Integrand size = 18, antiderivative size = 242 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {1}{a^3 c^3 x}-\frac {b^4}{2 a^2 (b c-a d)^3 (a+b x)^2}-\frac {b^4 (2 b c-5 a d)}{a^3 (b c-a d)^4 (a+b x)}+\frac {d^4}{2 c^2 (b c-a d)^3 (c+d x)^2}+\frac {d^4 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)}-\frac {3 (b c+a d) \log (x)}{a^4 c^4}+\frac {3 b^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac {3 d^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^5} \]
-1/a^3/c^3/x-1/2*b^4/a^2/(-a*d+b*c)^3/(b*x+a)^2-b^4*(-5*a*d+2*b*c)/a^3/(-a *d+b*c)^4/(b*x+a)+1/2*d^4/c^2/(-a*d+b*c)^3/(d*x+c)^2+d^4*(-2*a*d+5*b*c)/c^ 3/(-a*d+b*c)^4/(d*x+c)-3*(a*d+b*c)*ln(x)/a^4/c^4+3*b^4*(5*a^2*d^2-4*a*b*c* d+b^2*c^2)*ln(b*x+a)/a^4/(-a*d+b*c)^5-3*d^4*(a^2*d^2-4*a*b*c*d+5*b^2*c^2)* ln(d*x+c)/c^4/(-a*d+b*c)^5
Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {1}{a^3 c^3 x}+\frac {b^4}{2 a^2 (-b c+a d)^3 (a+b x)^2}+\frac {b^4 (-2 b c+5 a d)}{a^3 (b c-a d)^4 (a+b x)}+\frac {d^4}{2 c^2 (b c-a d)^3 (c+d x)^2}+\frac {d^4 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)}-\frac {3 (b c+a d) \log (x)}{a^4 c^4}-\frac {3 b^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (a+b x)}{a^4 (-b c+a d)^5}-\frac {3 d^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^5} \]
-(1/(a^3*c^3*x)) + b^4/(2*a^2*(-(b*c) + a*d)^3*(a + b*x)^2) + (b^4*(-2*b*c + 5*a*d))/(a^3*(b*c - a*d)^4*(a + b*x)) + d^4/(2*c^2*(b*c - a*d)^3*(c + d *x)^2) + (d^4*(5*b*c - 2*a*d))/(c^3*(b*c - a*d)^4*(c + d*x)) - (3*(b*c + a *d)*Log[x])/(a^4*c^4) - (3*b^4*(b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)*Log[a + b *x])/(a^4*(-(b*c) + a*d)^5) - (3*d^4*(5*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log [c + d*x])/(c^4*(b*c - a*d)^5)
Time = 0.53 (sec) , antiderivative size = 242, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {3 (a d+b c)}{a^4 c^4 x}-\frac {b^5 (5 a d-2 b c)}{a^3 (a+b x)^2 (a d-b c)^4}+\frac {1}{a^3 c^3 x^2}-\frac {b^5}{a^2 (a+b x)^3 (a d-b c)^3}-\frac {3 d^5 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right )}{c^4 (c+d x) (b c-a d)^5}-\frac {3 b^5 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right )}{a^4 (a+b x) (a d-b c)^5}-\frac {d^5 (5 b c-2 a d)}{c^3 (c+d x)^2 (b c-a d)^4}-\frac {d^5}{c^2 (c+d x)^3 (b c-a d)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \log (x) (a d+b c)}{a^4 c^4}-\frac {b^4 (2 b c-5 a d)}{a^3 (a+b x) (b c-a d)^4}-\frac {1}{a^3 c^3 x}-\frac {b^4}{2 a^2 (a+b x)^2 (b c-a d)^3}-\frac {3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}+\frac {3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}+\frac {d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac {d^4}{2 c^2 (c+d x)^2 (b c-a d)^3}\) |
