Integrand size = 18, antiderivative size = 144 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=-\frac {1}{a^2 c^2 x}-\frac {b^3}{a^2 (b c-a d)^2 (a+b x)}-\frac {d^3}{c^2 (b c-a d)^2 (c+d x)}-\frac {2 (b c+a d) \log (x)}{a^3 c^3}+\frac {2 b^3 (b c-2 a d) \log (a+b x)}{a^3 (b c-a d)^3}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3} \]
-1/a^2/c^2/x-b^3/a^2/(-a*d+b*c)^2/(b*x+a)-d^3/c^2/(-a*d+b*c)^2/(d*x+c)-2*( a*d+b*c)*ln(x)/a^3/c^3+2*b^3*(-2*a*d+b*c)*ln(b*x+a)/a^3/(-a*d+b*c)^3+2*d^3 *(-a*d+2*b*c)*ln(d*x+c)/c^3/(-a*d+b*c)^3
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=-\frac {1}{a^2 c^2 x}-\frac {b^3}{a^2 (b c-a d)^2 (a+b x)}-\frac {d^3}{c^2 (b c-a d)^2 (c+d x)}-\frac {2 (b c+a d) \log (x)}{a^3 c^3}+\frac {2 b^3 (-b c+2 a d) \log (a+b x)}{a^3 (-b c+a d)^3}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3} \]
-(1/(a^2*c^2*x)) - b^3/(a^2*(b*c - a*d)^2*(a + b*x)) - d^3/(c^2*(b*c - a*d )^2*(c + d*x)) - (2*(b*c + a*d)*Log[x])/(a^3*c^3) + (2*b^3*(-(b*c) + 2*a*d )*Log[a + b*x])/(a^3*(-(b*c) + a*d)^3) + (2*d^3*(2*b*c - a*d)*Log[c + d*x] )/(c^3*(b*c - a*d)^3)
Time = 0.35 (sec) , antiderivative size = 144, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {2 b^4 (2 a d-b c)}{a^3 (a+b x) (a d-b c)^3}-\frac {2 (a d+b c)}{a^3 c^3 x}+\frac {b^4}{a^2 (a+b x)^2 (a d-b c)^2}+\frac {1}{a^2 c^2 x^2}+\frac {2 d^4 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}+\frac {d^4}{c^2 (c+d x)^2 (b c-a d)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b^3 (b c-2 a d) \log (a+b x)}{a^3 (b c-a d)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {b^3}{a^2 (a+b x) (b c-a d)^2}-\frac {1}{a^2 c^2 x}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {d^3}{c^2 (c+d x) (b c-a d)^2}\) |
-(1/(a^2*c^2*x)) - b^3/(a^2*(b*c - a*d)^2*(a + b*x)) - d^3/(c^2*(b*c - a*d )^2*(c + d*x)) - (2*(b*c + a*d)*Log[x])/(a^3*c^3) + (2*b^3*(b*c - 2*a*d)*L og[a + b*x])/(a^3*(b*c - a*d)^3) + (2*d^3*(2*b*c - a*d)*Log[c + d*x])/(c^3 *(b*c - a*d)^3)
3.3.88.3.1 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 = 0.54 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(-\frac {1}{a^{2} c^{2} x}+\frac {\left (-2 a d -2 b c \right ) \ln \left (x \right )}{c^{3} a^{3}}-\frac {d^{3}}{c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {2 d^{3} \left (a d -2 b c \right ) \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )^{3}}-\frac {b^{3}}{a^{2} \left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {2 b^{3} \left (2 a d -b c \right ) \ln \left (b x +a \right )}{a^{3} \left (a d -b c \right )^{3}}\) | \(146\) |
| norman | \(\frac {\frac {\left (2 a^{4} d^{4}-a^{3} b c \,d^{3}-a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x^{2}}{c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) b d \,x^{3}}{c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {1}{a c}}{x \left (d x +c \right ) \left (b x +a \right )}-\frac {2 \left (a d +b c \right ) \ln \left (x \right )}{a^{3} c^{3}}+\frac {2 b^{3} \left (2 a d -b c \right ) \ln \left (b x +a \right )}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {2 d^{3} \left (a d -2 b c \right ) \ln \left (d x +c \right )}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(309\) |
| risch | \(\frac {-\frac {2 b d \left (a^{2} d^{2}-a b c d +b^{2} c^{2}\right ) x^{2}}{c^{2} a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) x}{a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {1}{a c}}{x \left (d x +c \right ) \left (b x +a \right )}-\frac {2 \ln \left (-x \right ) d}{c^{3} a^{2}}-\frac {2 \ln \left (-x \right ) b}{c^{2} a^{3}}+\frac {4 b^{3} \ln \left (b x +a \right ) d}{a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 b^{4} \ln \left (b x +a \right ) c}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {2 d^{4} \ln \left (-d x -c \right ) a}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {4 d^{3} \ln \left (-d x -c \right ) b}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(399\) |
