Integrand size = 18, antiderivative size = 172 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=\frac {b^3}{a (b c-a d)^3 (a+b x)}+\frac {d^2}{2 c (b c-a d)^2 (c+d x)^2}+\frac {d^2 (3 b c-a d)}{c^2 (b c-a d)^3 (c+d x)}+\frac {\log (x)}{a^2 c^3}-\frac {b^3 (b c-4 a d) \log (a+b x)}{a^2 (b c-a d)^4}-\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^4} \]
b^3/a/(-a*d+b*c)^3/(b*x+a)+1/2*d^2/c/(-a*d+b*c)^2/(d*x+c)^2+d^2*(-a*d+3*b* c)/c^2/(-a*d+b*c)^3/(d*x+c)+ln(x)/a^2/c^3-b^3*(-4*a*d+b*c)*ln(b*x+a)/a^2/( -a*d+b*c)^4-d^2*(a^2*d^2-4*a*b*c*d+6*b^2*c^2)*ln(d*x+c)/c^3/(-a*d+b*c)^4
Time = 0.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=-\frac {b^3}{a (-b c+a d)^3 (a+b x)}+\frac {d^2}{2 c (b c-a d)^2 (c+d x)^2}+\frac {d^2 (3 b c-a d)}{c^2 (b c-a d)^3 (c+d x)}+\frac {\log (x)}{a^2 c^3}+\frac {b^3 (-b c+4 a d) \log (a+b x)}{a^2 (b c-a d)^4}-\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^4} \]
-(b^3/(a*(-(b*c) + a*d)^3*(a + b*x))) + d^2/(2*c*(b*c - a*d)^2*(c + d*x)^2 ) + (d^2*(3*b*c - a*d))/(c^2*(b*c - a*d)^3*(c + d*x)) + Log[x]/(a^2*c^3) + (b^3*(-(b*c) + 4*a*d)*Log[a + b*x])/(a^2*(b*c - a*d)^4) - (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x])/(c^3*(b*c - a*d)^4)
Time = 0.40 (sec) , antiderivative size = 172, 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.
| \(\displaystyle \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {b^4 (4 a d-b c)}{a^2 (a+b x) (a d-b c)^4}-\frac {d^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{c^3 (c+d x) (b c-a d)^4}+\frac {1}{a^2 c^3 x}+\frac {b^4}{a (a+b x)^2 (a d-b c)^3}-\frac {d^3 (3 b c-a d)}{c^2 (c+d x)^2 (b c-a d)^3}-\frac {d^3}{c (c+d x)^3 (b c-a d)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^3 (b c-4 a d) \log (a+b x)}{a^2 (b c-a d)^4}-\frac {d^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^4}+\frac {\log (x)}{a^2 c^3}+\frac {b^3}{a (a+b x) (b c-a d)^3}+\frac {d^2 (3 b c-a d)}{c^2 (c+d x) (b c-a d)^3}+\frac {d^2}{2 c (c+d x)^2 (b c-a d)^2}\) |
b^3/(a*(b*c - a*d)^3*(a + b*x)) + d^2/(2*c*(b*c - a*d)^2*(c + d*x)^2) + (d ^2*(3*b*c - a*d))/(c^2*(b*c - a*d)^3*(c + d*x)) + Log[x]/(a^2*c^3) - (b^3* (b*c - 4*a*d)*Log[a + b*x])/(a^2*(b*c - a*d)^4) - (d^2*(6*b^2*c^2 - 4*a*b* c*d + a^2*d^2)*Log[c + d*x])/(c^3*(b*c - a*d)^4)
3.3.98.3.1 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 = 171, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{a^{2} c^{3}}+\frac {d^{2}}{2 c \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {d^{2} \left (a d -3 b c \right )}{c^{2} \left (a d -b c \right )^{3} \left (d x +c \right )}-\frac {d^{2} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )^{4}}-\frac {b^{3}}{a \left (a d -b c \right )^{3} \left (b x +a \right )}+\frac {b^{3} \left (4 a d -b c \right ) \ln \left (b x +a \right )}{a^{2} \left (a d -b c \right )^{4}}\) | \(171\) |
| norman | \(\frac {\frac {\left (-2 a^{4} d^{4}+4 a^{3} b c \,d^{3}+b^{4} c^{4}\right ) x}{c^{2} a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (-3 a^{4} d^{4}+3 a^{3} b c \,d^{3}+8 a^{2} b^{2} c^{2} d^{2}+4 b^{4} c^{4}\right ) x^{2}}{2 c^{3} a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d^{2} b \left (-3 a^{3} d^{3}+7 a^{2} b c \,d^{2}+2 b^{3} c^{3}\right ) x^{3}}{2 c^{3} a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {\ln \left (x \right )}{a^{2} c^{3}}+\frac {b^{3} \left (4 a d -b c \right ) \ln \left (b x +a \right )}{a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {d^{2} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(430\) |