-(1/(a^3*c^3*x)) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x)^2) - (b^4*(2*b*c - 5 *a*d))/(a^3*(b*c - a*d)^4*(a + b*x)) + d^4/(2*c^2*(b*c - a*d)^3*(c + d*x)^ 2) + (d^4*(5*b*c - 2*a*d))/(c^3*(b*c - a*d)^4*(c + d*x)) - (3*(b*c + a*d)* Log[x])/(a^4*c^4) + (3*b^4*(b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)*Log[a + b*x]) /(a^4*(b*c - a*d)^5) - (3*d^4*(5*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d* x])/(c^4*(b*c - a*d)^5)
3.4.18.3.1 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 = 0.58 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {1}{a^{3} c^{3} x}+\frac {\left (-3 a d -3 b c \right ) \ln \left (x \right )}{a^{4} c^{4}}-\frac {d^{4}}{2 c^{2} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {d^{4} \left (2 a d -5 b c \right )}{c^{3} \left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {3 d^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{4} \left (a d -b c \right )^{5}}+\frac {b^{4}}{2 a^{2} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {b^{4} \left (5 a d -2 b c \right )}{a^{3} \left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {3 b^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4} \left (a d -b c \right )^{5}}\) | \(240\) |
| norman | \(\frac {\frac {\left (6 a^{6} d^{6}-14 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+5 a^{2} b^{4} c^{4} d^{2}-14 a \,b^{5} c^{5} d +6 b^{6} c^{6}\right ) x^{2}}{c^{3} a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {d b \left (9 a^{6} d^{6}-17 a^{5} b c \,d^{5}-6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}-6 a^{2} b^{4} c^{4} d^{2}-17 a \,b^{5} c^{5} d +9 b^{6} c^{6}\right ) x^{4}}{c^{4} a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {1}{a c}+\frac {\left (9 a^{7} d^{7}+a^{6} b c \,d^{6}-48 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+20 a^{3} b^{4} c^{4} d^{3}-48 a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +9 b^{7} c^{7}\right ) x^{3}}{2 c^{4} a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b^{2} d^{2} \left (9 a^{5} d^{5}-23 a^{4} b c \,d^{4}+8 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}-23 a \,b^{4} c^{4} d +9 b^{5} c^{5}\right ) x^{5}}{2 c^{4} a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{x \left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {3 \left (a d +b c \right ) \ln \left (x \right )}{a^{4} c^{4}}-\frac {3 b^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}+\frac {3 d^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{4} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(804\) |
| risch | \(\text {Expression too large to display}\) | \(1052\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1949\) |
-1/a^3/c^3/x+(-3*a*d-3*b*c)/a^4/c^4*ln(x)-1/2*d^4/c^2/(a*d-b*c)^3/(d*x+c)^ 2-d^4*(2*a*d-5*b*c)/c^3/(a*d-b*c)^4/(d*x+c)+3*d^4*(a^2*d^2-4*a*b*c*d+5*b^2 *c^2)/c^4/(a*d-b*c)^5*ln(d*x+c)+1/2*b^4/a^2/(a*d-b*c)^3/(b*x+a)^2+b^4*(5*a *d-2*b*c)/a^3/(a*d-b*c)^4/(b*x+a)-3*b^4*(5*a^2*d^2-4*a*b*c*d+b^2*c^2)/a^4/ (a*d-b*c)^5*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1798 vs. \(2 (238) = 476\).
