Integrand size = 16, antiderivative size = 88 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a}{(b c-a d)^2 (a+b x)}+\frac {c}{(b c-a d)^2 (c+d x)}+\frac {(b c+a d) \log (a+b x)}{(b c-a d)^3}-\frac {(b c+a d) \log (c+d x)}{(b c-a d)^3} \]
a/(-a*d+b*c)^2/(b*x+a)+c/(-a*d+b*c)^2/(d*x+c)+(a*d+b*c)*ln(b*x+a)/(-a*d+b* c)^3-(a*d+b*c)*ln(d*x+c)/(-a*d+b*c)^3
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a (b c-a d)}{a+b x}+\frac {c (b c-a d)}{c+d x}+(b c+a d) \log (a+b x)-(b c+a d) \log (c+d x)}{(b c-a d)^3} \]
((a*(b*c - a*d))/(a + b*x) + (c*(b*c - a*d))/(c + d*x) + (b*c + a*d)*Log[a + b*x] - (b*c + a*d)*Log[c + d*x])/(b*c - a*d)^3
Time = 0.24 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {b (a d+b c)}{(a+b x) (b c-a d)^3}-\frac {a b}{(a+b x)^2 (b c-a d)^2}-\frac {d (a d+b c)}{(c+d x) (b c-a d)^3}-\frac {c d}{(c+d x)^2 (b c-a d)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a}{(a+b x) (b c-a d)^2}+\frac {c}{(c+d x) (b c-a d)^2}+\frac {(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac {(a d+b c) \log (c+d x)}{(b c-a d)^3}\) |
a/((b*c - a*d)^2*(a + b*x)) + c/((b*c - a*d)^2*(c + d*x)) + ((b*c + a*d)*L og[a + b*x])/(b*c - a*d)^3 - ((b*c + a*d)*Log[c + d*x])/(b*c - a*d)^3
3.3.85.3.1 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 = 0.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {\left (a d +b c \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}+\frac {c}{\left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {a}{\left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {\left (a d +b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}\) | \(89\) |
norman | \(\frac {\frac {\left (a d +b c \right ) x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {2 a c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {\left (a d +b c \right ) \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {\left (a d +b c \right ) \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(182\) |
risch | \(\frac {\frac {\left (a d +b c \right ) x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {2 a c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {\ln \left (b x +a \right ) a d}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {\ln \left (b x +a \right ) b c}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {\ln \left (-d x -c \right ) a d}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {\ln \left (-d x -c \right ) b c}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(273\) |
parallelrisch | \(-\frac {\ln \left (b x +a \right ) x^{2} a \,b^{2} d^{3}+\ln \left (b x +a \right ) x^{2} b^{3} c \,d^{2}-\ln \left (d x +c \right ) x^{2} a \,b^{2} d^{3}-\ln \left (d x +c \right ) x^{2} b^{3} c \,d^{2}-\ln \left (d x +c \right ) a^{2} b c \,d^{2}-\ln \left (d x +c \right ) a \,b^{2} c^{2} d -x \,a^{2} b \,d^{3}+x \,b^{3} c^{2} d -2 \ln \left (d x +c \right ) x a \,b^{2} c \,d^{2}-2 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +2 \ln \left (b x +a \right ) x a \,b^{2} c \,d^{2}+\ln \left (b x +a \right ) x \,a^{2} b \,d^{3}+\ln \left (b x +a \right ) x \,b^{3} c^{2} d +\ln \left (b x +a \right ) a^{2} b c \,d^{2}+\ln \left (b x +a \right ) a \,b^{2} c^{2} d -\ln \left (d x +c \right ) x \,a^{2} b \,d^{3}-\ln \left (d x +c \right ) x \,b^{3} c^{2} d}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b d}\) | \(329\) |
(a*d+b*c)/(a*d-b*c)^3*ln(d*x+c)+c/(a*d-b*c)^2/(d*x+c)+a/(a*d-b*c)^2/(b*x+a )-(a*d+b*c)/(a*d-b*c)^3*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (88) = 176\).