| risch | \(\frac {\frac {b \,d^{2} \left (a^{2} d^{2}-3 a b c d -b^{2} c^{2}\right ) x^{2}}{a \,c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {d \left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x}{2 c^{2} a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 a^{3} d^{3}-7 a^{2} b c \,d^{2}-2 b^{3} c^{3}}{2 a c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {d^{4} \ln \left (-d x -c \right ) a^{2}}{c^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {4 d^{3} \ln \left (-d x -c \right ) a b}{c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {6 d^{2} \ln \left (-d x -c \right ) b^{2}}{c \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\ln \left (-x \right )}{a^{2} c^{3}}+\frac {4 b^{3} \ln \left (b x +a \right ) d}{a \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {b^{4} \ln \left (b x +a \right ) c}{a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(605\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1089\) |
ln(x)/a^2/c^3+1/2*d^2/c/(a*d-b*c)^2/(d*x+c)^2+d^2*(a*d-3*b*c)/c^2/(a*d-b*c )^3/(d*x+c)-d^2*(a^2*d^2-4*a*b*c*d+6*b^2*c^2)/c^3/(a*d-b*c)^4*ln(d*x+c)-b^ 3/a/(a*d-b*c)^3/(b*x+a)+b^3*(4*a*d-b*c)/a^2/(a*d-b*c)^4*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (170) = 340\).
Time = 7.37 (sec) , antiderivative size = 1044, normalized size of antiderivative = 6.07 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=\frac {2 \, a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 10 \, a^{4} b c^{3} d^{3} + 3 \, a^{5} c^{2} d^{4} + 2 \, {\left (a b^{4} c^{4} d^{2} + 2 \, a^{2} b^{3} c^{3} d^{3} - 4 \, a^{3} b^{2} c^{2} d^{4} + a^{4} b c d^{5}\right )} x^{2} + {\left (4 \, a b^{4} c^{5} d + 3 \, a^{2} b^{3} c^{4} d^{2} - 4 \, a^{3} b^{2} c^{3} d^{3} - 5 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x - 2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} - 4 \, a^{2} b^{3} c^{3} d^{3}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 8 \, a^{2} b^{3} c^{4} d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (12 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (6 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (d x + c\right ) + 2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{4} c^{9} - 4 \, a^{4} b^{3} c^{8} d + 6 \, a^{5} b^{2} c^{7} d^{2} - 4 \, a^{6} b c^{6} d^{3} + a^{7} c^{5} d^{4} + {\left (a^{2} b^{5} c^{7} d^{2} - 4 \, a^{3} b^{4} c^{6} d^{3} + 6 \, a^{4} b^{3} c^{5} d^{4} - 4 \, a^{5} b^{2} c^{4} d^{5} + a^{6} b c^{3} d^{6}\right )} x^{3} + {\left (2 \, a^{2} b^{5} c^{8} d - 7 \, a^{3} b^{4} c^{7} d^{2} + 8 \, a^{4} b^{3} c^{6} d^{3} - 2 \, a^{5} b^{2} c^{5} d^{4} - 2 \, a^{6} b c^{4} d^{5} + a^{7} c^{3} d^{6}\right )} x^{2} + {\left (a^{2} b^{5} c^{9} - 2 \, a^{3} b^{4} c^{8} d - 2 \, a^{4} b^{3} c^{7} d^{2} + 8 \, a^{5} b^{2} c^{6} d^{3} - 7 \, a^{6} b c^{5} d^{4} + 2 \, a^{7} c^{4} d^{5}\right )} x\right )}} \]