| parallelrisch | \(-\frac {2 \ln \left (d x +c \right ) x^{2} a^{4} b c \,d^{4}+2 x^{2} b^{5} c^{5}-2 x^{2} a^{5} d^{5}+d^{3} c^{2} a^{5}-3 d^{2} c^{3} b \,a^{4}+3 d \,c^{4} b^{2} a^{3}-c^{5} b^{3} a^{2}-4 \ln \left (x \right ) x^{3} a^{3} b^{2} c \,d^{4}+4 \ln \left (d x +c \right ) x^{2} a^{3} b^{2} c^{2} d^{3}-4 \ln \left (x \right ) x \,a^{4} b \,c^{2} d^{3}+4 \ln \left (x \right ) x \,a^{2} b^{3} c^{4} d -4 \ln \left (b x +a \right ) x \,a^{2} b^{3} c^{4} d +4 \ln \left (d x +c \right ) x \,a^{4} b \,c^{2} d^{3}+4 \ln \left (x \right ) x^{3} a \,b^{4} c^{3} d^{2}-4 \ln \left (b x +a \right ) x^{3} a \,b^{4} c^{3} d^{2}+4 \ln \left (d x +c \right ) x^{3} a^{3} b^{2} c \,d^{4}-2 \ln \left (x \right ) x^{2} a^{4} b c \,d^{4}-4 \ln \left (x \right ) x^{2} a^{3} b^{2} c^{2} d^{3}+4 \ln \left (x \right ) x^{2} a^{2} b^{3} c^{3} d^{2}+2 \ln \left (x \right ) x^{2} a \,b^{4} c^{4} d -4 \ln \left (b x +a \right ) x^{2} a^{2} b^{3} c^{3} d^{2}-2 \ln \left (b x +a \right ) x^{2} a \,b^{4} c^{4} d -2 x^{3} a^{4} b \,d^{5}+2 x^{3} b^{5} c^{4} d +2 \ln \left (x \right ) x^{2} a^{5} d^{5}-2 \ln \left (x \right ) x^{2} b^{5} c^{5}+2 \ln \left (b x +a \right ) x^{2} b^{5} c^{5}-2 \ln \left (d x +c \right ) x^{2} a^{5} d^{5}+3 x^{3} a^{3} b^{2} c \,d^{4}-3 x^{3} a \,b^{4} c^{3} d^{2}+3 x^{2} a^{4} b c \,d^{4}-x^{2} a^{3} b^{2} c^{2} d^{3}+x^{2} a^{2} b^{3} c^{3} d^{2}-3 x^{2} a \,b^{4} c^{4} d +2 \ln \left (x \right ) x^{3} a^{4} b \,d^{5}-2 \ln \left (x \right ) x^{3} b^{5} c^{4} d +2 \ln \left (b x +a \right ) x^{3} b^{5} c^{4} d -2 \ln \left (d x +c \right ) x^{3} a^{4} b \,d^{5}+2 \ln \left (x \right ) x \,a^{5} c \,d^{4}-2 \ln \left (x \right ) x a \,b^{4} c^{5}+2 \ln \left (b x +a \right ) x a \,b^{4} c^{5}-2 \ln \left (d x +c \right ) x \,a^{5} c \,d^{4}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) x \,c^{3} a^{3}}\) | \(721\) |
-1/a^2/c^2/x+(-2*a*d-2*b*c)/c^3/a^3*ln(x)-d^3/c^2/(a*d-b*c)^2/(d*x+c)+2*d^ 3*(a*d-2*b*c)/c^3/(a*d-b*c)^3*ln(d*x+c)-b^3/a^2/(a*d-b*c)^2/(b*x+a)+2*b^3* (2*a*d-b*c)/a^3/(a*d-b*c)^3*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (144) = 288\).
Time = 7.49 (sec) , antiderivative size = 653, normalized size of antiderivative = 4.53 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=-\frac {a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \, {\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x - 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{3} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{2} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{3} + {\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{2} + {\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x\right )} \log \left (d x + c\right ) + 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{3} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{2} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x\right )} \log \left (x\right )}{{\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{3} + {\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{2} + {\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x} \]
-(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + 2*(a*b^4 *c^4*d - 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*x^2 + (2*a*b ^4*c^5 - 3*a^2*b^3*c^4*d + 3*a^4*b*c^2*d^3 - 2*a^5*c*d^4)*x - 2*((b^5*c^4* d - 2*a*b^4*c^3*d^2)*x^3 + (b^5*c^5 - a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2)*x^2 + (a*b^4*c^5 - 2*a^2*b^3*c^4*d)*x)*log(b*x + a) - 2*((2*a^3*b^2*c*d^4 - a ^4*b*d^5)*x^3 + (2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x^2 + (2*a^4*b *c^2*d^3 - a^5*c*d^4)*x)*log(d*x + c) + 2*((b^5*c^4*d - 2*a*b^4*c^3*d^2 + 2*a^3*b^2*c*d^4 - a^4*b*d^5)*x^3 + (b^5*c^5 - a*b^4*c^4*d - 2*a^2*b^3*c^3* d^2 + 2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x^2 + (a*b^4*c^5 - 2*a^2* b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x)*log(x))/((a^3*b^4*c^6*d - 3*a^ 4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^3 + (a^3*b^4*c^7 - 2* a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^2 + (a^4*b^3*c^7 - 3*a^5* b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x)
Timed out. \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (144) = 288\).
