Integrand size = 18, antiderivative size = 169 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {a^3 (4 b c-a d)}{b^2 (b c-a d)^4 (a+b x)}+\frac {c^4}{2 d^2 (b c-a d)^3 (c+d x)^2}-\frac {c^3 (b c-4 a d)}{d^2 (b c-a d)^4 (c+d x)}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5} \]
-1/2*a^4/b^2/(-a*d+b*c)^3/(b*x+a)^2+a^3*(-a*d+4*b*c)/b^2/(-a*d+b*c)^4/(b*x +a)+1/2*c^4/d^2/(-a*d+b*c)^3/(d*x+c)^2-c^3*(-4*a*d+b*c)/d^2/(-a*d+b*c)^4/( d*x+c)+6*a^2*c^2*ln(b*x+a)/(-a*d+b*c)^5-6*a^2*c^2*ln(d*x+c)/(-a*d+b*c)^5
Time = 0.16 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {4 a^3 b c-a^4 d}{b^2 (b c-a d)^4 (a+b x)}-\frac {c^4}{2 d^2 (-b c+a d)^3 (c+d x)^2}-\frac {c^3 (b c-4 a d)}{d^2 (-b c+a d)^4 (c+d x)}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5} \]
-1/2*a^4/(b^2*(b*c - a*d)^3*(a + b*x)^2) + (4*a^3*b*c - a^4*d)/(b^2*(b*c - a*d)^4*(a + b*x)) - c^4/(2*d^2*(-(b*c) + a*d)^3*(c + d*x)^2) - (c^3*(b*c - 4*a*d))/(d^2*(-(b*c) + a*d)^4*(c + d*x)) + (6*a^2*c^2*Log[a + b*x])/(b*c - a*d)^5 - (6*a^2*c^2*Log[c + d*x])/(b*c - a*d)^5
Time = 0.40 (sec) , antiderivative size = 169, 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.
| \(\displaystyle \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^4}{b (a+b x)^3 (b c-a d)^3}+\frac {a^3 (a d-4 b c)}{b (a+b x)^2 (b c-a d)^4}+\frac {6 a^2 b c^2}{(a+b x) (b c-a d)^5}+\frac {6 a^2 c^2 d}{(c+d x) (a d-b c)^5}+\frac {c^4}{d (c+d x)^3 (a d-b c)^3}+\frac {c^3 (b c-4 a d)}{d (c+d x)^2 (a d-b c)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac {a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac {c^4}{2 d^2 (c+d x)^2 (b c-a d)^3}-\frac {c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4}\) |
-1/2*a^4/(b^2*(b*c - a*d)^3*(a + b*x)^2) + (a^3*(4*b*c - a*d))/(b^2*(b*c - a*d)^4*(a + b*x)) + c^4/(2*d^2*(b*c - a*d)^3*(c + d*x)^2) - (c^3*(b*c - 4 *a*d))/(d^2*(b*c - a*d)^4*(c + d*x)) + (6*a^2*c^2*Log[a + b*x])/(b*c - a*d )^5 - (6*a^2*c^2*Log[c + d*x])/(b*c - a*d)^5
3.4.12.3.1 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 = 166, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {c^{4}}{2 d^{2} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {6 c^{2} a^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {c^{3} \left (4 a d -b c \right )}{\left (a d -b c \right )^{4} d^{2} \left (d x +c \right )}-\frac {a^{3} \left (a d -4 b c \right )}{\left (a d -b c \right )^{4} b^{2} \left (b x +a \right )}+\frac {a^{4}}{2 b^{2} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {6 c^{2} a^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}\) | \(166\) |
| risch | \(\frac {-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{3}}{b d \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {\left (a^{5} d^{5}-3 a^{4} b c \,d^{4}-16 a^{3} b^{2} c^{2} d^{3}-16 a^{2} b^{3} c^{3} d^{2}-3 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{2}}{2 b^{2} d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {a c \left (a^{4} d^{4}-6 a^{3} b c \,d^{3}-8 a^{2} b^{2} c^{2} d^{2}-6 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{b^{2} d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {c^{2} a^{2} \left (a^{3} d^{3}-7 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {6 c^{2} a^{2} \ln \left (-d x -c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {6 c^{2} a^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(614\) |
