Integrand size = 16, antiderivative size = 121 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=\frac {a b}{(b c-a d)^3 (a+b x)}+\frac {c}{2 (b c-a d)^2 (c+d x)^2}+\frac {b c+a d}{(b c-a d)^3 (c+d x)}+\frac {b (b c+2 a d) \log (a+b x)}{(b c-a d)^4}-\frac {b (b c+2 a d) \log (c+d x)}{(b c-a d)^4} \]
a*b/(-a*d+b*c)^3/(b*x+a)+1/2*c/(-a*d+b*c)^2/(d*x+c)^2+(a*d+b*c)/(-a*d+b*c) ^3/(d*x+c)+b*(2*a*d+b*c)*ln(b*x+a)/(-a*d+b*c)^4-b*(2*a*d+b*c)*ln(d*x+c)/(- a*d+b*c)^4
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {2 a b (b c-a d)}{a+b x}+\frac {c (b c-a d)^2}{(c+d x)^2}+\frac {2 (b c-a d) (b c+a d)}{c+d x}+2 b (b c+2 a d) \log (a+b x)-2 b (b c+2 a d) \log (c+d x)}{2 (b c-a d)^4} \]
((2*a*b*(b*c - a*d))/(a + b*x) + (c*(b*c - a*d)^2)/(c + d*x)^2 + (2*(b*c - a*d)*(b*c + a*d))/(c + d*x) + 2*b*(b*c + 2*a*d)*Log[a + b*x] - 2*b*(b*c + 2*a*d)*Log[c + d*x])/(2*(b*c - a*d)^4)
Time = 0.30 (sec) , antiderivative size = 121, 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.
| \(\displaystyle \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^2 (2 a d+b c)}{(a+b x) (b c-a d)^4}-\frac {a b^2}{(a+b x)^2 (b c-a d)^3}-\frac {b d (2 a d+b c)}{(c+d x) (b c-a d)^4}-\frac {d (a d+b c)}{(c+d x)^2 (b c-a d)^3}-\frac {c d}{(c+d x)^3 (b c-a d)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a b}{(a+b x) (b c-a d)^3}+\frac {a d+b c}{(c+d x) (b c-a d)^3}+\frac {c}{2 (c+d x)^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log (a+b x)}{(b c-a d)^4}-\frac {b (2 a d+b c) \log (c+d x)}{(b c-a d)^4}\) |
(a*b)/((b*c - a*d)^3*(a + b*x)) + c/(2*(b*c - a*d)^2*(c + d*x)^2) + (b*c + a*d)/((b*c - a*d)^3*(c + d*x)) + (b*(b*c + 2*a*d)*Log[a + b*x])/(b*c - a* d)^4 - (b*(b*c + 2*a*d)*Log[c + d*x])/(b*c - a*d)^4
3.3.96.3.1 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 = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(-\frac {a d +b c}{\left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {c}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}-\frac {b \left (2 a d +b c \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {a b}{\left (a d -b c \right )^{3} \left (b x +a \right )}+\frac {b \left (2 a d +b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\) | \(122\) |
| norman | \(\frac {\frac {\left (-2 a \,b^{2} d^{3}-b^{3} c \,d^{2}\right ) x^{2}}{d b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {a c \left (-a b \,d^{3}-5 b^{2} c \,d^{2}\right )}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 a d +b c \right ) \left (-a b \,d^{3}-3 b^{2} c \,d^{2}\right ) x}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {b \left (2 a d +b c \right ) \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {b \left (2 a d +b c \right ) \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(357\) |
| risch | \(\frac {-\frac {b d \left (2 a d +b c \right ) x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {\left (a d +3 b c \right ) \left (2 a d +b c \right ) x}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {a c \left (a d +5 b c \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {2 b \ln \left (d x +c \right ) a d}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {b^{2} \ln \left (d x +c \right ) c}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {2 b \ln \left (-b x -a \right ) a d}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {b^{2} \ln \left (-b x -a \right ) c}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(433\) |
