Integrand size = 18, antiderivative size = 228 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=-\frac {1}{2 a^2 c^3 x^2}+\frac {2 b c+3 a d}{a^3 c^4 x}+\frac {b^5}{a^3 (b c-a d)^3 (a+b x)}+\frac {d^4}{2 c^3 (b c-a d)^2 (c+d x)^2}+\frac {d^4 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)}+\frac {3 \left (b^2 c^2+2 a b c d+2 a^2 d^2\right ) \log (x)}{a^4 c^5}-\frac {3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}-\frac {3 d^4 \left (5 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log (c+d x)}{c^5 (b c-a d)^4} \]
-1/2/a^2/c^3/x^2+(3*a*d+2*b*c)/a^3/c^4/x+b^5/a^3/(-a*d+b*c)^3/(b*x+a)+1/2* d^4/c^3/(-a*d+b*c)^2/(d*x+c)^2+d^4*(-3*a*d+5*b*c)/c^4/(-a*d+b*c)^3/(d*x+c) +3*(2*a^2*d^2+2*a*b*c*d+b^2*c^2)*ln(x)/a^4/c^5-3*b^5*(-2*a*d+b*c)*ln(b*x+a )/a^4/(-a*d+b*c)^4-3*d^4*(2*a^2*d^2-6*a*b*c*d+5*b^2*c^2)*ln(d*x+c)/c^5/(-a *d+b*c)^4
Time = 0.18 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=-\frac {1}{2 a^2 c^3 x^2}+\frac {2 b c+3 a d}{a^3 c^4 x}-\frac {b^5}{a^3 (-b c+a d)^3 (a+b x)}+\frac {d^4}{2 c^3 (b c-a d)^2 (c+d x)^2}+\frac {d^4 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)}+\frac {3 \left (b^2 c^2+2 a b c d+2 a^2 d^2\right ) \log (x)}{a^4 c^5}+\frac {3 b^5 (-b c+2 a d) \log (a+b x)}{a^4 (b c-a d)^4}-\frac {3 d^4 \left (5 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log (c+d x)}{c^5 (b c-a d)^4} \]
-1/2*1/(a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) - b^5/(a^3*(-(b*c) + a* d)^3*(a + b*x)) + d^4/(2*c^3*(b*c - a*d)^2*(c + d*x)^2) + (d^4*(5*b*c - 3* a*d))/(c^4*(b*c - a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2) *Log[x])/(a^4*c^5) + (3*b^5*(-(b*c) + 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d )^4) - (3*d^4*(5*b^2*c^2 - 6*a*b*c*d + 2*a^2*d^2)*Log[c + d*x])/(c^5*(b*c - a*d)^4)
Time = 0.52 (sec) , antiderivative size = 228, 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.
| \(\displaystyle \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {3 b^6 (2 a d-b c)}{a^4 (a+b x) (a d-b c)^4}+\frac {b^6}{a^3 (a+b x)^2 (a d-b c)^3}+\frac {-3 a d-2 b c}{a^3 c^4 x^2}-\frac {3 d^5 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{c^5 (c+d x) (b c-a d)^4}+\frac {1}{a^2 c^3 x^3}+\frac {3 \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5 x}-\frac {d^5 (5 b c-3 a d)}{c^4 (c+d x)^2 (b c-a d)^3}-\frac {d^5}{c^3 (c+d x)^3 (b c-a d)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac {b^5}{a^3 (a+b x) (b c-a d)^3}+\frac {3 a d+2 b c}{a^3 c^4 x}-\frac {3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}-\frac {1}{2 a^2 c^3 x^2}+\frac {3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}+\frac {d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac {d^4}{2 c^3 (c+d x)^2 (b c-a d)^2}\) |
-1/2*1/(a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) + b^5/(a^3*(b*c - a*d)^ 3*(a + b*x)) + d^4/(2*c^3*(b*c - a*d)^2*(c + d*x)^2) + (d^4*(5*b*c - 3*a*d ))/(c^4*(b*c - a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*Lo g[x])/(a^4*c^5) - (3*b^5*(b*c - 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d)^4) - (3*d^4*(5*b^2*c^2 - 6*a*b*c*d + 2*a^2*d^2)*Log[c + d*x])/(c^5*(b*c - a*d) ^4)
