Integrand size = 18, antiderivative size = 179 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) x}{b^4 d^4}-\frac {(b c+a d) x^2}{b^3 d^3}+\frac {x^3}{3 b^2 d^2}-\frac {a^6}{b^5 (b c-a d)^2 (a+b x)}-\frac {c^6}{d^5 (b c-a d)^2 (c+d x)}-\frac {2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}-\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3} \]
(3*a^2*d^2+4*a*b*c*d+3*b^2*c^2)*x/b^4/d^4-(a*d+b*c)*x^2/b^3/d^3+1/3*x^3/b^ 2/d^2-a^6/b^5/(-a*d+b*c)^2/(b*x+a)-c^6/d^5/(-a*d+b*c)^2/(d*x+c)-2*a^5*(-2* a*d+3*b*c)*ln(b*x+a)/b^5/(-a*d+b*c)^3-2*c^5*(-3*a*d+2*b*c)*ln(d*x+c)/d^5/( -a*d+b*c)^3
Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) x}{b^4 d^4}-\frac {(b c+a d) x^2}{b^3 d^3}+\frac {x^3}{3 b^2 d^2}-\frac {a^6}{b^5 (b c-a d)^2 (a+b x)}-\frac {c^6}{d^5 (b c-a d)^2 (c+d x)}+\frac {2 a^5 (-3 b c+2 a d) \log (a+b x)}{b^5 (b c-a d)^3}+\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (-b c+a d)^3} \]
((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x)/(b^4*d^4) - ((b*c + a*d)*x^2)/(b^3 *d^3) + x^3/(3*b^2*d^2) - a^6/(b^5*(b*c - a*d)^2*(a + b*x)) - c^6/(d^5*(b* c - a*d)^2*(c + d*x)) + (2*a^5*(-3*b*c + 2*a*d)*Log[a + b*x])/(b^5*(b*c - a*d)^3) + (2*c^5*(2*b*c - 3*a*d)*Log[c + d*x])/(d^5*(-(b*c) + a*d)^3)
Time = 0.43 (sec) , antiderivative size = 179, 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.
| \(\displaystyle \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^6}{b^4 (a+b x)^2 (b c-a d)^2}+\frac {2 a^5 (2 a d-3 b c)}{b^4 (a+b x) (b c-a d)^3}+\frac {3 a^2 d^2+4 a b c d+3 b^2 c^2}{b^4 d^4}-\frac {2 x (a d+b c)}{b^3 d^3}+\frac {c^6}{d^4 (c+d x)^2 (a d-b c)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^4 (c+d x) (a d-b c)^3}+\frac {x^2}{b^2 d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^6}{b^5 (a+b x) (b c-a d)^2}-\frac {2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}+\frac {x \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{b^4 d^4}-\frac {x^2 (a d+b c)}{b^3 d^3}-\frac {c^6}{d^5 (c+d x) (b c-a d)^2}-\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3}+\frac {x^3}{3 b^2 d^2}\) |
((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x)/(b^4*d^4) - ((b*c + a*d)*x^2)/(b^3 *d^3) + x^3/(3*b^2*d^2) - a^6/(b^5*(b*c - a*d)^2*(a + b*x)) - c^6/(d^5*(b* c - a*d)^2*(c + d*x)) - (2*a^5*(3*b*c - 2*a*d)*Log[a + b*x])/(b^5*(b*c - a *d)^3) - (2*c^5*(2*b*c - 3*a*d)*Log[c + d*x])/(d^5*(b*c - a*d)^3)