Time = 38.53 (sec) , antiderivative size = 1798, normalized size of antiderivative = 7.43 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]
-1/2*(2*a^3*b^5*c^8 - 10*a^4*b^4*c^7*d + 20*a^5*b^3*c^6*d^2 - 20*a^6*b^2*c ^5*d^3 + 10*a^7*b*c^4*d^4 - 2*a^8*c^3*d^5 + 6*(a*b^7*c^6*d^2 - 4*a^2*b^6*c ^5*d^3 + 5*a^3*b^5*c^4*d^4 - 5*a^4*b^4*c^3*d^5 + 4*a^5*b^3*c^2*d^6 - a^6*b ^2*c*d^7)*x^4 + 3*(4*a*b^7*c^7*d - 13*a^2*b^6*c^6*d^2 + 8*a^3*b^5*c^5*d^3 - 8*a^5*b^3*c^3*d^5 + 13*a^6*b^2*c^2*d^6 - 4*a^7*b*c*d^7)*x^3 + 2*(3*a*b^7 *c^8 - 3*a^2*b^6*c^7*d - 20*a^3*b^5*c^6*d^2 + 36*a^4*b^4*c^5*d^3 - 36*a^5* b^3*c^4*d^4 + 20*a^6*b^2*c^3*d^5 + 3*a^7*b*c^2*d^6 - 3*a^8*c*d^7)*x^2 + (9 *a^2*b^6*c^8 - 32*a^3*b^5*c^7*d + 31*a^4*b^4*c^6*d^2 - 31*a^6*b^2*c^4*d^4 + 32*a^7*b*c^3*d^5 - 9*a^8*c^2*d^6)*x - 6*((b^8*c^6*d^2 - 4*a*b^7*c^5*d^3 + 5*a^2*b^6*c^4*d^4)*x^5 + 2*(b^8*c^7*d - 3*a*b^7*c^6*d^2 + a^2*b^6*c^5*d^ 3 + 5*a^3*b^5*c^4*d^4)*x^4 + (b^8*c^8 - 10*a^2*b^6*c^6*d^2 + 16*a^3*b^5*c^ 5*d^3 + 5*a^4*b^4*c^4*d^4)*x^3 + 2*(a*b^7*c^8 - 3*a^2*b^6*c^7*d + a^3*b^5* c^6*d^2 + 5*a^4*b^4*c^5*d^3)*x^2 + (a^2*b^6*c^8 - 4*a^3*b^5*c^7*d + 5*a^4* b^4*c^6*d^2)*x)*log(b*x + a) + 6*((5*a^4*b^4*c^2*d^6 - 4*a^5*b^3*c*d^7 + a ^6*b^2*d^8)*x^5 + 2*(5*a^4*b^4*c^3*d^5 + a^5*b^3*c^2*d^6 - 3*a^6*b^2*c*d^7 + a^7*b*d^8)*x^4 + (5*a^4*b^4*c^4*d^4 + 16*a^5*b^3*c^3*d^5 - 10*a^6*b^2*c ^2*d^6 + a^8*d^8)*x^3 + 2*(5*a^5*b^3*c^4*d^4 + a^6*b^2*c^3*d^5 - 3*a^7*b*c ^2*d^6 + a^8*c*d^7)*x^2 + (5*a^6*b^2*c^4*d^4 - 4*a^7*b*c^3*d^5 + a^8*c^2*d ^6)*x)*log(d*x + c) + 6*((b^8*c^6*d^2 - 4*a*b^7*c^5*d^3 + 5*a^2*b^6*c^4*d^ 4 - 5*a^4*b^4*c^2*d^6 + 4*a^5*b^3*c*d^7 - a^6*b^2*d^8)*x^5 + 2*(b^8*c^7...
Timed out. \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 936 vs. \(2 (238) = 476\).
Time = 0.24 (sec) , antiderivative size = 936, normalized size of antiderivative = 3.87 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (b^{6} c^{2} - 4 \, a b^{5} c d + 5 \, a^{2} b^{4} d^{2}\right )} \log \left (b x + a\right )}{a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{4} - 4 \, a b c d^{5} + a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{5} c^{9} - 5 \, a b^{4} c^{8} d + 10 \, a^{2} b^{3} c^{7} d^{2} - 10 \, a^{3} b^{2} c^{6} d^{3} + 5 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}} - \frac {2 \, a^{2} b^{4} c^{6} - 8 \, a^{3} b^{3} c^{5} d + 12 \, a^{4} b^{2} c^{4} d^{2} - 8 \, a^{5} b c^{3} d^{3} + 2 \, a^{6} c^{2} d^{4} + 6 \, {\left (b^{6} c^{4} d^{2} - 3 \, a b^{5} c^{3} d^{3} + 2 \, a^{2} b^{4} c^{2} d^{4} - 3 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 3 \, {\left (4 \, b^{6} c^{5} d - 9 \, a b^{5} c^{4} d^{2} - a^{2} b^{4} c^{3} d^{3} - a^{3} b^{3} c^{2} d^{4} - 9 \, a^{4} b^{2} c d^{5} + 4 \, a^{5} b d^{6}\right )} x^{3} + 2 \, {\left (3 \, b^{6} c^{6} - 20 