3.3.100.3.1 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 = 0.57 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {1}{2 a^{2} c^{3} x^{2}}-\frac {-3 a d -2 b c}{x \,a^{3} c^{4}}+\frac {\left (6 a^{2} d^{2}+6 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{5}}+\frac {d^{4}}{2 c^{3} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {d^{4} \left (3 a d -5 b c \right )}{c^{4} \left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {3 d^{4} \left (2 a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{5} \left (a d -b c \right )^{4}}-\frac {b^{5}}{a^{3} \left (a d -b c \right )^{3} \left (b x +a \right )}+\frac {3 b^{5} \left (2 a d -b c \right ) \ln \left (b x +a \right )}{a^{4} \left (a d -b c \right )^{4}}\) | \(228\) |
| norman | \(\frac {\frac {\left (-12 a^{6} d^{6}+20 a^{5} b c \,d^{5}-6 a^{3} b^{3} c^{3} d^{3}-2 a^{2} b^{4} c^{4} d^{2}+3 b^{6} c^{6}\right ) x^{3}}{c^{4} a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {1}{2 a c}+\frac {\left (4 a d +3 b c \right ) x}{2 c^{2} a^{2}}+\frac {d \left (-18 a^{6} d^{6}+8 a^{5} b c \,d^{5}+39 a^{4} b^{2} c^{2} d^{4}-9 a^{3} b^{3} c^{3} d^{3}-20 a^{2} b^{4} c^{4} d^{2}+12 b^{6} c^{6}\right ) x^{4}}{2 c^{5} a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{2} b \left (-18 a^{5} d^{5}+32 a^{4} b c \,d^{4}-a^{3} b^{2} c^{2} d^{3}-13 a^{2} b^{3} c^{3} d^{2}+6 b^{5} c^{5}\right ) x^{5}}{2 c^{5} a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{x^{2} \left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {3 \left (2 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{5}}+\frac {3 b^{5} \left (2 a d -b c \right ) \ln \left (b x +a \right )}{a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {3 d^{4} \left (2 a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(571\) |
| risch | \(\frac {\frac {3 b \,d^{2} \left (2 a^{4} d^{4}-4 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d -b^{4} c^{4}\right ) x^{4}}{a^{3} c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d \left (4 a^{5} d^{5}-2 a^{4} b c \,d^{4}-10 a^{3} b^{2} c^{2} d^{3}+5 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -4 b^{5} c^{5}\right ) x^{3}}{2 a^{3} c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (18 a^{5} d^{5}-32 a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+13 a^{2} b^{3} c^{3} d^{2}-6 b^{5} c^{5}\right ) x^{2}}{2 c^{3} a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (4 a d +3 b c \right ) x}{2 c^{2} a^{2}}-\frac {1}{2 a c}}{x^{2} \left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {6 \ln \left (-x \right ) d^{2}}{a^{2} c^{5}}+\frac {6 \ln \left (-x \right ) b d}{a^{3} c^{4}}+\frac {3 \ln \left (-x \right ) b^{2}}{a^{4} c^{3}}-\frac {6 d^{6} \ln \left (-d x -c \right ) a^{2}}{c^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {18 d^{5} \ln \left (-d x -c \right ) a b}{c^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {15 d^{4} \ln \left (-d x -c \right ) b^{2}}{c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {6 b^{5} \ln \left (b x +a \right ) d}{a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {3 b^{6} \ln \left (b x +a \right ) c}{a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(755\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1421\) |
-1/2/a^2/c^3/x^2-(-3*a*d-2*b*c)/x/a^3/c^4+(6*a^2*d^2+6*a*b*c*d+3*b^2*c^2)/ a^4/c^5*ln(x)+1/2/c^3*d^4/(a*d-b*c)^2/(d*x+c)^2+d^4*(3*a*d-5*b*c)/c^4/(a*d -b*c)^3/(d*x+c)-3*d^4*(2*a^2*d^2-6*a*b*c*d+5*b^2*c^2)/c^5/(a*d-b*c)^4*ln(d *x+c)-b^5/a^3/(a*d-b*c)^3/(b*x+a)+3*b^5*(2*a*d-b*c)/a^4/(a*d-b*c)^4*ln(b*x +a)
Leaf count of result is larger than twice the leaf count of optimal. 1302 vs. \(2 (224) = 448\).