Time = 0.23 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=\frac {2 \, {\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {2 \, {\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x}{{\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{3} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{2} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x} - \frac {2 \, {\left (b c + a d\right )} \log \left (x\right )}{a^{3} c^{3}} \]
2*(b^4*c - 2*a*b^3*d)*log(b*x + a)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5* b*c*d^2 - a^6*d^3) + 2*(2*b*c*d^3 - a*d^4)*log(d*x + c)/(b^3*c^6 - 3*a*b^2 *c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3 *c*d^2 + 2*(b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (2*b^3*c^3 - a*b^2* c^2*d - a^2*b*c*d^2 + 2*a^3*d^3)*x)/((a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^3 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^ 2*d^3)*x^2 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*x) - 2*(b*c + a*d )*log(x)/(a^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (144) = 288\).
Time = 0.28 (sec) , antiderivative size = 553, normalized size of antiderivative = 3.84 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=-\frac {b^{7}}{{\left (a^{2} b^{6} c^{2} - 2 \, a^{3} b^{5} c d + a^{4} b^{4} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left ({\left | -\frac {b c}{b x + a} + \frac {a b c}{{\left (b x + a\right )}^{2}} + \frac {2 \, a d}{b x + a} - \frac {a^{2} d}{{\left (b x + a\right )}^{2}} - d \right |}\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {{\left (b^{6} c^{4} - 2 \, a b^{5} c^{3} d + 4 \, a^{3} b^{3} c d^{3} - 2 \, a^{4} b^{2} d^{4}\right )} \log \left (\frac {{\left | -\frac {2 \, a b^{2} c}{b x + a} + b^{2} c - 2 \, a b d + \frac {2 \, a^{2} b d}{b x + a} - b^{2} {\left | c \right |} \right |}}{{\left | -\frac {2 \, a b^{2} c}{b x + a} + b^{2} c - 2 \, a b d + \frac {2 \, a^{2} b d}{b x + a} + b^{2} {\left | c \right |} \right |}}\right )}{{\left (a^{3} b^{3} c^{5} - 3 \, a^{4} b^{2} c^{4} d + 3 \, a^{5} b c^{3} d^{2} - a^{6} c^{2} d^{3}\right )} b^{2} {\left | c \right |}} - \frac {\frac {b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}}{a b c - a^{2} d} + \frac {b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + 2 \, a^{4} b^{2} d^{4}}{{\left (a b c - a^{2} d\right )} {\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{2} a^{2} {\left (\frac {b c}{b x + a} - \frac {a b c}{{\left (b x + a\right )}^{2}} - \frac {2 \, a d}{b x + a} + \frac {a^{2} d}{{\left (b x + a\right )}^{2}} + d\right )} c^{2}} \]
-b^7/((a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*(b*x + a)) - (b^4*c - 2* a*b^3*d)*log(abs(-b*c/(b*x + a) + a*b*c/(b*x + a)^2 + 2*a*d/(b*x + a) - a^ 2*d/(b*x + a)^2 - d))/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6 *d^3) + (b^6*c^4 - 2*a*b^5*c^3*d + 4*a^3*b^3*c*d^3 - 2*a^4*b^2*d^4)*log(ab s(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a*b*d + 2*a^2*b*d/(b*x + a) - b^2*abs(c ))/abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a*b*d + 2*a^2*b*d/(b*x + a) + b^2* abs(c)))/((a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3)* b^2*abs(c)) - ((b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - 2*a^3*b*d^ 4)/(a*b*c - a^2*d) + (b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3* b^3*c*d^3 + 2*a^4*b^2*d^4)/((a*b*c - a^2*d)*(b*x + a)*b))/((b*c - a*d)^2*a ^2*(b*c/(b*x + a) - a*b*c/(b*x + a)^2 - 2*a*d/(b*x + a) + a^2*d/(b*x + a)^ 2 + d)*c^2)
Time = 0.90 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.11 \[ \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx=-\frac {\frac {1}{a\,c}+\frac {2\,x^2\,\left (a^2\,b\,d^3-a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^3+\left (a\,d+b\,c\right )\,x^2+a\,c\,x}-\frac {\ln \left (a+b\,x\right )\,\left (2\,b^4\,c-4\,a\,b^3\,d\right )}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (2\,a\,d^4-4\,b\,c\,d^3\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}-\frac {2\,\ln \left (x\right )\,\left (a\,d+b\,c\right )}{a^3\,c^3} \]
- (1/(a*c) + (2*x^2*(a^2*b*d^3 + b^3*c^2*d - a*b^2*c*d^2))/(a^2*c^2*(a^2*d ^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(a*d + b*c)*(2*a^2*d^2 + 2*b^2*c^2 - 3*a*b *c*d))/(a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^2*(a*d + b*c) + a*c*x + b*d*x^3) - (log(a + b*x)*(2*b^4*c - 4*a*b^3*d))/(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2) - (log(c + d*x)*(2*a*d^4 - 4*b*c*d^3))/ (b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d) - (2*log(x)*(a*d + b*c))/(a^3*c^3)