1/2*(2*a*b^4*c^6 - 2*a^2*b^3*c^5*d + 7*a^3*b^2*c^4*d^2 - 10*a^4*b*c^3*d^3 + 3*a^5*c^2*d^4 + 2*(a*b^4*c^4*d^2 + 2*a^2*b^3*c^3*d^3 - 4*a^3*b^2*c^2*d^4 + a^4*b*c*d^5)*x^2 + (4*a*b^4*c^5*d + 3*a^2*b^3*c^4*d^2 - 4*a^3*b^2*c^3*d ^3 - 5*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x - 2*(a*b^4*c^6 - 4*a^2*b^3*c^5*d + ( b^5*c^4*d^2 - 4*a*b^4*c^3*d^3)*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 - 4*a^ 2*b^3*c^3*d^3)*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 8*a^2*b^3*c^4*d^2)*x)*log( b*x + a) - 2*(6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (6*a^2*b ^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d^6)*x^3 + (12*a^2*b^3*c^3*d^3 - 2*a^ 3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^2 + (6*a^2*b^3*c^4*d^2 + 8*a^3* b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x)*log(d*x + c) + 2*(a*b^4*c^ 6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4 *b*d^6)*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b ^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a ^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x)*log (x))/(a^3*b^4*c^9 - 4*a^4*b^3*c^8*d + 6*a^5*b^2*c^7*d^2 - 4*a^6*b*c^6*d^3 + a^7*c^5*d^4 + (a^2*b^5*c^7*d^2 - 4*a^3*b^4*c^6*d^3 + 6*a^4*b^3*c^5*d^4 - 4*a^5*b^2*c^4*d^5 + a^6*b*c^3*d^6)*x^3 + (2*a^2*b^5*c^8*d - 7*a^3*b^4*c^7 *d^2 + 8*a^4*b^3*c^6*d^3 - 2*a^5*b^2*c^5*d^4 - 2*a^6*b*c^4*d^5 + a^7*c^3*d ^6)*x^2 + (a^2*b^5*c^9 - 2*a^3*b^4*c^8*d - 2*a^4*b^3*c^7*d^2 + 8*a^5*b^...
Timed out. \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (170) = 340\).
Time = 0.23 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.00 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=-\frac {{\left (b^{4} c - 4 \, a b^{3} d\right )} \log \left (b x + a\right )}{a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}} - \frac {{\left (6 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}} + \frac {2 \, b^{3} c^{4} + 7 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} + 2 \, {\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} c d^{3} - a^{2} b d^{4}\right )} x^{2} + {\left (4 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x}{2 \, {\left (a^{2} b^{3} c^{7} - 3 \, a^{3} b^{2} c^{6} d + 3 \, a^{4} b c^{5} d^{2} - a^{5} c^{4} d^{3} + {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5}\right )} x^{3} + {\left (2 \, a b^{4} c^{6} d - 5 \, a^{2} b^{3} c^{5} d^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} x^{2} + {\left (a b^{4} c^{7} - a^{2} b^{3} c^{6} d - 3 \, a^{3} b^{2} c^{5} d^{2} + 5 \, a^{4} b c^{4} d^{3} - 2 \, a^{5} c^{3} d^{4}\right )} x\right )}} + \frac {\log \left (x\right )}{a^{2} c^{3}} \]
-(b^4*c - 4*a*b^3*d)*log(b*x + a)/(a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b ^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4) - (6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2 *d^4)*log(d*x + c)/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b* c^4*d^3 + a^4*c^3*d^4) + 1/2*(2*b^3*c^4 + 7*a^2*b*c^2*d^2 - 3*a^3*c*d^3 + 2*(b^3*c^2*d^2 + 3*a*b^2*c*d^3 - a^2*b*d^4)*x^2 + (4*b^3*c^3*d + 7*a*b^2*c ^2*d^2 + 3*a^2*b*c*d^3 - 2*a^3*d^4)*x)/(a^2*b^3*c^7 - 3*a^3*b^2*c^6*d + 3* a^4*b*c^5*d^2 - a^5*c^4*d^3 + (a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b ^2*c^3*d^4 - a^4*b*c^2*d^5)*x^3 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5*d^2 + 3*a ^3*b^2*c^4*d^3 + a^4*b*c^3*d^4 - a^5*c^2*d^5)*x^2 + (a*b^4*c^7 - a^2*b^3*c ^6*d - 3*a^3*b^2*c^5*d^2 + 5*a^4*b*c^4*d^3 - 2*a^5*c^3*d^4)*x) + log(x)/(a ^2*c^3)