Time = 31.02 (sec) , antiderivative size = 1302, normalized size of antiderivative = 5.71 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=\text {Too large to display} \]
-1/2*(a^3*b^4*c^8 - 4*a^4*b^3*c^7*d + 6*a^5*b^2*c^6*d^2 - 4*a^6*b*c^5*d^3 + a^7*c^4*d^4 - 6*(a*b^6*c^6*d^2 - 2*a^2*b^5*c^5*d^3 + 5*a^4*b^3*c^3*d^5 - 6*a^5*b^2*c^2*d^6 + 2*a^6*b*c*d^7)*x^4 - 3*(4*a*b^6*c^7*d - 7*a^2*b^5*c^6 *d^2 - 2*a^3*b^4*c^5*d^3 + 15*a^4*b^3*c^4*d^4 - 8*a^5*b^2*c^3*d^5 - 6*a^6* b*c^2*d^6 + 4*a^7*c*d^7)*x^3 - (6*a*b^6*c^8 - 6*a^2*b^5*c^7*d - 13*a^3*b^4 *c^6*d^2 + 12*a^4*b^3*c^5*d^3 + 33*a^5*b^2*c^4*d^4 - 50*a^6*b*c^3*d^5 + 18 *a^7*c^2*d^6)*x^2 - (3*a^2*b^5*c^8 - 8*a^3*b^4*c^7*d + 2*a^4*b^3*c^6*d^2 + 12*a^5*b^2*c^5*d^3 - 13*a^6*b*c^4*d^4 + 4*a^7*c^3*d^5)*x + 6*((b^7*c^6*d^ 2 - 2*a*b^6*c^5*d^3)*x^5 + (2*b^7*c^7*d - 3*a*b^6*c^6*d^2 - 2*a^2*b^5*c^5* d^3)*x^4 + (b^7*c^8 - 4*a^2*b^5*c^6*d^2)*x^3 + (a*b^6*c^8 - 2*a^2*b^5*c^7* d)*x^2)*log(b*x + a) + 6*((5*a^4*b^3*c^2*d^6 - 6*a^5*b^2*c*d^7 + 2*a^6*b*d ^8)*x^5 + (10*a^4*b^3*c^3*d^5 - 7*a^5*b^2*c^2*d^6 - 2*a^6*b*c*d^7 + 2*a^7* d^8)*x^4 + (5*a^4*b^3*c^4*d^4 + 4*a^5*b^2*c^3*d^5 - 10*a^6*b*c^2*d^6 + 4*a ^7*c*d^7)*x^3 + (5*a^5*b^2*c^4*d^4 - 6*a^6*b*c^3*d^5 + 2*a^7*c^2*d^6)*x^2) *log(d*x + c) - 6*((b^7*c^6*d^2 - 2*a*b^6*c^5*d^3 + 5*a^4*b^3*c^2*d^6 - 6* a^5*b^2*c*d^7 + 2*a^6*b*d^8)*x^5 + (2*b^7*c^7*d - 3*a*b^6*c^6*d^2 - 2*a^2* b^5*c^5*d^3 + 10*a^4*b^3*c^3*d^5 - 7*a^5*b^2*c^2*d^6 - 2*a^6*b*c*d^7 + 2*a ^7*d^8)*x^4 + (b^7*c^8 - 4*a^2*b^5*c^6*d^2 + 5*a^4*b^3*c^4*d^4 + 4*a^5*b^2 *c^3*d^5 - 10*a^6*b*c^2*d^6 + 4*a^7*c*d^7)*x^3 + (a*b^6*c^8 - 2*a^2*b^5*c^ 7*d + 5*a^5*b^2*c^4*d^4 - 6*a^6*b*c^3*d^5 + 2*a^7*c^2*d^6)*x^2)*log(x))...