Time = 0.23 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.22 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {2 \, a b c^{2} - 2 \, a^{2} c d + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x + {\left (a b c^{2} + a^{2} c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2} + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) - {\left (a b c^{2} + a^{2} c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2} + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \]
(2*a*b*c^2 - 2*a^2*c*d + (b^2*c^2 - a^2*d^2)*x + (a*b*c^2 + a^2*c*d + (b^2 *c*d + a*b*d^2)*x^2 + (b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)*log(b*x + a) - (a *b*c^2 + a^2*c*d + (b^2*c*d + a*b*d^2)*x^2 + (b^2*c^2 + 2*a*b*c*d + a^2*d^ 2)*x)*log(d*x + c))/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c *d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + ( b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (73) = 146\).
Time = 0.73 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.49 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {2 a c + x \left (a d + b c\right )}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} + \frac {\left (a d + b c\right ) \log {\left (x + \frac {- \frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} - \frac {\left (a d + b c\right ) \log {\left (x + \frac {\frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} \]
(2*a*c + x*(a*d + b*c))/(a**3*c*d**2 - 2*a**2*b*c**2*d + a*b**2*c**3 + x** 2*(a**2*b*d**3 - 2*a*b**2*c*d**2 + b**3*c**2*d) + x*(a**3*d**3 - a**2*b*c* d**2 - a*b**2*c**2*d + b**3*c**3)) + (a*d + b*c)*log(x + (-a**4*d**4*(a*d + b*c)/(a*d - b*c)**3 + 4*a**3*b*c*d**3*(a*d + b*c)/(a*d - b*c)**3 - 6*a** 2*b**2*c**2*d**2*(a*d + b*c)/(a*d - b*c)**3 + a**2*d**2 + 4*a*b**3*c**3*d* (a*d + b*c)/(a*d - b*c)**3 + 2*a*b*c*d - b**4*c**4*(a*d + b*c)/(a*d - b*c) **3 + b**2*c**2)/(2*a*b*d**2 + 2*b**2*c*d))/(a*d - b*c)**3 - (a*d + b*c)*l og(x + (a**4*d**4*(a*d + b*c)/(a*d - b*c)**3 - 4*a**3*b*c*d**3*(a*d + b*c) /(a*d - b*c)**3 + 6*a**2*b**2*c**2*d**2*(a*d + b*c)/(a*d - b*c)**3 + a**2* d**2 - 4*a*b**3*c**3*d*(a*d + b*c)/(a*d - b*c)**3 + 2*a*b*c*d + b**4*c**4* (a*d + b*c)/(a*d - b*c)**3 + b**2*c**2)/(2*a*b*d**2 + 2*b**2*c*d))/(a*d - b*c)**3
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (88) = 176\).
Time = 0.22 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.48 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {{\left (b c + a d\right )} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {{\left (b c + a d\right )} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, a c + {\left (b c + a d\right )} x}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]
(b*c + a*d)*log(b*x + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^ 3) - (b*c + a*d)*log(d*x + c)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a ^3*d^3) + (2*a*c + (b*c + a*d)*x)/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^ 2*b*c*d^2 + a^3*d^3)*x)
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.90 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (b^{3} c + a b^{2} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {b^{2} c d}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}}{b} \]
(a*b^3/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(b*x + a)) - (b^3*c + a*b^2* d)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - b^2*c*d/((b*c - a*d)^3*(b*c/(b*x + a) - a*d /(b*x + a) + d)))/b
Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.67 \[ \int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {2\,a\,c}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {x\,\left (a\,d+b\,c\right )}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}-\frac {2\,\mathrm {atanh}\left (\frac {\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a\,d+b\,c+2\,b\,d\,x\right )}{{\left (a\,d-b\,c\right )}^3}\right )\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^3} \]