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 20 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{6} d^{6}\right )} x^{2} + {\left (9 \, a b^{5} c^{6} - 23 \, a^{2} b^{4} c^{5} d + 8 \, a^{3} b^{3} c^{4} d^{2} + 8 \, a^{4} b^{2} c^{3} d^{3} - 23 \, a^{5} b c^{2} d^{4} + 9 \, a^{6} c d^{5}\right )} x}{2 \, {\left ({\left (a^{3} b^{6} c^{7} d^{2} - 4 \, a^{4} b^{5} c^{6} d^{3} + 6 \, a^{5} b^{4} c^{5} d^{4} - 4 \, a^{6} b^{3} c^{4} d^{5} + a^{7} b^{2} c^{3} d^{6}\right )} x^{5} + 2 \, {\left (a^{3} b^{6} c^{8} d - 3 \, a^{4} b^{5} c^{7} d^{2} + 2 \, a^{5} b^{4} c^{6} d^{3} + 2 \, a^{6} b^{3} c^{5} d^{4} - 3 \, a^{7} b^{2} c^{4} d^{5} + a^{8} b c^{3} d^{6}\right )} x^{4} + {\left (a^{3} b^{6} c^{9} - 9 \, a^{5} b^{4} c^{7} d^{2} + 16 \, a^{6} b^{3} c^{6} d^{3} - 9 \, a^{7} b^{2} c^{5} d^{4} + a^{9} c^{3} d^{6}\right )} x^{3} + 2 \, {\left (a^{4} b^{5} c^{9} - 3 \, a^{5} b^{4} c^{8} d + 2 \, a^{6} b^{3} c^{7} d^{2} + 2 \, a^{7} b^{2} c^{6} d^{3} - 3 \, a^{8} b c^{5} d^{4} + a^{9} c^{4} d^{5}\right )} x^{2} + {\left (a^{5} b^{4} c^{9} - 4 \, a^{6} b^{3} c^{8} d + 6 \, a^{7} b^{2} c^{7} d^{2} - 4 \, a^{8} b c^{6} d^{3} + a^{9} c^{5} d^{4}\right )} x\right )}} - \frac {3 \, {\left (b c + a d\right )} \log \left (x\right )}{a^{4} c^{4}} \]
3*(b^6*c^2 - 4*a*b^5*c*d + 5*a^2*b^4*d^2)*log(b*x + a)/(a^4*b^5*c^5 - 5*a^ 5*b^4*c^4*d + 10*a^6*b^3*c^3*d^2 - 10*a^7*b^2*c^2*d^3 + 5*a^8*b*c*d^4 - a^ 9*d^5) - 3*(5*b^2*c^2*d^4 - 4*a*b*c*d^5 + a^2*d^6)*log(d*x + c)/(b^5*c^9 - 5*a*b^4*c^8*d + 10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 + 5*a^4*b*c^5*d^4 - a^5*c^4*d^5) - 1/2*(2*a^2*b^4*c^6 - 8*a^3*b^3*c^5*d + 12*a^4*b^2*c^4*d^ 2 - 8*a^5*b*c^3*d^3 + 2*a^6*c^2*d^4 + 6*(b^6*c^4*d^2 - 3*a*b^5*c^3*d^3 + 2 *a^2*b^4*c^2*d^4 - 3*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 3*(4*b^6*c^5*d - 9 *a*b^5*c^4*d^2 - a^2*b^4*c^3*d^3 - a^3*b^3*c^2*d^4 - 9*a^4*b^2*c*d^5 + 4*a ^5*b*d^6)*x^3 + 2*(3*b^6*c^6 - 20*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 2 0*a^4*b^2*c^2*d^4 + 3*a^6*d^6)*x^2 + (9*a*b^5*c^6 - 23*a^2*b^4*c^5*d + 8*a ^3*b^3*c^4*d^2 + 8*a^4*b^2*c^3*d^3 - 23*a^5*b*c^2*d^4 + 9*a^6*c*d^5)*x)/(( a^3*b^6*c^7*d^2 - 4*a^4*b^5*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^3*c^4*d^ 5 + a^7*b^2*c^3*d^6)*x^5 + 2*(a^3*b^6*c^8*d - 3*a^4*b^5*c^7*d^2 + 2*a^5*b^ 4*c^6*d^3 + 2*a^6*b^3*c^5*d^4 - 3*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6)*x^4 + ( a^3*b^6*c^9 - 9*a^5*b^4*c^7*d^2 + 16*a^6*b^3*c^6*d^3 - 9*a^7*b^2*c^5*d^4 + a^9*c^3*d^6)*x^3 + 2*(a^4*b^5*c^9 - 3*a^5*b^4*c^8*d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8*b*c^5*d^4 + a^9*c^4*d^5)*x^2 + (a^5*b^4*c^9 - 4 *a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6*d^3 + a^9*c^5*d^4)*x) - 3 *(b*c + a*d)*log(x)/(a^4*c^4)
Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (238) = 476\).
