Integrand size = 18, antiderivative size = 142 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 (b c+a d) x}{b^3 d^3}+\frac {x^2}{2 b^2 d^2}+\frac {a^5}{b^4 (b c-a d)^2 (a+b x)}+\frac {c^5}{d^4 (b c-a d)^2 (c+d x)}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}+\frac {c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3} \]
-2*(a*d+b*c)*x/b^3/d^3+1/2*x^2/b^2/d^2+a^5/b^4/(-a*d+b*c)^2/(b*x+a)+c^5/d^ 4/(-a*d+b*c)^2/(d*x+c)+a^4*(-3*a*d+5*b*c)*ln(b*x+a)/b^4/(-a*d+b*c)^3+c^4*( -5*a*d+3*b*c)*ln(d*x+c)/d^4/(-a*d+b*c)^3
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 (b c+a d) x}{b^3 d^3}+\frac {x^2}{2 b^2 d^2}+\frac {a^5}{b^4 (b c-a d)^2 (a+b x)}+\frac {c^5}{d^4 (b c-a d)^2 (c+d x)}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}+\frac {c^4 (-3 b c+5 a d) \log (c+d x)}{d^4 (-b c+a d)^3} \]
(-2*(b*c + a*d)*x)/(b^3*d^3) + x^2/(2*b^2*d^2) + a^5/(b^4*(b*c - a*d)^2*(a + b*x)) + c^5/(d^4*(b*c - a*d)^2*(c + d*x)) + (a^4*(5*b*c - 3*a*d)*Log[a + b*x])/(b^4*(b*c - a*d)^3) + (c^4*(-3*b*c + 5*a*d)*Log[c + d*x])/(d^4*(-( b*c) + a*d)^3)
Time = 0.37 (sec) , antiderivative size = 142, 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.
| \(\displaystyle \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^5}{b^3 (a+b x)^2 (b c-a d)^2}-\frac {a^4 (3 a d-5 b c)}{b^3 (a+b x) (b c-a d)^3}-\frac {2 (a d+b c)}{b^3 d^3}-\frac {c^5}{d^3 (c+d x)^2 (a d-b c)^2}-\frac {c^4 (3 b c-5 a d)}{d^3 (c+d x) (a d-b c)^3}+\frac {x}{b^2 d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5}{b^4 (a+b x) (b c-a d)^2}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}-\frac {2 x (a d+b c)}{b^3 d^3}+\frac {c^5}{d^4 (c+d x) (b c-a d)^2}+\frac {c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3}+\frac {x^2}{2 b^2 d^2}\) |
(-2*(b*c + a*d)*x)/(b^3*d^3) + x^2/(2*b^2*d^2) + a^5/(b^4*(b*c - a*d)^2*(a + b*x)) + c^5/(d^4*(b*c - a*d)^2*(c + d*x)) + (a^4*(5*b*c - 3*a*d)*Log[a + b*x])/(b^4*(b*c - a*d)^3) + (c^4*(3*b*c - 5*a*d)*Log[c + d*x])/(d^4*(b*c - a*d)^3)
3.3.81.3.1 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 = 140, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {-\frac {1}{2} b d \,x^{2}+2 a d x +2 b c x}{b^{3} d^{3}}+\frac {c^{4} \left (5 a d -3 b c \right ) \ln \left (d x +c \right )}{d^{4} \left (a d -b c \right )^{3}}+\frac {c^{5}}{d^{4} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {a^{4} \left (3 a d -5 b c \right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{3}}+\frac {a^{5}}{b^{4} \left (a d -b c \right )^{2} \left (b x +a \right )}\) | \(140\) |