| parallelrisch | \(\frac {-4 x^{2} a^{2} b^{2} d^{5}+2 x^{2} b^{4} c^{2} d^{3}-2 x \,a^{3} b \,d^{5}+3 x \,b^{4} c^{3} d^{2}-d^{4} c b \,a^{3}-4 d^{3} c^{2} b^{2} a^{2}+5 d^{2} c^{3} b^{3} a -2 \ln \left (d x +c \right ) a \,b^{3} c^{3} d^{2}+2 x^{2} a \,b^{3} c \,d^{4}-5 x \,a^{2} b^{2} c \,d^{4}+4 x a \,b^{3} c^{2} d^{3}+4 \ln \left (b x +a \right ) x^{3} a \,b^{3} d^{5}+2 \ln \left (b x +a \right ) x^{3} b^{4} c \,d^{4}-4 \ln \left (d x +c \right ) x^{3} a \,b^{3} d^{5}-2 \ln \left (d x +c \right ) x^{3} b^{4} c \,d^{4}+4 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} d^{5}+4 \ln \left (b x +a \right ) x^{2} b^{4} c^{2} d^{3}-4 \ln \left (d x +c \right ) x^{2} a^{2} b^{2} d^{5}-4 \ln \left (d x +c \right ) x^{2} b^{4} c^{2} d^{3}+2 \ln \left (b x +a \right ) x \,b^{4} c^{3} d^{2}-2 \ln \left (d x +c \right ) x \,b^{4} c^{3} d^{2}+4 \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{3}+2 \ln \left (b x +a \right ) a \,b^{3} c^{3} d^{2}-4 \ln \left (d x +c \right ) a^{2} b^{2} c^{2} d^{3}+10 \ln \left (b x +a \right ) x^{2} a \,b^{3} c \,d^{4}-10 \ln \left (d x +c \right ) x^{2} a \,b^{3} c \,d^{4}+8 \ln \left (b x +a \right ) x \,a^{2} b^{2} c \,d^{4}+8 \ln \left (b x +a \right ) x a \,b^{3} c^{2} d^{3}-8 \ln \left (d x +c \right ) x \,a^{2} b^{2} c \,d^{4}-8 \ln \left (d x +c \right ) x a \,b^{3} c^{2} d^{3}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right )^{2} \left (b x +a \right ) b \,d^{2}}\) | \(578\) |
-(a*d+b*c)/(a*d-b*c)^3/(d*x+c)+1/2*c/(a*d-b*c)^2/(d*x+c)^2-b*(2*a*d+b*c)/( a*d-b*c)^4*ln(d*x+c)-a*b/(a*d-b*c)^3/(b*x+a)+b*(2*a*d+b*c)/(a*d-b*c)^4*ln( b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (119) = 238\).
Time = 0.23 (sec) , antiderivative size = 586, normalized size of antiderivative = 4.84 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=\frac {5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x + 2 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \]
1/2*(5*a*b^2*c^3 - 4*a^2*b*c^2*d - a^3*c*d^2 + 2*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*x^2 + (3*b^3*c^3 + 4*a*b^2*c^2*d - 5*a^2*b*c*d^2 - 2*a^3*d^ 3)*x + 2*(a*b^2*c^3 + 2*a^2*b*c^2*d + (b^3*c*d^2 + 2*a*b^2*d^3)*x^3 + (2*b ^3*c^2*d + 5*a*b^2*c*d^2 + 2*a^2*b*d^3)*x^2 + (b^3*c^3 + 4*a*b^2*c^2*d + 4 *a^2*b*c*d^2)*x)*log(b*x + a) - 2*(a*b^2*c^3 + 2*a^2*b*c^2*d + (b^3*c*d^2 + 2*a*b^2*d^3)*x^3 + (2*b^3*c^2*d + 5*a*b^2*c*d^2 + 2*a^2*b*d^3)*x^2 + (b^ 3*c^3 + 4*a*b^2*c^2*d + 4*a^2*b*c*d^2)*x)*log(d*x + c))/(a*b^4*c^6 - 4*a^2 *b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (b^5*c^4* d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d^6)*x ^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^ 4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^ 4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (105) = 210\).