Time = 0.31 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (b^{7} c^{2} - 4 \, a b^{6} c d + 5 \, a^{2} b^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{5} - 4 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{9} d - 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{7} d^{3} - 10 \, a^{3} b^{2} c^{6} d^{4} + 5 \, a^{4} b c^{5} d^{5} - a^{5} c^{4} d^{6}} - \frac {3 \, {\left (b c + a d\right )} \log \left ({\left | x \right |}\right )}{a^{4} c^{4}} - \frac {2 \, a^{3} b^{4} c^{7} - 8 \, a^{4} b^{3} c^{6} d + 12 \, a^{5} b^{2} c^{5} d^{2} - 8 \, a^{6} b c^{4} d^{3} + 2 \, a^{7} c^{3} d^{4} + 6 \, {\left (a b^{6} c^{5} d^{2} - 3 \, a^{2} b^{5} c^{4} d^{3} + 2 \, a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6}\right )} x^{4} + 3 \, {\left (4 \, a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} - a^{3} b^{4} c^{4} d^{3} - a^{4} b^{3} c^{3} d^{4} - 9 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6}\right )} x^{3} + 2 \, {\left (3 \, a b^{6} c^{7} - 20 \, a^{3} b^{4} c^{5} d^{2} + 16 \, a^{4} b^{3} c^{4} d^{3} - 20 \, a^{5} b^{2} c^{3} d^{4} + 3 \, a^{7} c d^{6}\right )} x^{2} + {\left (9 \, a^{2} b^{5} c^{7} - 23 \, a^{3} b^{4} c^{6} d + 8 \, a^{4} b^{3} c^{5} d^{2} + 8 \, a^{5} b^{2} c^{4} d^{3} - 23 \, a^{6} b c^{3} d^{4} + 9 \, a^{7} c^{2} d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} a^{4} c^{4} x} \]
3*(b^7*c^2 - 4*a*b^6*c*d + 5*a^2*b^5*d^2)*log(abs(b*x + a))/(a^4*b^6*c^5 - 5*a^5*b^5*c^4*d + 10*a^6*b^4*c^3*d^2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d ^4 - a^9*b*d^5) - 3*(5*b^2*c^2*d^5 - 4*a*b*c*d^6 + a^2*d^7)*log(abs(d*x + c))/(b^5*c^9*d - 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^7*d^3 - 10*a^3*b^2*c^6*d^4 + 5*a^4*b*c^5*d^5 - a^5*c^4*d^6) - 3*(b*c + a*d)*log(abs(x))/(a^4*c^4) - 1/2*(2*a^3*b^4*c^7 - 8*a^4*b^3*c^6*d + 12*a^5*b^2*c^5*d^2 - 8*a^6*b*c^4*d^ 3 + 2*a^7*c^3*d^4 + 6*(a*b^6*c^5*d^2 - 3*a^2*b^5*c^4*d^3 + 2*a^3*b^4*c^3*d ^4 - 3*a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6)*x^4 + 3*(4*a*b^6*c^6*d - 9*a^2*b^5 *c^5*d^2 - a^3*b^4*c^4*d^3 - a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + 4*a^6*b *c*d^6)*x^3 + 2*(3*a*b^6*c^7 - 20*a^3*b^4*c^5*d^2 + 16*a^4*b^3*c^4*d^3 - 2 0*a^5*b^2*c^3*d^4 + 3*a^7*c*d^6)*x^2 + (9*a^2*b^5*c^7 - 23*a^3*b^4*c^6*d + 8*a^4*b^3*c^5*d^2 + 8*a^5*b^2*c^4*d^3 - 23*a^6*b*c^3*d^4 + 9*a^7*c^2*d^5) *x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*a^4*c^4*x)