Timed out. \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (224) = 448\).
Time = 0.24 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.30 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=-\frac {3 \, {\left (b^{6} c - 2 \, a b^{5} d\right )} \log \left (b x + a\right )}{a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{4} - 6 \, a b c d^{5} + 2 \, a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4}} - \frac {a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - 6 \, {\left (b^{5} c^{4} d^{2} - a b^{4} c^{3} d^{3} - a^{2} b^{3} c^{2} d^{4} + 4 \, a^{3} b^{2} c d^{5} - 2 \, a^{4} b d^{6}\right )} x^{4} - 3 \, {\left (4 \, b^{5} c^{5} d - 3 \, a b^{4} c^{4} d^{2} - 5 \, a^{2} b^{3} c^{3} d^{3} + 10 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 4 \, a^{5} d^{6}\right )} x^{3} - {\left (6 \, b^{5} c^{6} - 13 \, a^{2} b^{3} c^{4} d^{2} - a^{3} b^{2} c^{3} d^{3} + 32 \, a^{4} b c^{2} d^{4} - 18 \, a^{5} c d^{5}\right )} x^{2} - {\left (3 \, a b^{4} c^{6} - 5 \, a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 9 \, a^{4} b c^{3} d^{3} - 4 \, a^{5} c^{2} d^{4}\right )} x}{2 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{5} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{4} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{3} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{2}\right )}} + \frac {3 \, {\left (b^{2} c^{2} + 2 \, a b c d + 2 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{5}} \]
-3*(b^6*c - 2*a*b^5*d)*log(b*x + a)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6 *b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4) - 3*(5*b^2*c^2*d^4 - 6*a*b*c*d^5 + 2*a^2*d^6)*log(d*x + c)/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4* a^3*b*c^6*d^3 + a^4*c^5*d^4) - 1/2*(a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4* b*c^4*d^2 - a^5*c^3*d^3 - 6*(b^5*c^4*d^2 - a*b^4*c^3*d^3 - a^2*b^3*c^2*d^4 + 4*a^3*b^2*c*d^5 - 2*a^4*b*d^6)*x^4 - 3*(4*b^5*c^5*d - 3*a*b^4*c^4*d^2 - 5*a^2*b^3*c^3*d^3 + 10*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - 4*a^5*d^6)*x^3 - (6*b^5*c^6 - 13*a^2*b^3*c^4*d^2 - a^3*b^2*c^3*d^3 + 32*a^4*b*c^2*d^4 - 18 *a^5*c*d^5)*x^2 - (3*a*b^4*c^6 - 5*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 9*a ^4*b*c^3*d^3 - 4*a^5*c^2*d^4)*x)/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3 *a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^5 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d ^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^4 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^3 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2) + 3 *(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*log(x)/(a^4*c^5)
Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (224) = 448\).