| norman | \(\frac {\frac {x^{4}}{2 b d}-\frac {3 \left (a d +b c \right ) x^{3}}{2 b^{2} d^{2}}+\frac {\left (6 a^{5} d^{5}-a^{4} b c \,d^{4}-3 a^{3} b^{2} c^{2} d^{3}-3 a^{2} b^{3} c^{3} d^{2}-a \,b^{4} c^{4} d +6 b^{5} c^{5}\right ) x}{2 d^{4} b^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (6 a^{4} d^{4}-a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-a \,b^{3} c^{3} d +6 b^{4} c^{4}\right ) a c}{2 d^{4} b^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {a^{4} \left (3 a d -5 b c \right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4}}+\frac {c^{4} \left (5 a d -3 b c \right ) \ln \left (d x +c \right )}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(348\) |
| risch | \(\frac {x^{2}}{2 b^{2} d^{2}}-\frac {2 a x}{b^{3} d^{2}}-\frac {2 c x}{b^{2} d^{3}}+\frac {\frac {\left (a^{5} d^{5}+b^{5} c^{5}\right ) x}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a c \left (a^{4} d^{4}+b^{4} c^{4}\right )}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{b^{3} d^{3} \left (b x +a \right ) \left (d x +c \right )}+\frac {5 c^{4} \ln \left (d x +c \right ) a}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {3 c^{5} \ln \left (d x +c \right ) b}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 a^{5} \ln \left (-b x -a \right ) d}{b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {5 a^{4} \ln \left (-b x -a \right ) c}{b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(366\) |
| parallelrisch | \(\frac {7 a \,b^{5} c^{5} d x -6 b^{6} c^{6} x +6 a^{6} d^{6} x -6 \ln \left (d x +c \right ) a \,b^{5} c^{6}+6 \ln \left (b x +a \right ) x \,a^{6} d^{6}-6 \ln \left (d x +c \right ) x \,b^{6} c^{6}+6 \ln \left (b x +a \right ) a^{6} c \,d^{5}-7 a^{5} b \,c^{2} d^{4}-5 a^{4} b^{2} c^{3} d^{3}+5 a^{3} b^{3} c^{4} d^{2}+7 a^{2} b^{4} c^{5} d -10 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} c \,d^{5}+10 \ln \left (d x +c \right ) x^{2} a \,b^{5} c^{4} d^{2}-4 \ln \left (b x +a \right ) x \,a^{5} b c \,d^{5}-10 \ln \left (b x +a \right ) x \,a^{4} b^{2} c^{2} d^{4}+10 \ln \left (d x +c \right ) x \,a^{2} b^{4} c^{4} d^{2}+4 \ln \left (d x +c \right ) x a \,b^{5} c^{5} d +6 a^{6} c \,d^{5}-6 a \,b^{5} c^{6}-6 a \,b^{5} c^{3} d^{3} x^{3}-7 a^{5} b c \,d^{5} x -2 a^{4} b^{2} c^{2} d^{4} x +2 a^{2} b^{4} c^{4} d^{2} x -b^{6} c^{3} d^{3} x^{4}-3 a^{4} b^{2} d^{6} x^{3}+3 b^{6} c^{4} d^{2} x^{3}+a^{3} b^{3} d^{6} x^{4}-3 a^{2} b^{4} c \,d^{5} x^{4}+3 a \,b^{5} c^{2} d^{4} x^{4}+6 a^{3} b^{3} c \,d^{5} x^{3}+6 \ln \left (b x +a \right ) x^{2} a^{5} b \,d^{6}-6 \ln \left (d x +c \right ) x^{2} b^{6} c^{5} d -10 \ln \left (b x +a \right ) a^{5} b \,c^{2} d^{4}+10 \ln \left (d x +c \right ) a^{2} b^{4} c^{5} d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b^{4} d^{4}}\) | \(567\) |
-1/b^3/d^3*(-1/2*b*d*x^2+2*a*d*x+2*b*c*x)+1/d^4*c^4*(5*a*d-3*b*c)/(a*d-b*c )^3*ln(d*x+c)+1/d^4*c^5/(a*d-b*c)^2/(d*x+c)+1/b^4*a^4*(3*a*d-5*b*c)/(a*d-b *c)^3*ln(b*x+a)+1/b^4*a^5/(a*d-b*c)^2/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (140) = 280\).