| norman | \(\frac {\frac {\left (-a^{4} d^{4}+4 a^{3} b c \,d^{3}+4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) x^{3}}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b d}+\frac {a c \left (-a^{4} d^{4}+6 a^{3} b c \,d^{3}+8 a^{2} b^{2} c^{2} d^{2}+6 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (-a^{5} d^{5}+3 a^{4} b c \,d^{4}+16 a^{3} b^{2} c^{2} d^{3}+16 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) x^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a^{2} c^{2} \left (-a^{3} d^{3}+7 a^{2} b c \,d^{2}+7 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {6 c^{2} a^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {6 c^{2} a^{2} \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(617\) |
| parallelrisch | \(-\frac {12 \ln \left (b x +a \right ) x^{4} a^{2} b^{4} c^{2} d^{4}-12 \ln \left (d x +c \right ) x^{4} a^{2} b^{4} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a^{3} b^{3} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} c^{3} d^{3}-24 \ln \left (d x +c \right ) x^{3} a^{3} b^{3} c^{2} d^{4}-24 \ln \left (d x +c \right ) x^{3} a^{2} b^{4} c^{3} d^{3}+12 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} c^{2} d^{4}+2 x \,a^{6} c \,d^{5}-2 x a \,b^{5} c^{6}-x^{2} b^{6} c^{6}+a^{6} c^{2} d^{4}-a^{2} b^{4} c^{6}+x^{2} a^{6} d^{6}+2 x^{3} a^{5} b \,d^{6}-2 x^{3} b^{6} c^{5} d -10 x^{3} a^{4} b^{2} c \,d^{5}+8 x^{3} a^{3} b^{3} c^{2} d^{4}-8 x^{3} a^{2} b^{4} c^{3} d^{3}+10 x^{3} a \,b^{5} c^{4} d^{2}-4 x^{2} a^{5} b c \,d^{5}-13 x^{2} a^{4} b^{2} c^{2} d^{4}+13 x^{2} a^{2} b^{4} c^{4} d^{2}+4 x^{2} a \,b^{5} c^{5} d -14 x \,a^{5} b \,c^{2} d^{4}-4 x \,a^{4} b^{2} c^{3} d^{3}+4 x \,a^{3} b^{3} c^{4} d^{2}+14 x \,a^{2} b^{4} c^{5} d +12 \ln \left (b x +a \right ) a^{4} b^{2} c^{4} d^{2}-12 \ln \left (d x +c \right ) a^{4} b^{2} c^{4} d^{2}+24 \ln \left (b x +a \right ) x \,a^{4} b^{2} c^{3} d^{3}+24 \ln \left (b x +a \right ) x \,a^{3} b^{3} c^{4} d^{2}-24 \ln \left (d x +c \right ) x \,a^{4} b^{2} c^{3} d^{3}-24 \ln \left (d x +c \right ) x \,a^{3} b^{3} c^{4} d^{2}-8 a^{5} b \,c^{3} d^{3}+8 a^{3} b^{3} c^{5} d +48 \ln \left (b x +a \right ) x^{2} a^{3} b^{3} c^{3} d^{3}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} c^{4} d^{2}-12 \ln \left (d x +c \right ) x^{2} a^{4} b^{2} c^{2} d^{4}-48 \ln \left (d x +c \right ) x^{2} a^{3} b^{3} c^{3} d^{3}-12 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} c^{4} d^{2}}{2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \left (d x +c \right )^{2} \left (b x +a \right )^{2} b^{2} d^{2}}\) | \(780\) |
-1/2*c^4/d^2/(a*d-b*c)^3/(d*x+c)^2+6*c^2*a^2/(a*d-b*c)^5*ln(d*x+c)+c^3*(4* a*d-b*c)/(a*d-b*c)^4/d^2/(d*x+c)-a^3*(a*d-4*b*c)/(a*d-b*c)^4/b^2/(b*x+a)+1 /2*a^4/b^2/(a*d-b*c)^3/(b*x+a)^2-6*c^2*a^2/(a*d-b*c)^5*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 985 vs. \(2 (165) = 330\).
