, a^{6} c d^{6}\right )} x}{2 \, {\left (a b^{7} c^{5} d^{5} - 3 \, a^{2} b^{6} c^{4} d^{6} + 3 \, a^{3} b^{5} c^{3} d^{7} - a^{4} b^{4} c^{2} d^{8} + {\left (b^{8} c^{3} d^{7} - 3 \, a b^{7} c^{2} d^{8} + 3 \, a^{2} b^{6} c d^{9} - a^{3} b^{5} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{4} d^{6} - 5 \, a b^{7} c^{3} d^{7} + 3 \, a^{2} b^{6} c^{2} d^{8} + a^{3} b^{5} c d^{9} - a^{4} b^{4} d^{10}\right )} x^{2} + {\left (b^{8} c^{5} d^{5} - a b^{7} c^{4} d^{6} - 3 \, a^{2} b^{6} c^{3} d^{7} + 5 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9}\right )} x\right )}} + \frac {b d x^{2} - 2 \, {\left (3 \, b c + 2 \, a d\right )} x}{2 \, b^{3} d^{4}} \]
-3*(2*a^5*b*c - a^6*d)*log(b*x + a)/(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c ^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4) + 3*(2*b^2*c^6 - 6*a*b*c^5*d + 5*a ^2*c^4*d^2)*log(d*x + c)/(b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^ 7 - 4*a^3*b*c*d^8 + a^4*d^9) + 1/2*(7*a*b^5*c^7 - 11*a^2*b^4*c^6*d - 2*a^6 *c^2*d^5 + 2*(4*b^6*c^6*d - 6*a*b^5*c^5*d^2 - a^6*d^7)*x^2 + (7*b^6*c^7 - 3*a*b^5*c^6*d - 12*a^2*b^4*c^5*d^2 - 4*a^6*c*d^6)*x)/(a*b^7*c^5*d^5 - 3*a^ 2*b^6*c^4*d^6 + 3*a^3*b^5*c^3*d^7 - a^4*b^4*c^2*d^8 + (b^8*c^3*d^7 - 3*a*b ^7*c^2*d^8 + 3*a^2*b^6*c*d^9 - a^3*b^5*d^10)*x^3 + (2*b^8*c^4*d^6 - 5*a*b^ 7*c^3*d^7 + 3*a^2*b^6*c^2*d^8 + a^3*b^5*c*d^9 - a^4*b^4*d^10)*x^2 + (b^8*c ^5*d^5 - a*b^7*c^4*d^6 - 3*a^2*b^6*c^3*d^7 + 5*a^3*b^5*c^2*d^8 - 2*a^4*b^4 *c*d^9)*x) + 1/2*(b*d*x^2 - 2*(3*b*c + 2*a*d)*x)/(b^3*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (194) = 388\).
Time = 0.27 (sec) , antiderivative size = 621, normalized size of antiderivative = 3.14 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^{6} b^{5}}{{\left (b^{12} c^{3} - 3 \, a b^{11} c^{2} d + 3 \, a^{2} b^{10} c d^{2} - a^{3} b^{9} d^{3}\right )} {\left (b x + a\right )}} + \frac {3 \, {\left (2 \, b^{3} c^{6} - 6 \, a b^{2} c^{5} d + 5 \, a^{2} b c^{4} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{5} - 4 \, a b^{4} c^{3} d^{6} + 6 \, a^{2} b^{3} c^{2} d^{7} - 4 \, a^{3} b^{2} c d^{8} + a^{4} b d^{9}} - \frac {3 \, {\left (2 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} d^{5}} + \frac {{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7} - \frac {4 \, {\left (b^{6} c^{5} d^{2} - 2 \, a b^{5} c^{4} d^{3} - 2 \, a^{2} b^{4} c^{3} d^{4} + 8 \, a^{3} b^{3} c^{2} d^{5} - 7 \, a^{4} b^{2} c d^{6} + 2 \, a^{5} b d^{7}\right )}}{{\left (b x + a\right )} b} - \frac {18 \, b^{8} c^{6} d - 54 \, a b^{7} c^{5} d^{2} + 45 \, a^{2} b^{6} c^{4} d^{3} + 20 \, a^{3} b^{5} c^{3} d^{4} - 75 \, a^{4} b^{4} c^{2} d^{5} + 54 \, a^{5} b^{3} c d^{6} - 13 \, a^{6} b^{2} d^{7}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {6 \, {\left (2 \, b^{10} c^{7} - 8 \, a b^{9} c^{6} d + 11 \, a^{2} b^{8} c^{5} d^{2} - 5 \, a^{3} b^{7} c^{4} d^{3} - 5 \, a^{4} b^{6} c^{3} d^{4} + 9 \, a^{5} b^{5} c^{2} d^{5} - 5 \, a^{6} b^{4} c d^{6} + a^{7} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{2}}{2 \, {\left (b c - a d\right )}^{4} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{4}} \]