Time = 1.75 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.40 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {\frac {1}{a\,c}+\frac {3\,x^4\,\left (a^4\,b^2\,d^6-3\,a^3\,b^3\,c\,d^5+2\,a^2\,b^4\,c^2\,d^4-3\,a\,b^5\,c^3\,d^3+b^6\,c^4\,d^2\right )}{a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x^2\,\left (3\,a^6\,d^6-20\,a^4\,b^2\,c^2\,d^4+16\,a^3\,b^3\,c^3\,d^3-20\,a^2\,b^4\,c^4\,d^2+3\,b^6\,c^6\right )}{a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {3\,x^3\,\left (-4\,a^5\,b\,d^6+9\,a^4\,b^2\,c\,d^5+a^3\,b^3\,c^2\,d^4+a^2\,b^4\,c^3\,d^3+9\,a\,b^5\,c^4\,d^2-4\,b^6\,c^5\,d\right )}{2\,a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x\,\left (9\,a^5\,d^5-23\,a^4\,b\,c\,d^4+8\,a^3\,b^2\,c^2\,d^3+8\,a^2\,b^3\,c^3\,d^2-23\,a\,b^4\,c^4\,d+9\,b^5\,c^5\right )}{2\,a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^2\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^4\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2\,x+b^2\,d^2\,x^5}-\frac {\ln \left (a+b\,x\right )\,\left (15\,a^2\,b^4\,d^2-12\,a\,b^5\,c\,d+3\,b^6\,c^2\right )}{a^9\,d^5-5\,a^8\,b\,c\,d^4+10\,a^7\,b^2\,c^2\,d^3-10\,a^6\,b^3\,c^3\,d^2+5\,a^5\,b^4\,c^4\,d-a^4\,b^5\,c^5}-\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,d^6-12\,a\,b\,c\,d^5+15\,b^2\,c^2\,d^4\right )}{-a^5\,c^4\,d^5+5\,a^4\,b\,c^5\,d^4-10\,a^3\,b^2\,c^6\,d^3+10\,a^2\,b^3\,c^7\,d^2-5\,a\,b^4\,c^8\,d+b^5\,c^9}-\frac {3\,\ln \left (x\right )\,\left (a\,d+b\,c\right )}{a^4\,c^4} \]
- (1/(a*c) + (3*x^4*(a^4*b^2*d^6 + b^6*c^4*d^2 - 3*a*b^5*c^3*d^3 - 3*a^3*b ^3*c*d^5 + 2*a^2*b^4*c^2*d^4))/(a^3*c^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2 *d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x^2*(3*a^6*d^6 + 3*b^6*c^6 - 20* a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 20*a^4*b^2*c^2*d^4))/(a^3*c^3*(a^4* d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (3*x ^3*(9*a*b^5*c^4*d^2 - 4*b^6*c^5*d - 4*a^5*b*d^6 + 9*a^4*b^2*c*d^5 + a^2*b^ 4*c^3*d^3 + a^3*b^3*c^2*d^4))/(2*a^3*c^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^ 2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x*(9*a^5*d^5 + 9*b^5*c^5 + 8*a^ 2*b^3*c^3*d^2 + 8*a^3*b^2*c^2*d^3 - 23*a*b^4*c^4*d - 23*a^4*b*c*d^4))/(2*a ^2*c^2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c* d^3)))/(x^3*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^2*(2*a*b*c^2 + 2*a^2*c*d) + x^4*(2*a*b*d^2 + 2*b^2*c*d) + a^2*c^2*x + b^2*d^2*x^5) - (log(a + b*x)*( 3*b^6*c^2 + 15*a^2*b^4*d^2 - 12*a*b^5*c*d))/(a^9*d^5 - a^4*b^5*c^5 + 5*a^5 *b^4*c^4*d - 10*a^6*b^3*c^3*d^2 + 10*a^7*b^2*c^2*d^3 - 5*a^8*b*c*d^4) - (l og(c + d*x)*(3*a^2*d^6 + 15*b^2*c^2*d^4 - 12*a*b*c*d^5))/(b^5*c^9 - a^5*c^ 4*d^5 + 5*a^4*b*c^5*d^4 + 10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 - 5*a*b^ 4*c^8*d) - (3*log(x)*(a*d + b*c))/(a^4*c^4)