Time = 0.25 (sec) , antiderivative size = 985, normalized size of antiderivative = 5.83 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^{2} b^{4} c^{6} - 8 \, a^{3} b^{3} c^{5} d + 8 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4} + 2 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 4 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d - 13 \, a^{2} b^{4} c^{4} d^{2} + 13 \, a^{4} b^{2} c^{2} d^{4} + 4 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} + 7 \, a^{5} b c^{2} d^{4} - a^{6} c d^{5}\right )} x - 12 \, {\left (a^{2} b^{4} c^{2} d^{4} x^{4} + a^{4} b^{2} c^{4} d^{2} + 2 \, {\left (a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (a^{3} b^{3} c^{4} d^{2} + a^{4} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (a^{2} b^{4} c^{2} d^{4} x^{4} + a^{4} b^{2} c^{4} d^{2} + 2 \, {\left (a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (a^{3} b^{3} c^{4} d^{2} + a^{4} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{7} c^{7} d^{2} - 5 \, a^{3} b^{6} c^{6} d^{3} + 10 \, a^{4} b^{5} c^{5} d^{4} - 10 \, a^{5} b^{4} c^{4} d^{5} + 5 \, a^{6} b^{3} c^{3} d^{6} - a^{7} b^{2} c^{2} d^{7} + {\left (b^{9} c^{5} d^{4} - 5 \, a b^{8} c^{4} d^{5} + 10 \, a^{2} b^{7} c^{3} d^{6} - 10 \, a^{3} b^{6} c^{2} d^{7} + 5 \, a^{4} b^{5} c d^{8} - a^{5} b^{4} d^{9}\right )} x^{4} + 2 \, {\left (b^{9} c^{6} d^{3} - 4 \, a b^{8} c^{5} d^{4} + 5 \, a^{2} b^{7} c^{4} d^{5} - 5 \, a^{4} b^{5} c^{2} d^{7} + 4 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + {\left (b^{9} c^{7} d^{2} - a b^{8} c^{6} d^{3} - 9 \, a^{2} b^{7} c^{5} d^{4} + 25 \, a^{3} b^{6} c^{4} d^{5} - 25 \, a^{4} b^{5} c^{3} d^{6} + 9 \, a^{5} b^{4} c^{2} d^{7} + a^{6} b^{3} c d^{8} - a^{7} b^{2} d^{9}\right )} x^{2} + 2 \, {\left (a b^{8} c^{7} d^{2} - 4 \, a^{2} b^{7} c^{6} d^{3} + 5 \, a^{3} b^{6} c^{5} d^{4} - 5 \, a^{5} b^{4} c^{3} d^{6} + 4 \, a^{6} b^{3} c^{2} d^{7} - a^{7} b^{2} c d^{8}\right )} x\right )}} \]
-1/2*(a^2*b^4*c^6 - 8*a^3*b^3*c^5*d + 8*a^5*b*c^3*d^3 - a^6*c^2*d^4 + 2*(b ^6*c^5*d - 5*a*b^5*c^4*d^2 + 4*a^2*b^4*c^3*d^3 - 4*a^3*b^3*c^2*d^4 + 5*a^4 *b^2*c*d^5 - a^5*b*d^6)*x^3 + (b^6*c^6 - 4*a*b^5*c^5*d - 13*a^2*b^4*c^4*d^ 2 + 13*a^4*b^2*c^2*d^4 + 4*a^5*b*c*d^5 - a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 7*a ^2*b^4*c^5*d - 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 + 7*a^5*b*c^2*d^4 - a ^6*c*d^5)*x - 12*(a^2*b^4*c^2*d^4*x^4 + a^4*b^2*c^4*d^2 + 2*(a^2*b^4*c^3*d ^3 + a^3*b^3*c^2*d^4)*x^3 + (a^2*b^4*c^4*d^2 + 4*a^3*b^3*c^3*d^3 + a^4*b^2 *c^2*d^4)*x^2 + 2*(a^3*b^3*c^4*d^2 + a^4*b^2*c^3*d^3)*x)*log(b*x + a) + 12 *(a^2*b^4*c^2*d^4*x^4 + a^4*b^2*c^4*d^2 + 2*(a^2*b^4*c^3*d^3 + a^3*b^3*c^2 *d^4)*x^3 + (a^2*b^4*c^4*d^2 + 4*a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4)*x^2 + 2*(a^3*b^3*c^4*d^2 + a^4*b^2*c^3*d^3)*x)*log(d*x + c))/(a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^ 3*d^6 - a^7*b^2*c^2*d^7 + (b^9*c^5*d^4 - 5*a*b^8*c^4*d^5 + 10*a^2*b^7*c^3* d^6 - 10*a^3*b^6*c^2*d^7 + 5*a^4*b^5*c*d^8 - a^5*b^4*d^9)*x^4 + 2*(b^9*c^6 *d^3 - 4*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5 - 5*a^4*b^5*c^2*d^7 + 4*a^5*b^4 *c*d^8 - a^6*b^3*d^9)*x^3 + (b^9*c^7*d^2 - a*b^8*c^6*d^3 - 9*a^2*b^7*c^5*d ^4 + 25*a^3*b^6*c^4*d^5 - 25*a^4*b^5*c^3*d^6 + 9*a^5*b^4*c^2*d^7 + a^6*b^3 *c*d^8 - a^7*b^2*d^9)*x^2 + 2*(a*b^8*c^7*d^2 - 4*a^2*b^7*c^6*d^3 + 5*a^3*b ^6*c^5*d^4 - 5*a^5*b^4*c^3*d^6 + 4*a^6*b^3*c^2*d^7 - a^7*b^2*c*d^8)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (151) = 302\).
Time = 1.55 (sec) , antiderivative size = 1046, normalized size of antiderivative = 6.19 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 a^{2} c^{2} \log {\left (x + \frac {- \frac {6 a^{8} c^{2} d^{6}}{\left (a d - b c\right )^{5}} + \frac {36 a^{7} b c^{3} d^{5}}{\left (a d - b c\right )^{5}} - \frac {90 a^{6} b^{2} c^{4} d^{4}}{\left (a d - b c\right )^{5}} + \frac {120 a^{5} b^{3} c^{5} d^{3}}{\left (a d - b c\right )^{5}} - \frac {90 a^{4} b^{4} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + \frac {36 a^{3} b^{5} c^{7} d}{\left (a d - b c\right )^{5}} + 6 a^{3} c^{2} d - \frac {6 a^{2} b^{6} c^{8}}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{3}}{12 a^{2} b c^{2} d} \right )}}{\left (a d - b c\right )^{5}} - \frac {6 a^{2} c^{2} \log {\left (x + \frac {\frac {6 a^{8} c^{2} d^{6}}{\left (a d - b c\right )^{5}} - \frac {36 a^{7} b c^{3} d^{5}}{\left (a d - b c\right )^{5}} + \frac {90 a^{6} b^{2} c^{4} d^{4}}{\left (a d - b c\right )^{5}} - \frac {120 a^{5} b^{3} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + \frac {90 a^{4} b^{4} c^{6} d^{2}}{\left (a d - b c\right )^{5}} - \frac {36 a^{3} b^{5} c^{7} d}{\left (a d - b c\right )^{5}} + 6 a^{3} c^{2} d + \frac {6 a^{2} b^{6} c^{8}}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{3}}{12 a^{2} b c^{2} d} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{5} c^{2} d^{3} + 7 a^{4} b c^{3} d^{2} + 7 a^{3} b^{2} c^{4} d - a^{2} b^{3} c^{5} + x^{3} \left (- 2 a^{4} b d^{5} + 8 a^{3} b^{2} c d^{4} + 8 a b^{4} c^{3} d^{2} - 2 b^{5} c^{4} d\right ) + x^{2} \left (- a^{5} d^{5} + 3 a^{4} b c d^{4} + 