Time = 0.26 (sec) , antiderivative size = 623, normalized size of antiderivative = 4.39 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {2 \, a b^{5} c^{6} - 2 \, a^{2} b^{4} c^{5} d + 2 \, a^{5} b c^{2} d^{4} - 2 \, a^{6} c d^{5} + {\left (b^{6} c^{3} d^{3} - 3 \, a b^{5} c^{2} d^{4} + 3 \, a^{2} b^{4} c d^{5} - a^{3} b^{3} d^{6}\right )} x^{4} - 3 \, {\left (b^{6} c^{4} d^{2} - 2 \, a b^{5} c^{3} d^{3} + 2 \, a^{3} b^{3} c d^{5} - a^{4} b^{2} d^{6}\right )} x^{3} - {\left (4 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{2} b^{4} c^{3} d^{3} + 5 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - 4 \, a^{5} b d^{6}\right )} x^{2} + 2 \, {\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 4 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} x + 2 \, {\left (5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5} + {\left (5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{2} + {\left (5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d + {\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2}\right )} x^{2} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{7} c^{4} d^{4} - 3 \, a^{2} b^{6} c^{3} d^{5} + 3 \, a^{3} b^{5} c^{2} d^{6} - a^{4} b^{4} c d^{7} + {\left (b^{8} c^{3} d^{5} - 3 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} - a^{3} b^{5} d^{8}\right )} x^{2} + {\left (b^{8} c^{4} d^{4} - 2 \, a b^{7} c^{3} d^{5} + 2 \, a^{3} b^{5} c d^{7} - a^{4} b^{4} d^{8}\right )} x\right )}} \]
1/2*(2*a*b^5*c^6 - 2*a^2*b^4*c^5*d + 2*a^5*b*c^2*d^4 - 2*a^6*c*d^5 + (b^6* c^3*d^3 - 3*a*b^5*c^2*d^4 + 3*a^2*b^4*c*d^5 - a^3*b^3*d^6)*x^4 - 3*(b^6*c^ 4*d^2 - 2*a*b^5*c^3*d^3 + 2*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^3 - (4*b^6*c^5* d - 5*a*b^5*c^4*d^2 - 5*a^2*b^4*c^3*d^3 + 5*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c* d^5 - 4*a^5*b*d^6)*x^2 + 2*(b^6*c^6 - 3*a*b^5*c^5*d + 4*a^2*b^4*c^4*d^2 - 4*a^4*b^2*c^2*d^4 + 3*a^5*b*c*d^5 - a^6*d^6)*x + 2*(5*a^5*b*c^2*d^4 - 3*a^ 6*c*d^5 + (5*a^4*b^2*c*d^5 - 3*a^5*b*d^6)*x^2 + (5*a^4*b^2*c^2*d^4 + 2*a^5 *b*c*d^5 - 3*a^6*d^6)*x)*log(b*x + a) + 2*(3*a*b^5*c^6 - 5*a^2*b^4*c^5*d + (3*b^6*c^5*d - 5*a*b^5*c^4*d^2)*x^2 + (3*b^6*c^6 - 2*a*b^5*c^5*d - 5*a^2* b^4*c^4*d^2)*x)*log(d*x + c))/(a*b^7*c^4*d^4 - 3*a^2*b^6*c^3*d^5 + 3*a^3*b ^5*c^2*d^6 - a^4*b^4*c*d^7 + (b^8*c^3*d^5 - 3*a*b^7*c^2*d^6 + 3*a^2*b^6*c* d^7 - a^3*b^5*d^8)*x^2 + (b^8*c^4*d^4 - 2*a*b^7*c^3*d^5 + 2*a^3*b^5*c*d^7 - a^4*b^4*d^8)*x)
Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (136) = 272\).
Time = 120.78 (sec) , antiderivative size = 731, normalized size of antiderivative = 5.15 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^{4} \cdot \left (3 a d - 5 b c\right ) \log {\left (x + \frac {\frac {a^{8} d^{7} \cdot \left (3 a d - 5 b c\right )}{b \left (a d - b c\right )^{3}} - \frac {4 a^{7} c d^{6} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{6} b c^{2} d^{5} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{5} b^{2} c^{3} d^{4} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} + 3 a^{5} c d^{4} + \frac {a^{4} b^{3} c^{4} d^{3} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{4} b c^{2} d^{3} - 5 a^{2} b^{3} c^{4} d + 3 a b^{4} c^{5}}{3 a^{5} d^{5} - 5 a^{4} b c d^{4} - 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} \right )}}{b^{4} \left (a d - b c\right )^{3}} + \frac {c^{4} \cdot \left (5 a d - 3 b c\right ) \log {\left (x + \frac {3 a^{5} c d^{4} + \frac {a^{4} b^{3} c^{4} d^{3} \cdot \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{4} b c^{2} d^{3} - \frac {4 a^{3} b^{4} c^{5} d^{2} \cdot \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{5} c^{6} d \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{2} b^{3} c^{4} d - \frac {4 a b^{6} c^{7} \cdot \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + 3 a b^{4} c^{5} + \frac {b^{7} c^{8} \cdot \left (5 a d - 3 b c\right )}{d \left (a d - b c\right )^{3}}}{3 a^{5} d^{5} - 5 a^{4} b c d^{4} - 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} \right )}}{d^{4} \left (a d - b c\right )^{3}} + x \left (- \frac {2 a}{b^{3} d^{2}} - \frac {2 c}{b^{2} d^{3}}\right ) + \frac {a^{5} c d^{4} + a b^{4} c^{5} + x \left (a^{5} d^{5} + b^{5} c^{5}\right )}{a^{3} b^{4} c d^{6} - 2 a^{2} b^{5} c^{2} d^{5} + a b^{6} c^{3} d^{4} + x^{2} \left (a^{2} b^{5} d^{7} - 2 a b^{6} c d^{6} + b^{7} c^{2} d^{5}\right ) + x \left (a^{3} b^{4} d^{7} - a^{2} b^{5} c d^{6} - a b^{6} c^{2} d^{5} + b^{7} c^{3} d^{4}\right )} + \frac {x^{2}}{2 b^{2} d^{2}} \]
a**4*(3*a*d - 5*b*c)*log(x + (a**8*d**7*(3*a*d - 5*b*c)/(b*(a*d - b*c)**3) - 4*a**7*c*d**6*(3*a*d - 5*b*c)/(a*d - b*c)**3 + 6*a**6*b*c**2*d**5*(3*a* d - 5*b*c)/(a*d - b*c)**3 - 4*a**5*b**2*c**3*d**4*(3*a*d - 5*b*c)/(a*d - b *c)**3 + 3*a**5*c*d**4 + a**4*b**3*c**4*d**3*(3*a*d - 5*b*c)/(a*d - b*c)** 3 - 5*a**4*b*c**2*d**3 - 5*a**2*b**3*c**4*d + 3*a*b**4*c**5)/(3*a**5*d**5 - 5*a**4*b*c*d**4 - 5*a*b**4*c**4*d + 3*b**5*c**5))/(b**4*(a*d - b*c)**3) + c**4*(5*a*d - 3*b*c)*log(x + (3*a**5*c*d**4 + a**4*b**3*c**4*d**3*(5*a*d - 3*b*c)/(a*d - b*c)**3 - 5*a**4*b*c**2*d**3 - 4*a**3*b**4*c**5*d**2*(5*a *d - 3*b*c)/(a*d - b*c)**3 + 6*a**2*b**5*c**6*d*(5*a*d - 3*b*c)/(a*d - b*c )**3 - 5*a**2*b**3*c**4*d - 4*a*b**6*c**7*(5*a*d - 3*b*c)/(a*d - b*c)**3 + 3*a*b**4*c**5 + b**7*c**8*(5*a*d - 3*b*c)/(d*(a*d - b*c)**3))/(3*a**5*d** 5 - 5*a**4*b*c*d**4 - 5*a*b**4*c**4*d + 3*b**5*c**5))/(d**4*(a*d - b*c)**3 ) + x*(-2*a/(b**3*d**2) - 2*c/(b**2*d**3)) + (a**5*c*d**4 + a*b**4*c**5 + x*(a**5*d**5 + b**5*c**5))/(a**3*b**4*c*d**6 - 2*a**2*b**5*c**2*d**5 + a*b **6*c**3*d**4 + x**2*(a**2*b**5*d**7 - 2*a*b**6*c*d**6 + b**7*c**2*d**5) + x*(a**3*b**4*d**7 - a**2*b**5*c*d**6 - a*b**6*c**2*d**5 + b**7*c**3*d**4) ) + x**2/(2*b**2*d**2)
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (140) = 280\).
Time = 0.21 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.18 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {{\left (5 \, a^{4} b c - 3 \, a^{5} d\right )} \log \left (b x + a\right )}{b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}} + \frac {{\left (3 \, b c^{5} - 5 \, a c^{4} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac {a b^{4} c^{5} + a^{5} c d^{4} + {\left (b^{5} c^{5} + a^{5} d^{5}\right )} x}{a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x} + \frac {b d x^{2} - 4 \, {\left (b c + a d\right )} x}{2 \, b^{3} d^{3}} \]
(5*a^4*b*c - 3*a^5*d)*log(b*x + a)/(b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c* d^2 - a^3*b^4*d^3) + (3*b*c^5 - 5*a*c^4*d)*log(d*x + c)/(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) + (a*b^4*c^5 + a^5*c*d^4 + (b^5*c^ 5 + a^5*d^5)*x)/(a*b^6*c^3*d^4 - 2*a^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + (b^7* c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^2 + (b^7*c^3*d^4 - a*b^6*c^2*d^5 - a^2*b^5*c*d^6 + a^3*b^4*d^7)*x) + 1/2*(b*d*x^2 - 4*(b*c + a*d)*x)/(b^3*d ^3)
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (140) = 280\).
