Integrand size = 18, antiderivative size = 178 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {1}{2 a^2 c^2 x^2}+\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^3 (b c-a d)^2 (a+b x)}+\frac {d^4}{c^3 (b c-a d)^2 (c+d x)}+\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \log (x)}{a^4 c^4}-\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \] Output:
-1/2/a^2/c^2/x^2+2*(a*d+b*c)/a^3/c^3/x+b^4/a^3/(-a*d+b*c)^2/(b*x+a)+d^4/c^ 3/(-a*d+b*c)^2/(d*x+c)+(3*a^2*d^2+4*a*b*c*d+3*b^2*c^2)*ln(x)/a^4/c^4-b^4*( -5*a*d+3*b*c)*ln(b*x+a)/a^4/(-a*d+b*c)^3-d^4*(-3*a*d+5*b*c)*ln(d*x+c)/c^4/ (-a*d+b*c)^3
Time = 0.11 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {1}{2 a^2 c^2 x^2}+\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^3 (b c-a d)^2 (a+b x)}+\frac {d^4}{c^3 (b c-a d)^2 (c+d x)}+\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \log (x)}{a^4 c^4}+\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (-b c+a d)^3}+\frac {d^4 (-5 b c+3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \] Input:
Integrate[1/(x^3*(a + b*x)^2*(c + d*x)^2),x]
Output:
-1/2*1/(a^2*c^2*x^2) + (2*(b*c + a*d))/(a^3*c^3*x) + b^4/(a^3*(b*c - a*d)^ 2*(a + b*x)) + d^4/(c^3*(b*c - a*d)^2*(c + d*x)) + ((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^4*c^4) + (b^4*(3*b*c - 5*a*d)*Log[a + b*x])/(a^4* (-(b*c) + a*d)^3) + (d^4*(-5*b*c + 3*a*d)*Log[c + d*x])/(c^4*(b*c - a*d)^3 )
Time = 0.40 (sec) , antiderivative size = 178, 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.
| \(\displaystyle \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {b^5 (5 a d-3 b c)}{a^4 (a+b x) (a d-b c)^3}-\frac {b^5}{a^3 (a+b x)^2 (a d-b c)^2}-\frac {2 (a d+b c)}{a^3 c^3 x^2}+\frac {1}{a^2 c^2 x^3}+\frac {3 a^2 d^2+4 a b c d+3 b^2 c^2}{a^4 c^4 x}-\frac {d^5 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}-\frac {d^5}{c^3 (c+d x)^2 (b c-a d)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}+\frac {b^4}{a^3 (a+b x) (b c-a d)^2}+\frac {2 (a d+b c)}{a^3 c^3 x}-\frac {1}{2 a^2 c^2 x^2}+\frac {\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3}+\frac {d^4}{c^3 (c+d x) (b c-a d)^2}\) |
Input:
Int[1/(x^3*(a + b*x)^2*(c + d*x)^2),x]
Output:
-1/2*1/(a^2*c^2*x^2) + (2*(b*c + a*d))/(a^3*c^3*x) + b^4/(a^3*(b*c - a*d)^ 2*(a + b*x)) + d^4/(c^3*(b*c - a*d)^2*(c + d*x)) + ((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^4*c^4) - (b^4*(3*b*c - 5*a*d)*Log[a + b*x])/(a^4* (b*c - a*d)^3) - (d^4*(5*b*c - 3*a*d)*Log[c + d*x])/(c^4*(b*c - a*d)^3)
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 = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(\frac {d^{4}}{c^{3} \left (a d -b c \right )^{2} \left (x d +c \right )}-\frac {d^{4} \left (3 a d -5 b c \right ) \ln \left (x d +c \right )}{c^{4} \left (a d -b c \right )^{3}}-\frac {1}{2 a^{2} c^{2} x^{2}}-\frac {-2 a d -2 b c}{x \,c^{3} a^{3}}+\frac {\left (3 a^{2} d^{2}+4 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{4}}+\frac {b^{4}}{\left (a d -b c \right )^{2} a^{3} \left (b x +a \right )}-\frac {b^{4} \left (5 a d -3 b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{4}}\) | \(179\) |