Time = 0.29 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.80 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=\frac {1}{2} \, {\left (\frac {2 \, b^{6}}{{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (6 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{7} - 4 \, a b^{4} c^{6} d + 6 \, a^{2} b^{3} c^{5} d^{2} - 4 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4}} + \frac {2 \, \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b c^{3}} - \frac {7 \, b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + \frac {2 \, {\left (4 \, b^{4} c^{3} d^{3} - 5 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5}\right )}}{{\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{4} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} c^{3}}\right )} b \]
1/2*(2*b^6/((a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)* (b*x + a)) - 2*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^7 - 4*a*b^4*c^6*d + 6*a^2*b^3*c^5*d^2 - 4* a^3*b^2*c^4*d^3 + a^4*b*c^3*d^4) + 2*log(abs(-a/(b*x + a) + 1))/(a^2*b*c^3 ) - (7*b^2*c^2*d^4 - 2*a*b*c*d^5 + 2*(4*b^4*c^3*d^3 - 5*a*b^3*c^2*d^4 + a^ 2*b^2*c*d^5)/((b*x + a)*b))/((b*c - a*d)^4*b*(b*c/(b*x + a) - a*d/(b*x + a ) + d)^2*c^3))*b
Time = 1.07 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.67 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^3} \, dx=\frac {\ln \left (x\right )}{a^2\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^4-4\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{a^4\,c^3\,d^4-4\,a^3\,b\,c^4\,d^3+6\,a^2\,b^2\,c^5\,d^2-4\,a\,b^3\,c^6\,d+b^4\,c^7}-\frac {\frac {-3\,a^3\,d^3+7\,a^2\,b\,c\,d^2+2\,b^3\,c^3}{2\,a\,c\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (-2\,a^3\,d^4+3\,a^2\,b\,c\,d^3+7\,a\,b^2\,c^2\,d^2+4\,b^3\,c^3\,d\right )}{2\,a\,c^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d^2\,x^2\,\left (-a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{a\,c^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}-\frac {\ln \left (a+b\,x\right )\,\left (b^4\,c-4\,a\,b^3\,d\right )}{a^6\,d^4-4\,a^5\,b\,c\,d^3+6\,a^4\,b^2\,c^2\,d^2-4\,a^3\,b^3\,c^3\,d+a^2\,b^4\,c^4} \]
log(x)/(a^2*c^3) - (log(c + d*x)*(a^2*d^4 + 6*b^2*c^2*d^2 - 4*a*b*c*d^3))/ (b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6 *d) - ((2*b^3*c^3 - 3*a^3*d^3 + 7*a^2*b*c*d^2)/(2*a*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(4*b^3*c^3*d - 2*a^3*d^4 + 7*a*b^2*c ^2*d^2 + 3*a^2*b*c*d^3))/(2*a*c^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a ^2*b*c*d^2)) + (b*d^2*x^2*(b^2*c^2 - a^2*d^2 + 3*a*b*c*d))/(a*c^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(x*(b*c^2 + 2*a*c*d) + a*c^2 + x^2*(a*d^2 + 2*b*c*d) + b*d^2*x^3) - (log(a + b*x)*(b^4*c - 4*a*b^3*d)) /(a^6*d^4 + a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c* d^3)