Time = 0.27 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.92 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^{5} b^{4}}{{\left (b^{10} c^{2} - 2 \, a b^{9} c d + a^{2} b^{8} d^{2}\right )} {\left (b x + a\right )}} + \frac {{\left (3 \, b^{2} c^{5} - 5 \, a b c^{4} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{4} - 3 \, a b^{3} c^{2} d^{5} + 3 \, a^{2} b^{2} c d^{6} - a^{3} b d^{7}} - \frac {{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} d^{4}} + \frac {{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6} - \frac {3 \, b^{5} c^{4} d^{2} - 2 \, a b^{4} c^{3} d^{3} - 12 \, a^{2} b^{3} c^{2} d^{4} + 18 \, a^{3} b^{2} c d^{5} - 7 \, a^{4} b d^{6}}{{\left (b x + a\right )} b} - \frac {2 \, {\left (3 \, b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{3} b^{4} c^{2} d^{4} - 10 \, a^{4} b^{3} c d^{5} + 3 \, a^{5} b^{2} d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{2}}{2 \, {\left (b c - a d\right )}^{3} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{4}} \]
a^5*b^4/((b^10*c^2 - 2*a*b^9*c*d + a^2*b^8*d^2)*(b*x + a)) + (3*b^2*c^5 - 5*a*b*c^4*d)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^4*c^3*d^4 - 3* a*b^3*c^2*d^5 + 3*a^2*b^2*c*d^6 - a^3*b*d^7) - (3*b^2*c^2 + 4*a*b*c*d + 3* a^2*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^4*d^4) + 1/2*(b^3*c^3*d ^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6 - (3*b^5*c^4*d^2 - 2*a*b^4* c^3*d^3 - 12*a^2*b^3*c^2*d^4 + 18*a^3*b^2*c*d^5 - 7*a^4*b*d^6)/((b*x + a)* b) - 2*(3*b^7*c^5*d - 5*a*b^6*c^4*d^2 + 10*a^3*b^4*c^2*d^4 - 10*a^4*b^3*c* d^5 + 3*a^5*b^2*d^6)/((b*x + a)^2*b^2))*(b*x + a)^2/((b*c - a*d)^3*b^4*(b* c/(b*x + a) - a*d/(b*x + a) + d)*d^4)
Time = 0.67 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.35 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a^5\,c\,d^4+a\,b^4\,c^5}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (a^4\,d^4-a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2-a\,b^3\,c^3\,d+b^4\,c^4\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^4\,d^3+a\,b^3\,d^4\right )+b^4\,d^4\,x^2+a\,b^3\,c\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (3\,a^5\,d-5\,a^4\,b\,c\right )}{-a^3\,b^4\,d^3+3\,a^2\,b^5\,c\,d^2-3\,a\,b^6\,c^2\,d+b^7\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,b\,c^5-5\,a\,c^4\,d\right )}{a^3\,d^7-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^2\,d^5-b^3\,c^3\,d^4}+\frac {x^2}{2\,b^2\,d^2}-\frac {2\,x\,\left (a\,d+b\,c\right )}{b^3\,d^3} \]
((a*b^4*c^5 + a^5*c*d^4)/(b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(a*d + b*c)*(a^4*d^4 + b^4*c^4 + a^2*b^2*c^2*d^2 - a*b^3*c^3*d - a^3*b*c*d^3))/( b*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(a*b^3*d^4 + b^4*c*d^3) + b^4*d^4 *x^2 + a*b^3*c*d^3) - (log(a + b*x)*(3*a^5*d - 5*a^4*b*c))/(b^7*c^3 - a^3* b^4*d^3 + 3*a^2*b^5*c*d^2 - 3*a*b^6*c^2*d) - (log(c + d*x)*(3*b*c^5 - 5*a* c^4*d))/(a^3*d^7 - b^3*c^3*d^4 + 3*a*b^2*c^2*d^5 - 3*a^2*b*c*d^6) + x^2/(2 *b^2*d^2) - (2*x*(a*d + b*c))/(b^3*d^3)