3.3.80.3.1 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 = 0.51 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(\frac {\frac {1}{3} d^{2} x^{3} b^{2}-x^{2} a b \,d^{2}-x^{2} b^{2} c d +3 a^{2} d^{2} x +4 a b c d x +3 b^{2} c^{2} x}{b^{4} d^{4}}-\frac {c^{6}}{d^{5} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {2 c^{5} \left (3 a d -2 b c \right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{3}}-\frac {a^{6}}{b^{5} \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {2 a^{5} \left (2 a d -3 b c \right ) \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )^{3}}\) | \(182\) |
| norman | \(\frac {\frac {x^{5}}{3 b d}-\frac {2 \left (a d +b c \right ) x^{4}}{3 b^{2} d^{2}}+\frac {\left (6 a^{2} d^{2}+7 a b c d +6 b^{2} c^{2}\right ) x^{3}}{3 b^{3} d^{3}}-\frac {\left (4 a^{6} d^{6}-a^{4} b^{2} c^{2} d^{4}-4 a^{3} b^{3} c^{3} d^{3}-a^{2} b^{4} c^{4} d^{2}+4 b^{6} c^{6}\right ) x}{d^{5} b^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (4 a^{5} d^{5}-3 a^{3} b^{2} c^{2} d^{3}-3 a^{2} b^{3} c^{3} d^{2}+4 b^{5} c^{5}\right ) a c}{d^{5} b^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {2 a^{5} \left (2 a d -3 b c \right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{5}}-\frac {2 c^{5} \left (3 a d -2 b c \right ) \ln \left (d x +c \right )}{d^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(372\) |
| risch | \(\frac {x^{3}}{3 b^{2} d^{2}}-\frac {x^{2} a}{d^{2} b^{3}}-\frac {x^{2} c}{d^{3} b^{2}}+\frac {3 a^{2} x}{d^{2} b^{4}}+\frac {4 a c x}{d^{3} b^{3}}+\frac {3 c^{2} x}{d^{4} b^{2}}+\frac {-\frac {\left (a^{6} d^{6}+b^{6} c^{6}\right ) x}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a c \left (a^{5} d^{5}+b^{5} c^{5}\right )}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{b^{4} d^{4} \left (b x +a \right ) \left (d x +c \right )}-\frac {6 c^{5} \ln \left (-d x -c \right ) a}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {4 c^{6} \ln \left (-d x -c \right ) b}{d^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {4 a^{6} \ln \left (b x +a \right ) d}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{5}}+\frac {6 a^{5} \ln \left (b x +a \right ) c}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4}}\) | \(407\) |
| parallelrisch | \(-\frac {12 \ln \left (b x +a \right ) x^{2} a^{6} b \,d^{7}-12 \ln \left (d x +c \right ) x^{2} b^{7} c^{6} d -12 \ln \left (d x +c \right ) a \,b^{6} c^{7}+18 \ln \left (d x +c \right ) x^{2} a \,b^{6} c^{5} d^{2}-6 \ln \left (b x +a \right ) x \,a^{6} b c \,d^{6}-18 \ln \left (b x +a \right ) x \,a^{5} b^{2} c^{2} d^{5}+18 \ln \left (d x +c \right ) x \,a^{2} b^{5} c^{5} d^{2}+6 \ln \left (d x +c \right ) x a \,b^{6} c^{6} d -18 \ln \left (b x +a \right ) a^{6} b \,c^{2} d^{5}+18 \ln \left (d x +c \right ) a^{2} b^{5} c^{6} d +12 a^{7} c \,d^{6}-12 a \,b^{6} c^{7}-12 a^{6} b \,c^{2} d^{5}-9 a^{5} b^{2} c^{3} d^{4}+9 a^{3} b^{4} c^{5} d^{2}+12 a^{2} b^{5} c^{6} d +12 \ln \left (b x +a \right ) x \,a^{7} d^{7}-12 \ln \left (d x +c \right ) x \,b^{7} c^{7}+12 \ln \left (b x +a \right ) a^{7} c \,d^{6}-12 b^{7} c^{7} x +12 a^{7} d^{7} x +12 a \,b^{6} c^{6} d x -12 a^{6} b c \,d^{6} x -3 a^{5} b^{2} c^{2} d^{5} x -9 a^{4} b^{3} c^{3} d^{4} x +9 a^{3} b^{4} c^{4} d^{3} x +3 a^{2} b^{5} c^{5} d^{2} x -4 x^{4} a^{3} b^{4} c \,d^{6}+4 x^{4} a \,b^{6} c^{3} d^{4}+11 x^{3} a^{4} b^{3} c \,d^{6}-3 x^{3} a^{3} b^{4} c^{2} d^{5}+3 x^{3} a^{2} b^{5} c^{3} d^{4}-11 x^{3} a \,b^{6} c^{4} d^{3}+3 x^{5} a^{2} b^{5} c \,d^{6}-3 x^{5} a \,b^{6} c^{2} d^{5}+6 x^{3} b^{7} c^{5} d^{2}-x^{5} a^{3} b^{4} d^{7}+x^{5} b^{7} c^{3} d^{4}+2 x^{4} a^{4} b^{3} d^{7}-2 x^{4} b^{7} c^{4} d^{3}-6 x^{3} a^{5} b^{2} d^{7}-18 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} c \,d^{6}}{3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b^{5} d^{5}}\) | \(689\) |
1/b^4/d^4*(1/3*d^2*x^3*b^2-x^2*a*b*d^2-x^2*b^2*c*d+3*a^2*d^2*x+4*a*b*c*d*x +3*b^2*c^2*x)-1/d^5*c^6/(a*d-b*c)^2/(d*x+c)-2/d^5*c^5*(3*a*d-2*b*c)/(a*d-b *c)^3*ln(d*x+c)-1/b^5*a^6/(a*d-b*c)^2/(b*x+a)-2/b^5*a^5*(2*a*d-3*b*c)/(a*d -b*c)^3*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (177) = 354\).
Time = 0.26 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.91 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {3 \, a b^{6} c^{7} - 3 \, a^{2} b^{5} c^{6} d + 3 \, a^{6} b c^{2} d^{5} - 3 \, a^{7} c d^{6} - {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{5} + 2 \, {\left (b^{7} c^{4} d^{3} - 2 \, a b^{6} c^{3} d^{4} + 2 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{4} - {\left (6 \, b^{7} c^{5} d^{2} - 11 \, a b^{6} c^{4} d^{3} + 3 \, a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 11 \, a^{4} b^{3} c d^{6} - 6 \, a^{5} b^{2} d^{7}\right )} x^{3} - 9 \, {\left (b^{7} c^{6} d - a b^{6} c^{5} d^{2} - a^{2} b^{5} c^{4} d^{3} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{2} + 3 \, {\left (b^{7} c^{7} - 4 \, a b^{6} c^{6} d + 5 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} x + 6 \, {\left (3 \, a^{6} b c^{2} d^{5} - 2 \, a^{7} c d^{6} + {\left (3 \, a^{5} b^{2} c d^{6} - 2 \, a^{6} b d^{7}\right )} x^{2} + {\left (3 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - 2 \, a^{7} d^{7}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (2 \, a b^{6} c^{7} - 3 \, a^{2} b^{5} c^{6} d + {\left (2 \, b^{7} c^{6} d - 3 \, a b^{6} c^{5} d^{2}\right )} x^{2} + {\left (2 \, b^{7} c^{7} - a b^{6} c^{6} d - 3 \, a^{2} b^{5} c^{5} d^{2}\right )} x\right )} \log \left (d x + c\right )}{3 \, {\left (a b^{8} c^{4} d^{5} - 3 \, a^{2} b^{7} c^{3} d^{6} + 3 \, a^{3} b^{6} c^{2} d^{7} - a^{4} b^{5} c d^{8} + {\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{2} + {\left (b^{9} c^{4} d^{5} - 2 \, a b^{8} c^{3} d^{6} + 2 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x\right )}} \]