| norman | \(\frac {-\frac {1}{2 a c}+\frac {3 \left (a d +b c \right ) x}{2 a^{2} c^{2}}+\frac {\left (-6 a^{5} d^{5}+a^{4} b c \,d^{4}+3 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d -6 c^{5} b^{5}\right ) x^{3}}{2 c^{4} a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-6 d^{4} a^{4}+a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d -6 c^{4} b^{4}\right ) b d \,x^{4}}{2 c^{4} a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{x^{2} \left (b x +a \right ) \left (x d +c \right )}+\frac {\left (3 a^{2} d^{2}+4 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{4}}-\frac {b^{4} \left (5 a d -3 b c \right ) \ln \left (b x +a \right )}{a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{4} \left (3 a d -5 b c \right ) \ln \left (x d +c \right )}{c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(381\) |
| risch | \(\frac {\frac {d b \left (3 a^{3} d^{3}-2 a^{2} b c \,d^{2}-2 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) x^{3}}{c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (6 d^{4} a^{4}-a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-a \,b^{3} c^{3} d +6 c^{4} b^{4}\right ) x^{2}}{2 a^{3} c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {3 \left (a d +b c \right ) x}{2 a^{2} c^{2}}-\frac {1}{2 a c}}{x^{2} \left (b x +a \right ) \left (x d +c \right )}-\frac {3 d^{5} \ln \left (-x d -c \right ) a}{c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {5 d^{4} \ln \left (-x d -c \right ) b}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {5 b^{4} \ln \left (b x +a \right ) d}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 b^{5} \ln \left (b x +a \right ) c}{a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 \ln \left (-x \right ) d^{2}}{a^{2} c^{4}}+\frac {4 \ln \left (-x \right ) b d}{a^{3} c^{3}}+\frac {3 \ln \left (-x \right ) b^{2}}{a^{4} c^{2}}\) | \(464\) |
| parallelrisch | \(\frac {-10 \ln \left (x \right ) x^{3} a^{4} b^{2} c^{2} d^{4}+10 \ln \left (x \right ) x^{3} a^{2} b^{4} c^{4} d^{2}+4 \ln \left (x \right ) x^{3} a \,b^{5} c^{5} d -10 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} c^{4} d^{2}-4 \ln \left (b x +a \right ) x^{3} a \,b^{5} c^{5} d +4 \ln \left (x d +c \right ) x^{3} a^{5} b c \,d^{5}+10 \ln \left (x d +c \right ) x^{3} a^{4} b^{2} c^{2} d^{4}-10 \ln \left (x \right ) x^{2} a^{5} b \,c^{2} d^{4}+10 \ln \left (x \right ) x^{2} a^{2} b^{4} c^{5} d -10 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} c^{5} d +10 \ln \left (x d +c \right ) x^{2} a^{5} b \,c^{2} d^{4}-10 \ln \left (x \right ) x^{4} a^{4} b^{2} c \,d^{5}+10 \ln \left (x \right ) x^{4} a \,b^{5} c^{4} d^{2}-10 \ln \left (b x +a \right ) x^{4} a \,b^{5} c^{4} d^{2}+10 \ln \left (x d +c \right ) x^{4} a^{4} b^{2} c \,d^{5}-4 \ln \left (x \right ) x^{3} a^{5} b c \,d^{5}-6 x^{4} a^{5} b \,d^{6}+6 x^{4} b^{6} c^{5} d +3 x \,a^{6} c^{2} d^{4}-3 x \,a^{2} b^{4} c^{6}+6 \ln \left (x \right ) x^{3} a^{6} d^{6}-6 \ln \left (x \right ) x^{3} b^{6} c^{6}+6 \ln \left (b x +a \right ) x^{3} b^{6} c^{6}-6 \ln \left (x d +c \right ) x^{3} a^{6} d^{6}+7 x^{4} a^{4} b^{2} c \,d^{5}+5 x^{4} a^{3} b^{3} c^{2} d^{4}-5 x^{4} a^{2} b^{4} c^{3} d^{3}-7 x^{4} a \,b^{5} c^{4} d^{2}+7 x^{3} a^{5} b c \,d^{5}+2 x^{3} a^{4} b^{2} c^{2} d^{4}-2 x^{3} a^{2} b^{4} c^{4} d^{2}-7 x^{3} a \,b^{5} c^{5} d -6 x \,a^{5} b \,c^{3} d^{3}+6 x \,a^{3} b^{3} c^{5} d +6 \ln \left (x \right ) x^{4} a^{5} b \,d^{6}-6 \ln \left (x \right ) x^{4} b^{6} c^{5} d +6 \ln \left (b x +a \right ) x^{4} b^{6} c^{5} d -6 \ln \left (x d +c \right ) x^{4} a^{5} b \,d^{6}+6 \ln \left (x \right ) x^{2} a^{6} c \,d^{5}-6 \ln \left (x \right ) x^{2} a \,b^{5} c^{6}+6 \ln \left (b x +a \right ) x^{2} a \,b^{5} c^{6}-6 \ln \left (x d +c \right ) x^{2} a^{6} c \,d^{5}-d^{3} c^{3} a^{6}+3 d^{2} c^{4} b \,a^{5}-3 d \,c^{5} b^{2} a^{4}+c^{6} b^{3} a^{3}+6 x^{3} b^{6} c^{6}-6 x^{3} a^{6} d^{6}}{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 (x d +c \right ) \left (b x +a \right ) x^{2} a^{4} c^{4}}\) | \(822\) |
Input:
int(1/x^3/(b*x+a)^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/c^3*d^4/(a*d-b*c)^2/(d*x+c)-d^4*(3*a*d-5*b*c)/c^4/(a*d-b*c)^3*ln(d*x+c)- 1/2/a^2/c^2/x^2-(-2*a*d-2*b*c)/x/c^3/a^3+(3*a^2*d^2+4*a*b*c*d+3*b^2*c^2)*l n(x)/a^4/c^4+b^4/(a*d-b*c)^2/a^3/(b*x+a)-b^4*(5*a*d-3*b*c)/(a*d-b*c)^3/a^4 *ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (176) = 352\).
Time = 15.97 (sec) , antiderivative size = 750, normalized size of antiderivative = 4.21 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} - 2 \, {\left (3 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{2} d^{4} - 3 \, a^{5} b c d^{5}\right )} x^{3} - {\left (6 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{3} d^{3} + 7 \, a^{5} b c^{2} d^{4} - 6 \, a^{6} c d^{5}\right )} x^{2} - 3 \, {\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x + 2 \, {\left ({\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2}\right )} x^{4} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2}\right )} x^{3} + {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x^{3} + {\left (5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5}\right )} x^{2}\right )} \log \left (d x + c\right ) - 2 \, {\left ({\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{4} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x^{3} + {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d + 5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{4} b^{4} c^{7} d - 3 \, a^{5} b^{3} c^{6} d^{2} + 3 \, a^{6} b^{2} c^{5} d^{3} - a^{7} b c^{4} d^{4}\right )} x^{4} + {\left (a^{4} b^{4} c^{8} - 2 \, a^{5} b^{3} c^{7} d + 2 \, a^{7} b c^{5} d^{3} - a^{8} c^{4} d^{4}\right )} x^{3} + {\left (a^{5} b^{3} c^{8} - 3 \, a^{6} b^{2} c^{7} d + 3 \, a^{7} b c^{6} d^{2} - a^{8} c^{5} d^{3}\right )} x^{2}\right )}} \] Input:
integrate(1/x^3/(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")
Output:
-1/2*(a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3 - 2*(3 *a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 + 5*a^4*b^2*c^2*d^4 - 3*a^5*b*c*d^5)*x^3 - (6*a*b^5*c^6 - 7*a^2*b^4*c^5*d - 5*a^3*b^3*c^4*d^2 + 5*a^4*b^2*c^3*d^3 + 7*a^5*b*c^2*d^4 - 6*a^6*c*d^5)*x^2 - 3*(a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2 *a^5*b*c^3*d^3 - a^6*c^2*d^4)*x + 2*((3*b^6*c^5*d - 5*a*b^5*c^4*d^2)*x^4 + (3*b^6*c^6 - 2*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2)*x^3 + (3*a*b^5*c^6 - 5*a^ 2*b^4*c^5*d)*x^2)*log(b*x + a) + 2*((5*a^4*b^2*c*d^5 - 3*a^5*b*d^6)*x^4 + (5*a^4*b^2*c^2*d^4 + 2*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + (5*a^5*b*c^2*d^4 - 3 *a^6*c*d^5)*x^2)*log(d*x + c) - 2*((3*b^6*c^5*d - 5*a*b^5*c^4*d^2 + 5*a^4* b^2*c*d^5 - 3*a^5*b*d^6)*x^4 + (3*b^6*c^6 - 2*a*b^5*c^5*d - 5*a^2*b^4*c^4* d^2 + 5*a^4*b^2*c^2*d^4 + 2*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + (3*a*b^5*c^6 - 5*a^2*b^4*c^5*d + 5*a^5*b*c^2*d^4 - 3*a^6*c*d^5)*x^2)*log(x))/((a^4*b^4*c^ 7*d - 3*a^5*b^3*c^6*d^2 + 3*a^6*b^2*c^5*d^3 - a^7*b*c^4*d^4)*x^4 + (a^4*b^ 4*c^8 - 2*a^5*b^3*c^7*d + 2*a^7*b*c^5*d^3 - a^8*c^4*d^4)*x^3 + (a^5*b^3*c^ 8 - 3*a^6*b^2*c^7*d + 3*a^7*b*c^6*d^2 - a^8*c^5*d^3)*x^2)
Timed out. \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(b*x+a)**2/(d*x+c)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (176) = 352\).
Time = 0.05 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.65 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {{\left (3 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}} - \frac {{\left (5 \, b c d^{4} - 3 \, a d^{5}\right )} \log \left (d x + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}} - \frac {a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - 2 \, {\left (3 \, b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} - 2 \, a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x^{3} - {\left (6 \, b^{4} c^{4} - a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + 6 \, a^{4} d^{4}\right )} x^{2} - 3 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x}{2 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{4} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x^{2}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{4}} \] Input:
integrate(1/x^3/(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")
Output:
-(3*b^5*c - 5*a*b^4*d)*log(b*x + a)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6 *b*c*d^2 - a^7*d^3) - (5*b*c*d^4 - 3*a*d^5)*log(d*x + c)/(b^3*c^7 - 3*a*b^ 2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3) - 1/2*(a^2*b^2*c^4 - 2*a^3*b*c^3* d + a^4*c^2*d^2 - 2*(3*b^4*c^3*d - 2*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a ^3*b*d^4)*x^3 - (6*b^4*c^4 - a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 - a^3*b*c*d^3 + 6*a^4*d^4)*x^2 - 3*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d ^3)*x)/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^4 + (a^3*b^3 *c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + (a^4*b^2*c^6 - 2 *a^5*b*c^5*d + a^6*c^4*d^2)*x^2) + (3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*log (x)/(a^4*c^4)
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (176) = 352\).