16 a^{3} b^{2} c^{2} d^{3} + 16 a^{2} b^{3} c^{3} d^{2} + 3 a b^{4} c^{4} d - b^{5} c^{5}\right ) + x \left (- 2 a^{5} c d^{4} + 12 a^{4} b c^{2} d^{3} + 16 a^{3} b^{2} c^{3} d^{2} + 12 a^{2} b^{3} c^{4} d - 2 a b^{4} c^{5}\right )}{2 a^{6} b^{2} c^{2} d^{6} - 8 a^{5} b^{3} c^{3} d^{5} + 12 a^{4} b^{4} c^{4} d^{4} - 8 a^{3} b^{5} c^{5} d^{3} + 2 a^{2} b^{6} c^{6} d^{2} + x^{4} \cdot \left (2 a^{4} b^{4} d^{8} - 8 a^{3} b^{5} c d^{7} + 12 a^{2} b^{6} c^{2} d^{6} - 8 a b^{7} c^{3} d^{5} + 2 b^{8} c^{4} d^{4}\right ) + x^{3} \cdot \left (4 a^{5} b^{3} d^{8} - 12 a^{4} b^{4} c d^{7} + 8 a^{3} b^{5} c^{2} d^{6} + 8 a^{2} b^{6} c^{3} d^{5} - 12 a b^{7} c^{4} d^{4} + 4 b^{8} c^{5} d^{3}\right ) + x^{2} \cdot \left (2 a^{6} b^{2} d^{8} - 18 a^{4} b^{4} c^{2} d^{6} + 32 a^{3} b^{5} c^{3} d^{5} - 18 a^{2} b^{6} c^{4} d^{4} + 2 b^{8} c^{6} d^{2}\right ) + x \left (4 a^{6} b^{2} c d^{7} - 12 a^{5} b^{3} c^{2} d^{6} + 8 a^{4} b^{4} c^{3} d^{5} + 8 a^{3} b^{5} c^{4} d^{4} - 12 a^{2} b^{6} c^{5} d^{3} + 4 a b^{7} c^{6} d^{2}\right )} \]
6*a**2*c**2*log(x + (-6*a**8*c**2*d**6/(a*d - b*c)**5 + 36*a**7*b*c**3*d** 5/(a*d - b*c)**5 - 90*a**6*b**2*c**4*d**4/(a*d - b*c)**5 + 120*a**5*b**3*c **5*d**3/(a*d - b*c)**5 - 90*a**4*b**4*c**6*d**2/(a*d - b*c)**5 + 36*a**3* b**5*c**7*d/(a*d - b*c)**5 + 6*a**3*c**2*d - 6*a**2*b**6*c**8/(a*d - b*c)* *5 + 6*a**2*b*c**3)/(12*a**2*b*c**2*d))/(a*d - b*c)**5 - 6*a**2*c**2*log(x + (6*a**8*c**2*d**6/(a*d - b*c)**5 - 36*a**7*b*c**3*d**5/(a*d - b*c)**5 + 90*a**6*b**2*c**4*d**4/(a*d - b*c)**5 - 120*a**5*b**3*c**5*d**3/(a*d - b* c)**5 + 90*a**4*b**4*c**6*d**2/(a*d - b*c)**5 - 36*a**3*b**5*c**7*d/(a*d - b*c)**5 + 6*a**3*c**2*d + 6*a**2*b**6*c**8/(a*d - b*c)**5 + 6*a**2*b*c**3 )/(12*a**2*b*c**2*d))/(a*d - b*c)**5 + (-a**5*c**2*d**3 + 7*a**4*b*c**3*d* *2 + 7*a**3*b**2*c**4*d - a**2*b**3*c**5 + x**3*(-2*a**4*b*d**5 + 8*a**3*b **2*c*d**4 + 8*a*b**4*c**3*d**2 - 2*b**5*c**4*d) + x**2*(-a**5*d**5 + 3*a* *4*b*c*d**4 + 16*a**3*b**2*c**2*d**3 + 16*a**2*b**3*c**3*d**2 + 3*a*b**4*c **4*d - b**5*c**5) + x*(-2*a**5*c*d**4 + 12*a**4*b*c**2*d**3 + 16*a**3*b** 2*c**3*d**2 + 12*a**2*b**3*c**4*d - 2*a*b**4*c**5))/(2*a**6*b**2*c**2*d**6 - 8*a**5*b**3*c**3*d**5 + 12*a**4*b**4*c**4*d**4 - 8*a**3*b**5*c**5*d**3 + 2*a**2*b**6*c**6*d**2 + x**4*(2*a**4*b**4*d**8 - 8*a**3*b**5*c*d**7 + 12 *a**2*b**6*c**2*d**6 - 8*a*b**7*c**3*d**5 + 2*b**8*c**4*d**4) + x**3*(4*a* *5*b**3*d**8 - 12*a**4*b**4*c*d**7 + 8*a**3*b**5*c**2*d**6 + 8*a**2*b**6*c **3*d**5 - 12*a*b**7*c**4*d**4 + 4*b**8*c**5*d**3) + x**2*(2*a**6*b**2*...
Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (165) = 330\).
Time = 0.22 (sec) , antiderivative size = 740, normalized size of antiderivative = 4.38 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, a^{2} c^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {6 \, a^{2} c^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3} + 2 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x^{3} + {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 16 \, a^{2} b^{3} c^{3} d^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{2} + 2 \, {\left (a b^{4} c^{5} - 6 \, a^{2} b^{3} c^{4} d - 8 \, a^{3} b^{2} c^{3} d^{2} - 6 \, a^{4} b c^{2} d^{3} + a^{5} c d^{4}\right )} x}{2 \, {\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6} + {\left (b^{8} c^{4} d^{4} - 4 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} - 4 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 2 \, {\left (b^{8} c^{5} d^{3} - 3 \, a b^{7} c^{4} d^{4} + 2 \, a^{2} b^{6} c^{3} d^{5} + 2 \, a^{3} b^{5} c^{2} d^{6} - 3 \, a^{4} b^{4} c d^{7} + a^{5} b^{3} d^{8}\right )} x^{3} + {\left (b^{8} c^{6} d^{2} - 9 \, a^{2} b^{6} c^{4} d^{4} + 16 \, a^{3} b^{5} c^{3} d^{5} - 9 \, a^{4} b^{4} c^{2} d^{6} + a^{6} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (a b^{7} c^{6} d^{2} - 3 \, a^{2} b^{6} c^{5} d^{3} + 2 \, a^{3} b^{5} c^{4} d^{4} + 2 \, a^{4} b^{4} c^{3} d^{5} - 3 \, a^{5} b^{3} c^{2} d^{6} + a^{6} b^{2} c d^{7}\right )} x\right )}} \]
6*a^2*c^2*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10* a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 6*a^2*c^2*log(d*x + c)/(b^5*c ^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d ^4 - a^5*d^5) - 1/2*(a^2*b^3*c^5 - 7*a^3*b^2*c^4*d - 7*a^4*b*c^3*d^2 + a^5 *c^2*d^3 + 2*(b^5*c^4*d - 4*a*b^4*c^3*d^2 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x ^3 + (b^5*c^5 - 3*a*b^4*c^4*d - 16*a^2*b^3*c^3*d^2 - 16*a^3*b^2*c^2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*x^2 + 2*(a*b^4*c^5 - 6*a^2*b^3*c^4*d - 8*a^3*b^2* c^3*d^2 - 6*a^4*b*c^2*d^3 + a^5*c*d^4)*x)/(a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5 *d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6 + (b^8*c^4* d^4 - 4*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 - 4*a^3*b^5*c*d^7 + a^4*b^4*d^8) *x^4 + 2*(b^8*c^5*d^3 - 3*a*b^7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 2*a^3*b^5*c^ 2*d^6 - 3*a^4*b^4*c*d^7 + a^5*b^3*d^8)*x^3 + (b^8*c^6*d^2 - 9*a^2*b^6*c^4* d^4 + 16*a^3*b^5*c^3*d^5 - 9*a^4*b^4*c^2*d^6 + a^6*b^2*d^8)*x^2 + 2*(a*b^7 *c^6*d^2 - 3*a^2*b^6*c^5*d^3 + 2*a^3*b^5*c^4*d^4 + 2*a^4*b^4*c^3*d^5 - 3*a ^5*b^3*c^2*d^6 + a^6*b^2*c*d^7)*x)
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (165) = 330\).