Time = 1.10 (sec) , antiderivative size = 774, normalized size of antiderivative = 6.40 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=- \frac {b \left (2 a d + b c\right ) \log {\left (x + \frac {- \frac {a^{5} b d^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b^{2} c d^{4} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{3} c^{2} d^{3} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{4} c^{3} d^{2} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac {b^{6} c^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {b \left (2 a d + b c\right ) \log {\left (x + \frac {\frac {a^{5} b d^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b^{2} c d^{4} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{3} c^{2} d^{3} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{4} c^{3} d^{2} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac {b^{6} c^{5} \cdot \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {- a^{2} c d - 5 a b c^{2} + x^{2} \left (- 4 a b d^{2} - 2 b^{2} c d\right ) + x \left (- 2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \cdot \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \cdot \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \]
-b*(2*a*d + b*c)*log(x + (-a**5*b*d**5*(2*a*d + b*c)/(a*d - b*c)**4 + 5*a* *4*b**2*c*d**4*(2*a*d + b*c)/(a*d - b*c)**4 - 10*a**3*b**3*c**2*d**3*(2*a* d + b*c)/(a*d - b*c)**4 + 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d - b*c) **4 + 2*a**2*b*d**2 - 5*a*b**5*c**4*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b **2*c*d + b**6*c**5*(2*a*d + b*c)/(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2*d* *2 + 2*b**3*c*d))/(a*d - b*c)**4 + b*(2*a*d + b*c)*log(x + (a**5*b*d**5*(2 *a*d + b*c)/(a*d - b*c)**4 - 5*a**4*b**2*c*d**4*(2*a*d + b*c)/(a*d - b*c)* *4 + 10*a**3*b**3*c**2*d**3*(2*a*d + b*c)/(a*d - b*c)**4 - 10*a**2*b**4*c* *3*d**2*(2*a*d + b*c)/(a*d - b*c)**4 + 2*a**2*b*d**2 + 5*a*b**5*c**4*d*(2* a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d - b**6*c**5*(2*a*d + b*c)/(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2*d**2 + 2*b**3*c*d))/(a*d - b*c)**4 + (-a**2 *c*d - 5*a*b*c**2 + x**2*(-4*a*b*d**2 - 2*b**2*c*d) + x*(-2*a**2*d**2 - 7* a*b*c*d - 3*b**2*c**2))/(2*a**4*c**2*d**3 - 6*a**3*b*c**3*d**2 + 6*a**2*b* *2*c**4*d - 2*a*b**3*c**5 + x**3*(2*a**3*b*d**5 - 6*a**2*b**2*c*d**4 + 6*a *b**3*c**2*d**3 - 2*b**4*c**3*d**2) + x**2*(2*a**4*d**5 - 2*a**3*b*c*d**4 - 6*a**2*b**2*c**2*d**3 + 10*a*b**3*c**3*d**2 - 4*b**4*c**4*d) + x*(4*a**4 *c*d**4 - 10*a**3*b*c**2*d**3 + 6*a**2*b**2*c**3*d**2 + 2*a*b**3*c**4*d - 2*b**4*c**5))
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (119) = 238\).
Time = 0.23 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.36 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=\frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {5 \, a b c^{2} + a^{2} c d + 2 \, {\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{2} + {\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x}{2 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
(b^2*c + 2*a*b*d)*log(b*x + a)/(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^2*c + 2*a*b*d)*log(d*x + c)/(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) + 1/2*(5*a*b*c ^2 + a^2*c*d + 2*(b^2*c*d + 2*a*b*d^2)*x^2 + (3*b^2*c^2 + 7*a*b*c*d + 2*a^ 2*d^2)*x)/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + ( b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^3 + (2*b^4* c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^ 4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (119) = 238\).
Time = 0.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.11 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (b^{4} c + 2 \, a b^{3} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac {2 \, {\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}}}{2 \, b} \]
1/2*(2*a*b^5/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*(b *x + a)) - 2*(b^4*c + 2*a*b^3*d)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d ))/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b* d^4) - (3*b^3*c*d^2 + 2*a*b^2*d^3 + 2*(2*b^5*c^2*d - a*b^4*c*d^2 - a^2*b^3 *d^3)/((b*x + a)*b))/((b*c - a*d)^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2) )/b
Time = 0.66 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.12 \[ \int \frac {x}{(a+b x)^2 (c+d x)^3} \, dx=\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (2\,a\,d+b\,c\right )\,\left (a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{\left (c\,b^2+2\,a\,d\,b\right )\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,b^2\,d\,x\,\left (2\,a\,d+b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{\left (c\,b^2+2\,a\,d\,b\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {d\,a^2\,c+5\,b\,a\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (2\,a\,d+b\,c\right )\,\left (a\,d+3\,b\,c\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d\,x^2\,\left (2\,a\,d+b\,c\right )}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3} \]
(2*b*atanh((b*(2*a*d + b*c)*(a^4*d^4 - b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c *d^3))/((b^2*c + 2*a*b*d)*(a*d - b*c)^4) + (2*b^2*d*x*(2*a*d + b*c)*(a^3*d ^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/((b^2*c + 2*a*b*d)*(a*d - b *c)^4))*(2*a*d + b*c))/(a*d - b*c)^4 - ((5*a*b*c^2 + a^2*c*d)/(2*(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*d + b*c)*(a*d + 3*b* c))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*d*x^2*(2* a*d + b*c))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(x*(b*c^2 + 2*a*c*d) + a*c^2 + x^2*(a*d^2 + 2*b*c*d) + b*d^2*x^3)