Time = 0.14 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.58 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=\frac {b^{9}}{{\left (a^{3} b^{7} c^{2} - 2 \, a^{4} b^{6} c d + a^{5} b^{5} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (5 \, b^{2} c d^{4} - 3 \, a b d^{5}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{7} - 3 \, a b^{3} c^{6} d + 3 \, a^{2} b^{2} c^{5} d^{2} - a^{3} b c^{4} d^{3}} + \frac {{\left (3 \, b^{3} c^{2} + 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b c^{4}} + \frac {5 \, b^{5} c^{5} d - 11 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{3} d^{3} + 7 \, a^{3} b^{2} c^{2} d^{4} - 6 \, a^{4} b c d^{5} + \frac {5 \, b^{7} c^{6} - 22 \, a b^{6} c^{5} d + 28 \, a^{2} b^{5} c^{4} d^{2} - 2 \, a^{3} b^{4} c^{3} d^{3} - 17 \, a^{4} b^{3} c^{2} d^{4} + 12 \, a^{5} b^{2} c d^{5}}{{\left (b x + a\right )} b} - \frac {2 \, {\left (3 \, a b^{8} c^{6} - 10 \, a^{2} b^{7} c^{5} d + 10 \, a^{3} b^{6} c^{4} d^{2} - 5 \, a^{5} b^{4} c^{2} d^{4} + 3 \, a^{6} b^{3} c d^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \, {\left (b c - a d\right )}^{3} a^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} c^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \] Input:
integrate(1/x^3/(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")
Output:
b^9/((a^3*b^7*c^2 - 2*a^4*b^6*c*d + a^5*b^5*d^2)*(b*x + a)) - (5*b^2*c*d^4 - 3*a*b*d^5)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^4*c^7 - 3*a*b ^3*c^6*d + 3*a^2*b^2*c^5*d^2 - a^3*b*c^4*d^3) + (3*b^3*c^2 + 4*a*b^2*c*d + 3*a^2*b*d^2)*log(abs(-a/(b*x + a) + 1))/(a^4*b*c^4) + 1/2*(5*b^5*c^5*d - 11*a*b^4*c^4*d^2 + 3*a^2*b^3*c^3*d^3 + 7*a^3*b^2*c^2*d^4 - 6*a^4*b*c*d^5 + (5*b^7*c^6 - 22*a*b^6*c^5*d + 28*a^2*b^5*c^4*d^2 - 2*a^3*b^4*c^3*d^3 - 17 *a^4*b^3*c^2*d^4 + 12*a^5*b^2*c*d^5)/((b*x + a)*b) - 2*(3*a*b^8*c^6 - 10*a ^2*b^7*c^5*d + 10*a^3*b^6*c^4*d^2 - 5*a^5*b^4*c^2*d^4 + 3*a^6*b^3*c*d^5)/( (b*x + a)^2*b^2))/((b*c - a*d)^3*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c ^4*(a/(b*x + a) - 1)^2)
Time = 0.90 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (3\,b^5\,c-5\,a\,b^4\,d\right )}{a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d-a^4\,b^3\,c^3}-\frac {\frac {1}{2\,a\,c}-\frac {3\,x\,\left (a\,d+b\,c\right )}{2\,a^2\,c^2}+\frac {x^2\,\left (-6\,a^4\,d^4+a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d-6\,b^4\,c^4\right )}{2\,a^3\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {x^3\,\left (a\,d+b\,c\right )\,\left (3\,a^2\,b\,d^3-5\,a\,b^2\,c\,d^2+3\,b^3\,c^2\,d\right )}{a^3\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^4+\left (a\,d+b\,c\right )\,x^3+a\,c\,x^2}+\frac {\ln \left (c+d\,x\right )\,\left (3\,a\,d^5-5\,b\,c\,d^4\right )}{-a^3\,c^4\,d^3+3\,a^2\,b\,c^5\,d^2-3\,a\,b^2\,c^6\,d+b^3\,c^7}+\frac {\ln \left (x\right )\,\left (3\,a^2\,d^2+4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4\,c^4} \] Input:
int(1/(x^3*(a + b*x)^2*(c + d*x)^2),x)
Output:
(log(a + b*x)*(3*b^5*c - 5*a*b^4*d))/(a^7*d^3 - a^4*b^3*c^3 + 3*a^5*b^2*c^ 2*d - 3*a^6*b*c*d^2) - (1/(2*a*c) - (3*x*(a*d + b*c))/(2*a^2*c^2) + (x^2*( 6*a^2*b^2*c^2*d^2 - 6*b^4*c^4 - 6*a^4*d^4 + a*b^3*c^3*d + a^3*b*c*d^3))/(2 *a^3*c^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (x^3*(a*d + b*c)*(3*a^2*b*d^3 + 3*b^3*c^2*d - 5*a*b^2*c*d^2))/(a^3*c^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) /(x^3*(a*d + b*c) + a*c*x^2 + b*d*x^4) + (log(c + d*x)*(3*a*d^5 - 5*b*c*d^ 4))/(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b^2*c^6*d) + (log(x)*(3 *a^2*d^2 + 3*b^2*c^2 + 4*a*b*c*d))/(a^4*c^4)
Time = 0.18 (sec) , antiderivative size = 887, normalized size of antiderivative = 4.98 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx =\text {Too large to display} \] Input:
int(1/x^3/(b*x+a)^2/(d*x+c)^2,x)
Output:
( - 10*log(a + b*x)*a**2*b**4*c**5*d*x**2 - 10*log(a + b*x)*a**2*b**4*c**4 *d**2*x**3 + 6*log(a + b*x)*a*b**5*c**6*x**2 - 4*log(a + b*x)*a*b**5*c**5* d*x**3 - 10*log(a + b*x)*a*b**5*c**4*d**2*x**4 + 6*log(a + b*x)*b**6*c**6* x**3 + 6*log(a + b*x)*b**6*c**5*d*x**4 - 6*log(c + d*x)*a**6*c*d**5*x**2 - 6*log(c + d*x)*a**6*d**6*x**3 + 10*log(c + d*x)*a**5*b*c**2*d**4*x**2 + 4 *log(c + d*x)*a**5*b*c*d**5*x**3 - 6*log(c + d*x)*a**5*b*d**6*x**4 + 10*lo g(c + d*x)*a**4*b**2*c**2*d**4*x**3 + 10*log(c + d*x)*a**4*b**2*c*d**5*x** 4 + 6*log(x)*a**6*c*d**5*x**2 + 6*log(x)*a**6*d**6*x**3 - 10*log(x)*a**5*b *c**2*d**4*x**2 - 4*log(x)*a**5*b*c*d**5*x**3 + 6*log(x)*a**5*b*d**6*x**4 - 10*log(x)*a**4*b**2*c**2*d**4*x**3 - 10*log(x)*a**4*b**2*c*d**5*x**4 + 1 0*log(x)*a**2*b**4*c**5*d*x**2 + 10*log(x)*a**2*b**4*c**4*d**2*x**3 - 6*lo g(x)*a*b**5*c**6*x**2 + 4*log(x)*a*b**5*c**5*d*x**3 + 10*log(x)*a*b**5*c** 4*d**2*x**4 - 6*log(x)*b**6*c**6*x**3 - 6*log(x)*b**6*c**5*d*x**4 - a**6*c **3*d**3 + 3*a**6*c**2*d**4*x + 6*a**6*c*d**5*x**2 + 3*a**5*b*c**4*d**2 - 6*a**5*b*c**3*d**3*x - 13*a**5*b*c**2*d**4*x**2 - 3*a**4*b**2*c**5*d + 11* a**4*b**2*c**3*d**3*x**2 - 6*a**4*b**2*c*d**5*x**4 + a**3*b**3*c**6 + 6*a* *3*b**3*c**5*d*x - 11*a**3*b**3*c**4*d**2*x**2 + 16*a**3*b**3*c**2*d**4*x* *4 - 3*a**2*b**4*c**6*x + 13*a**2*b**4*c**5*d*x**2 - 16*a**2*b**4*c**3*d** 3*x**4 - 6*a*b**5*c**6*x**2 + 6*a*b**5*c**4*d**2*x**4)/(2*a**4*c**4*x**2*( a**4*c*d**3 + a**4*d**4*x - 3*a**3*b*c**2*d**2 - 2*a**3*b*c*d**3*x + a*...