Time = 0.31 (sec) , antiderivative size = 865, normalized size of antiderivative = 3.79 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=\frac {b^{11}}{{\left (a^{3} b^{9} c^{3} - 3 \, a^{4} b^{8} c^{2} d + 3 \, a^{5} b^{7} c d^{2} - a^{6} b^{6} d^{3}\right )} {\left (b x + a\right )}} + \frac {3 \, {\left (b^{6} c - 2 \, a b^{5} 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 )}{2 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4}\right )}} - \frac {3 \, {\left (b^{8} c^{6} - 2 \, a b^{7} c^{5} d + 10 \, a^{4} b^{4} c^{2} d^{4} - 12 \, a^{5} b^{3} c d^{5} + 4 \, a^{6} b^{2} d^{6}\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 )}{2 \, {\left (a^{4} b^{4} c^{8} - 4 \, a^{5} b^{3} c^{7} d + 6 \, a^{6} b^{2} c^{6} d^{2} - 4 \, a^{7} b c^{5} d^{3} + a^{8} c^{4} d^{4}\right )} b^{2} {\left | c \right |}} + \frac {5 \, b^{6} c^{5} d^{2} - 14 \, a b^{5} c^{4} d^{3} + 6 \, a^{2} b^{4} c^{3} d^{4} + 16 \, a^{3} b^{3} c^{2} d^{5} - 30 \, a^{4} b^{2} c d^{6} + 12 \, a^{5} b d^{7} + \frac {2 \, {\left (5 \, b^{8} c^{6} d - 22 \, a b^{7} c^{5} d^{2} + 29 \, a^{2} b^{6} c^{4} d^{3} + 4 \, a^{3} b^{5} c^{3} d^{4} - 47 \, a^{4} b^{4} c^{2} d^{5} + 54 \, a^{5} b^{3} c d^{6} - 18 \, a^{6} b^{2} d^{7}\right )}}{{\left (b x + a\right )} b} + \frac {5 \, b^{10} c^{7} - 36 \, a b^{9} c^{6} d + 87 \, a^{2} b^{8} c^{5} d^{2} - 70 \, a^{3} b^{7} c^{4} d^{3} - 45 \, a^{4} b^{6} c^{3} d^{4} + 144 \, a^{5} b^{5} c^{2} d^{5} - 126 \, a^{6} b^{4} c d^{6} + 36 \, a^{7} b^{3} d^{7}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {6 \, {\left (a b^{11} c^{7} - 5 \, a^{2} b^{10} c^{6} d + 9 \, a^{3} b^{9} c^{5} d^{2} - 5 \, a^{4} b^{8} c^{4} d^{3} - 5 \, a^{5} b^{7} c^{3} d^{4} + 11 \, a^{6} b^{6} c^{2} d^{5} - 8 \, a^{7} b^{5} c d^{6} + 2 \, a^{8} b^{4} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{2 \, {\left (b c - a d\right )}^{4} a^{4} {\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 )}^{2} c^{4}} \]
b^11/((a^3*b^9*c^3 - 3*a^4*b^8*c^2*d + 3*a^5*b^7*c*d^2 - a^6*b^6*d^3)*(b*x + a)) + 3/2*(b^6*c - 2*a*b^5*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^4*b^4*c^4 - 4*a^5*b^3*c^3 *d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4) - 3/2*(b^8*c^6 - 2*a*b^7 *c^5*d + 10*a^4*b^4*c^2*d^4 - 12*a^5*b^3*c*d^5 + 4*a^6*b^2*d^6)*log(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 bs(-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^4*b^4*c^8 - 4*a^5*b^3*c^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4)*b^2*abs(c)) + 1/2*(5*b^6*c^5*d^2 - 14*a*b^5*c^4*d^3 + 6*a^ 2*b^4*c^3*d^4 + 16*a^3*b^3*c^2*d^5 - 30*a^4*b^2*c*d^6 + 12*a^5*b*d^7 + 2*( 5*b^8*c^6*d - 22*a*b^7*c^5*d^2 + 29*a^2*b^6*c^4*d^3 + 