-a^6*b^5/((b^12*c^3 - 3*a*b^11*c^2*d + 3*a^2*b^10*c*d^2 - a^3*b^9*d^3)*(b* x + a)) + 3*(2*b^3*c^6 - 6*a*b^2*c^5*d + 5*a^2*b*c^4*d^2)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4*d^5 - 4*a*b^4*c^3*d^6 + 6*a^2*b^3*c^2 *d^7 - 4*a^3*b^2*c*d^8 + a^4*b*d^9) - 3*(2*b^2*c^2 + 2*a*b*c*d + a^2*d^2)* log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^4*d^5) + 1/2*(b^4*c^4*d^3 - 4*a* b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7 - 4*(b^6*c^5*d^2 - 2*a*b^5*c^4*d^3 - 2*a^2*b^4*c^3*d^4 + 8*a^3*b^3*c^2*d^5 - 7*a^4*b^2*c*d ^6 + 2*a^5*b*d^7)/((b*x + a)*b) - (18*b^8*c^6*d - 54*a*b^7*c^5*d^2 + 45*a^ 2*b^6*c^4*d^3 + 20*a^3*b^5*c^3*d^4 - 75*a^4*b^4*c^2*d^5 + 54*a^5*b^3*c*d^6 - 13*a^6*b^2*d^7)/((b*x + a)^2*b^2) - 6*(2*b^10*c^7 - 8*a*b^9*c^6*d + 11* a^2*b^8*c^5*d^2 - 5*a^3*b^7*c^4*d^3 - 5*a^4*b^6*c^3*d^4 + 9*a^5*b^5*c^2*d^ 5 - 5*a^6*b^4*c*d^6 + a^7*b^3*d^7)/((b*x + a)^3*b^3))*(b*x + a)^2/((b*c - a*d)^4*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^4)
Time = 0.80 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.57 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {2\,a^6\,c^2\,d^5+11\,a^2\,b^4\,c^6\,d-7\,a\,b^5\,c^7}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x^2\,\left (a^6\,d^6+6\,a\,b^5\,c^5\,d-4\,b^6\,c^6\right )}{b\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (4\,a^6\,c\,d^6+12\,a^2\,b^4\,c^5\,d^2+3\,a\,b^5\,c^6\,d-7\,b^6\,c^7\right )}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^2\,\left (2\,c\,b^4\,d^5+a\,b^3\,d^6\right )+x\,\left (b^4\,c^2\,d^4+2\,a\,b^3\,c\,d^5\right )+b^4\,d^6\,x^3+a\,b^3\,c^2\,d^4}+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^2\,c^4\,d^2-18\,a\,b\,c^5\,d+6\,b^2\,c^6\right )}{a^4\,d^9-4\,a^3\,b\,c\,d^8+6\,a^2\,b^2\,c^2\,d^7-4\,a\,b^3\,c^3\,d^6+b^4\,c^4\,d^5}+\frac {x^2}{2\,b^2\,d^3}+\frac {\ln \left (a+b\,x\right )\,\left (3\,a^6\,d-6\,a^5\,b\,c\right )}{a^4\,b^4\,d^4-4\,a^3\,b^5\,c\,d^3+6\,a^2\,b^6\,c^2\,d^2-4\,a\,b^7\,c^3\,d+b^8\,c^4}-\frac {x\,\left (2\,a\,d+3\,b\,c\right )}{b^3\,d^4} \]
((2*a^6*c^2*d^5 - 7*a*b^5*c^7 + 11*a^2*b^4*c^6*d)/(2*b*d*(a^3*d^3 - b^3*c^ 3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x^2*(a^6*d^6 - 4*b^6*c^6 + 6*a*b^5* c^5*d))/(b*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(4*a^ 6*c*d^6 - 7*b^6*c^7 + 12*a^2*b^4*c^5*d^2 + 3*a*b^5*c^6*d))/(2*b*d*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(x^2*(a*b^3*d^6 + 2*b^4*c*d^ 5) + x*(b^4*c^2*d^4 + 2*a*b^3*c*d^5) + b^4*d^6*x^3 + a*b^3*c^2*d^4) + (log (c + d*x)*(6*b^2*c^6 + 15*a^2*c^4*d^2 - 18*a*b*c^5*d))/(a^4*d^9 + b^4*c^4* d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c*d^8) + x^2/(2*b^2*d^ 3) + (log(a + b*x)*(3*a^6*d - 6*a^5*b*c))/(b^8*c^4 + a^4*b^4*d^4 - 4*a^3*b ^5*c*d^3 + 6*a^2*b^6*c^2*d^2 - 4*a*b^7*c^3*d) - (x*(2*a*d + 3*b*c))/(b^3*d ^4)