-1/3*(3*a*b^6*c^7 - 3*a^2*b^5*c^6*d + 3*a^6*b*c^2*d^5 - 3*a^7*c*d^6 - (b^7 *c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^5 + 2*(b^7*c ^4*d^3 - 2*a*b^6*c^3*d^4 + 2*a^3*b^4*c*d^6 - a^4*b^3*d^7)*x^4 - (6*b^7*c^5 *d^2 - 11*a*b^6*c^4*d^3 + 3*a^2*b^5*c^3*d^4 - 3*a^3*b^4*c^2*d^5 + 11*a^4*b ^3*c*d^6 - 6*a^5*b^2*d^7)*x^3 - 9*(b^7*c^6*d - a*b^6*c^5*d^2 - a^2*b^5*c^4 *d^3 + a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6 - a^6*b*d^7)*x^2 + 3*(b^7*c^7 - 4*a *b^6*c^6*d + 5*a^2*b^5*c^5*d^2 - 5*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 - a^7*d ^7)*x + 6*(3*a^6*b*c^2*d^5 - 2*a^7*c*d^6 + (3*a^5*b^2*c*d^6 - 2*a^6*b*d^7) *x^2 + (3*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - 2*a^7*d^7)*x)*log(b*x + a) + 6*( 2*a*b^6*c^7 - 3*a^2*b^5*c^6*d + (2*b^7*c^6*d - 3*a*b^6*c^5*d^2)*x^2 + (2*b ^7*c^7 - a*b^6*c^6*d - 3*a^2*b^5*c^5*d^2)*x)*log(d*x + c))/(a*b^8*c^4*d^5 - 3*a^2*b^7*c^3*d^6 + 3*a^3*b^6*c^2*d^7 - a^4*b^5*c*d^8 + (b^9*c^3*d^6 - 3 *a*b^8*c^2*d^7 + 3*a^2*b^7*c*d^8 - a^3*b^6*d^9)*x^2 + (b^9*c^4*d^5 - 2*a*b ^8*c^3*d^6 + 2*a^3*b^6*c*d^8 - a^4*b^5*d^9)*x)
Timed out. \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.96 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 \, {\left (3 \, a^{5} b c - 2 \, a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}} - \frac {2 \, {\left (2 \, b c^{6} - 3 \, a c^{5} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} - \frac {a b^{5} c^{6} + a^{6} c d^{5} + {\left (b^{6} c^{6} + a^{6} d^{6}\right )} x}{a b^{7} c^{3} d^{5} - 2 \, a^{2} b^{6} c^{2} d^{6} + a^{3} b^{5} c d^{7} + {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{2} + {\left (b^{8} c^{3} d^{5} - a b^{7} c^{2} d^{6} - a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x} + \frac {b^{2} d^{2} x^{3} - 3 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x}{3 \, b^{4} d^{4}} \]
-2*(3*a^5*b*c - 2*a^6*d)*log(b*x + a)/(b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6 *c*d^2 - a^3*b^5*d^3) - 2*(2*b*c^6 - 3*a*c^5*d)*log(d*x + c)/(b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) - (a*b^5*c^6 + a^6*c*d^5 + (b ^6*c^6 + a^6*d^6)*x)/(a*b^7*c^3*d^5 - 2*a^2*b^6*c^2*d^6 + a^3*b^5*c*d^7 + (b^8*c^2*d^6 - 2*a*b^7*c*d^7 + a^2*b^6*d^8)*x^2 + (b^8*c^3*d^5 - a*b^7*c^2 *d^6 - a^2*b^6*c*d^7 + a^3*b^5*d^8)*x) + 1/3*(b^2*d^2*x^3 - 3*(b^2*c*d + a *b*d^2)*x^2 + 3*(3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x)/(b^4*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (177) = 354\).