4*a^3*b^5*c^3*d^4 - 47*a^4*b^4*c^2*d^5 + 54*a^5*b^3*c*d^6 - 18*a^6*b^2*d^7)/((b*x + a)*b) + (5 *b^10*c^7 - 36*a*b^9*c^6*d + 87*a^2*b^8*c^5*d^2 - 70*a^3*b^7*c^4*d^3 - 45* a^4*b^6*c^3*d^4 + 144*a^5*b^5*c^2*d^5 - 126*a^6*b^4*c*d^6 + 36*a^7*b^3*d^7 )/((b*x + a)^2*b^2) - 6*(a*b^11*c^7 - 5*a^2*b^10*c^6*d + 9*a^3*b^9*c^5*d^2 - 5*a^4*b^8*c^4*d^3 - 5*a^5*b^7*c^3*d^4 + 11*a^6*b^6*c^2*d^5 - 8*a^7*b^5* c*d^6 + 2*a^8*b^4*d^7)/((b*x + a)^3*b^3))/((b*c - a*d)^4*a^4*(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)^2*c^4)
Time = 1.24 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {x\,\left (4\,a\,d+3\,b\,c\right )}{2\,a^2\,c^2}-\frac {1}{2\,a\,c}+\frac {x^2\,\left (18\,a^5\,d^5-32\,a^4\,b\,c\,d^4+a^3\,b^2\,c^2\,d^3+13\,a^2\,b^3\,c^3\,d^2-6\,b^5\,c^5\right )}{2\,a^3\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,x^3\,\left (4\,a^5\,d^6-2\,a^4\,b\,c\,d^5-10\,a^3\,b^2\,c^2\,d^4+5\,a^2\,b^3\,c^3\,d^3+3\,a\,b^4\,c^4\,d^2-4\,b^5\,c^5\,d\right )}{2\,a^3\,c^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b\,d^2\,x^4\,\left (2\,a^4\,d^4-4\,a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d-b^4\,c^4\right )}{a^3\,c^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^3\,\left (b\,c^2+2\,a\,d\,c\right )+x^4\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^2+b\,d^2\,x^5}-\frac {\ln \left (c+d\,x\right )\,\left (6\,a^2\,d^6-18\,a\,b\,c\,d^5+15\,b^2\,c^2\,d^4\right )}{a^4\,c^5\,d^4-4\,a^3\,b\,c^6\,d^3+6\,a^2\,b^2\,c^7\,d^2-4\,a\,b^3\,c^8\,d+b^4\,c^9}-\frac {\ln \left (a+b\,x\right )\,\left (3\,b^6\,c-6\,a\,b^5\,d\right )}{a^8\,d^4-4\,a^7\,b\,c\,d^3+6\,a^6\,b^2\,c^2\,d^2-4\,a^5\,b^3\,c^3\,d+a^4\,b^4\,c^4}+\frac {\ln \left (x\right )\,\left (6\,a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4\,c^5} \]
((x*(4*a*d + 3*b*c))/(2*a^2*c^2) - 1/(2*a*c) + (x^2*(18*a^5*d^5 - 6*b^5*c^ 5 + 13*a^2*b^3*c^3*d^2 + a^3*b^2*c^2*d^3 - 32*a^4*b*c*d^4))/(2*a^3*c^3*(a^ 3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (3*x^3*(4*a^5*d^6 - 4* b^5*c^5*d + 3*a*b^4*c^4*d^2 + 5*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 - 2*a ^4*b*c*d^5))/(2*a^3*c^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 )) + (3*b*d^2*x^4*(2*a^4*d^4 - b^4*c^4 + a^2*b^2*c^2*d^2 + a*b^3*c^3*d - 4 *a^3*b*c*d^3))/(a^3*c^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 )))/(x^3*(b*c^2 + 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) + a*c^2*x^2 + b*d^2*x^5 ) - (log(c + d*x)*(6*a^2*d^6 + 15*b^2*c^2*d^4 - 18*a*b*c*d^5))/(b^4*c^9 + a^4*c^5*d^4 - 4*a^3*b*c^6*d^3 + 6*a^2*b^2*c^7*d^2 - 4*a*b^3*c^8*d) - (log( a + b*x)*(3*b^6*c - 6*a*b^5*d))/(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3) + (log(x)*(6*a^2*d^2 + 3*b^2*c^2 + 6*a *b*c*d))/(a^4*c^5)