Time = 0.28 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.92 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, a^{2} b c^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {6 \, a^{2} c^{2} d \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {2 \, b^{5} c^{4} d x^{3} - 8 \, a b^{4} c^{3} d^{2} x^{3} - 8 \, a^{3} b^{2} c d^{4} x^{3} + 2 \, a^{4} b d^{5} x^{3} + b^{5} c^{5} x^{2} - 3 \, a b^{4} c^{4} d x^{2} - 16 \, a^{2} b^{3} c^{3} d^{2} x^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 3 \, a^{4} b c d^{4} x^{2} + a^{5} d^{5} x^{2} + 2 \, a b^{4} c^{5} x - 12 \, a^{2} b^{3} c^{4} d x - 16 \, a^{3} b^{2} c^{3} d^{2} x - 12 \, a^{4} b c^{2} d^{3} x + 2 \, a^{5} c d^{4} x + a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}}{2 \, {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]
6*a^2*b*c^2*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^ 2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) - 6*a^2*c^2*d*log(ab s(d*x + c))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2 *c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6) - 1/2*(2*b^5*c^4*d*x^3 - 8*a*b^4*c^3*d ^2*x^3 - 8*a^3*b^2*c*d^4*x^3 + 2*a^4*b*d^5*x^3 + b^5*c^5*x^2 - 3*a*b^4*c^4 *d*x^2 - 16*a^2*b^3*c^3*d^2*x^2 - 16*a^3*b^2*c^2*d^3*x^2 - 3*a^4*b*c*d^4*x ^2 + a^5*d^5*x^2 + 2*a*b^4*c^5*x - 12*a^2*b^3*c^4*d*x - 16*a^3*b^2*c^3*d^2 *x - 12*a^4*b*c^2*d^3*x + 2*a^5*c*d^4*x + a^2*b^3*c^5 - 7*a^3*b^2*c^4*d - 7*a^4*b*c^3*d^2 + a^5*c^2*d^3)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4 *c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*(b*d*x^2 + b*c*x + a*d*x + a*c)^ 2)
Time = 0.70 (sec) , antiderivative size = 678, normalized size of antiderivative = 4.01 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {x^2\,\left (-a^5\,d^5+3\,a^4\,b\,c\,d^4+16\,a^3\,b^2\,c^2\,d^3+16\,a^2\,b^3\,c^3\,d^2+3\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {a^2\,c^2\,\left (a^3\,d^3-7\,a^2\,b\,c\,d^2-7\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{2\,b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a\,c\,x\,\left (-a^4\,d^4+6\,a^3\,b\,c\,d^3+8\,a^2\,b^2\,c^2\,d^2+6\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {12\,a^2\,c^2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \]
((x^2*(16*a^2*b^3*c^3*d^2 - b^5*c^5 - a^5*d^5 + 16*a^3*b^2*c^2*d^3 + 3*a*b ^4*c^4*d + 3*a^4*b*c*d^4))/(2*b^2*d^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^4*d^4 + b^4*c^4 - 4*a*b^3*c ^3*d - 4*a^3*b*c*d^3))/(b*d*(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)) - (a^2*c^2*(a^3*d^3 + b^3*c^3 - 7*a*b^2*c^2*d - 7*a^2*b*c*d^2))/(2*b^2*d^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)) + (a*c*x*(8*a^2*b^2*c^2*d^2 - b^4*c^4 - a^4*d^4 + 6*a*b^3*c^3*d + 6*a^3*b*c*d^3))/(b^2*d^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*(2*a*b*c^2 + 2*a^2*c*d) + x ^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*a*b*d^2 + 2*b^2*c*d) + a^2*c^2 + b^2*d^2*x^4) - (12*a^2*c^2*atanh((a^5*d^5 + b^5*c^5 + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a*b^4*c^4*d - 3*a^4*b*c*d^4)/(a*d - b*c)^5 + (2*b *d*x*(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))/(a*d - b*c)^5))/(a*d - b*c)^5