Time = 0.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.89 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a^{6} b^{5}}{{\left (b^{12} c^{2} - 2 \, a b^{11} c d + a^{2} b^{10} d^{2}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (2 \, b^{2} c^{6} - 3 \, a b c^{5} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{5} - 3 \, a b^{3} c^{2} d^{6} + 3 \, a^{2} b^{2} c d^{7} - a^{3} b d^{8}} + \frac {2 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5} d^{5}} + \frac {{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7} - \frac {2 \, b^{5} c^{4} d^{3} + a b^{4} c^{3} d^{4} - 15 \, a^{2} b^{3} c^{2} d^{5} + 19 \, a^{3} b^{2} c d^{6} - 7 \, a^{4} b d^{7}}{{\left (b x + a\right )} b} + \frac {3 \, {\left (2 \, b^{7} c^{5} d^{2} - a b^{6} c^{4} d^{3} - a^{2} b^{5} c^{3} d^{4} - 11 \, a^{3} b^{4} c^{2} d^{5} + 19 \, a^{4} b^{3} c d^{6} - 8 \, a^{5} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {3 \, {\left (4 \, b^{9} c^{6} d - 6 \, a b^{8} c^{5} d^{2} + 15 \, a^{4} b^{5} c^{2} d^{5} - 18 \, a^{5} b^{4} c d^{6} + 6 \, a^{6} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{3}}{3 \, {\left (b c - a d\right )}^{3} b^{5} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{5}} \]
-a^6*b^5/((b^12*c^2 - 2*a*b^11*c*d + a^2*b^10*d^2)*(b*x + a)) - 2*(2*b^2*c ^6 - 3*a*b*c^5*d)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^4*c^3*d^5 - 3*a*b^3*c^2*d^6 + 3*a^2*b^2*c*d^7 - a^3*b*d^8) + 2*(2*b^3*c^3 + 3*a*b^2 *c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b))) /(b^5*d^5) + 1/3*(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7 - (2*b^5*c^4*d^3 + a*b^4*c^3*d^4 - 15*a^2*b^3*c^2*d^5 + 19*a^3*b^2*c*d^6 - 7*a^4*b*d^7)/((b*x + a)*b) + 3*(2*b^7*c^5*d^2 - a*b^6*c^4*d^3 - a^2*b^5*c ^3*d^4 - 11*a^3*b^4*c^2*d^5 + 19*a^4*b^3*c*d^6 - 8*a^5*b^2*d^7)/((b*x + a) ^2*b^2) + 3*(4*b^9*c^6*d - 6*a*b^8*c^5*d^2 + 15*a^4*b^5*c^2*d^5 - 18*a^5*b ^4*c*d^6 + 6*a^6*b^3*d^7)/((b*x + a)^3*b^3))*(b*x + a)^3/((b*c - a*d)^3*b^ 5*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^5)
Time = 0.69 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.92 \[ \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx=x\,\left (\frac {4\,{\left (a\,d+b\,c\right )}^2}{b^4\,d^4}-\frac {a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2}{b^4\,d^4}\right )-\frac {\frac {a^6\,c\,d^5+a\,b^5\,c^6}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a^6\,d^6+b^6\,c^6\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^5\,d^4+a\,b^4\,d^5\right )+b^5\,d^5\,x^2+a\,b^4\,c\,d^4}+\frac {\ln \left (a+b\,x\right )\,\left (4\,a^6\,d-6\,a^5\,b\,c\right )}{-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (4\,b\,c^6-6\,a\,c^5\,d\right )}{a^3\,d^8-3\,a^2\,b\,c\,d^7+3\,a\,b^2\,c^2\,d^6-b^3\,c^3\,d^5}+\frac {x^3}{3\,b^2\,d^2}-\frac {x^2\,\left (a\,d+b\,c\right )}{b^3\,d^3} \]
x*((4*(a*d + b*c)^2)/(b^4*d^4) - (a^2*d^2 + b^2*c^2 + 4*a*b*c*d)/(b^4*d^4) ) - ((a*b^5*c^6 + a^6*c*d^5)/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(a ^6*d^6 + b^6*c^6))/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(a*b^4*d^5 + b^5*c*d^4) + b^5*d^5*x^2 + a*b^4*c*d^4) + (log(a + b*x)*(4*a^6*d - 6*a^5*b *c))/(b^8*c^3 - a^3*b^5*d^3 + 3*a^2*b^6*c*d^2 - 3*a*b^7*c^2*d) + (log(c + d*x)*(4*b*c^6 - 6*a*c^5*d))/(a^3*d^8 - b^3*c^3*d^5 + 3*a*b^2*c^2*d^6 - 3*a ^2*b*c*d^7) + x^3/(3*b^2*d^2) - (x^2*(a*d + b*c))/(b^3*d^3)