Integrand size = 18, antiderivative size = 213 \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=\frac {a^5}{2 b^3 (b c-a d)^3 (a+b x)^2}-\frac {a^4 (5 b c-2 a d)}{b^3 (b c-a d)^4 (a+b x)}-\frac {c^5}{2 d^3 (b c-a d)^3 (c+d x)^2}+\frac {c^4 (2 b c-5 a d)}{d^3 (b c-a d)^4 (c+d x)}-\frac {a^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}+\frac {c^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^5} \]
1/2*a^5/b^3/(-a*d+b*c)^3/(b*x+a)^2-a^4*(-2*a*d+5*b*c)/b^3/(-a*d+b*c)^4/(b* x+a)-1/2*c^5/d^3/(-a*d+b*c)^3/(d*x+c)^2+c^4*(-5*a*d+2*b*c)/d^3/(-a*d+b*c)^ 4/(d*x+c)-a^3*(a^2*d^2-5*a*b*c*d+10*b^2*c^2)*ln(b*x+a)/b^3/(-a*d+b*c)^5+c^ 3*(10*a^2*d^2-5*a*b*c*d+b^2*c^2)*ln(d*x+c)/d^3/(-a*d+b*c)^5
Time = 0.21 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=\frac {a^5}{2 b^3 (b c-a d)^3 (a+b x)^2}+\frac {a^4 (-5 b c+2 a d)}{b^3 (b c-a d)^4 (a+b x)}+\frac {c^5}{2 d^3 (-b c+a d)^3 (c+d x)^2}+\frac {c^4 (2 b c-5 a d)}{d^3 (b c-a d)^4 (c+d x)}-\frac {a^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}-\frac {c^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^3 (-b c+a d)^5} \]
a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) + (a^4*(-5*b*c + 2*a*d))/(b^3*(b*c - a*d)^4*(a + b*x)) + c^5/(2*d^3*(-(b*c) + a*d)^3*(c + d*x)^2) + (c^4*(2*b* c - 5*a*d))/(d^3*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[a + b*x])/(b^3*(b*c - a*d)^5) - (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d^3*(-(b*c) + a*d)^5)
Time = 0.47 (sec) , antiderivative size = 213, 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.
| \(\displaystyle \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^5}{b^2 (a+b x)^3 (b c-a d)^3}-\frac {a^4 (2 a d-5 b c)}{b^2 (a+b x)^2 (b c-a d)^4}-\frac {c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right )}{d^2 (c+d x) (a d-b c)^5}-\frac {a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right )}{b^2 (a+b x) (b c-a d)^5}-\frac {c^5}{d^2 (c+d x)^3 (a d-b c)^3}-\frac {c^4 (2 b c-5 a d)}{d^2 (c+d x)^2 (a d-b c)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}-\frac {a^4 (5 b c-2 a d)}{b^3 (a+b x) (b c-a d)^4}+\frac {c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^5}-\frac {a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}-\frac {c^5}{2 d^3 (c+d x)^2 (b c-a d)^3}+\frac {c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4}\) |
a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) - (a^4*(5*b*c - 2*a*d))/(b^3*(b*c - a*d)^4*(a + b*x)) - c^5/(2*d^3*(b*c - a*d)^3*(c + d*x)^2) + (c^4*(2*b*c - 5*a*d))/(d^3*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d + a^2 *d^2)*Log[a + b*x])/(b^3*(b*c - a*d)^5) + (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a ^2*d^2)*Log[c + d*x])/(d^3*(b*c - a*d)^5)
3.4.11.3.1 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 = 0.55 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {c^{4} \left (5 a d -2 b c \right )}{d^{3} \left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {c^{5}}{2 d^{3} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {c^{3} \left (10 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{3}}-\frac {a^{5}}{2 b^{3} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {a^{3} \left (a^{2} d^{2}-5 a b c d +10 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{3}}+\frac {a^{4} \left (2 a d -5 b c \right )}{b^{3} \left (a d -b c \right )^{4} \left (b x +a \right )}\) | \(210\) |
| norman | \(\frac {\frac {\left (2 a^{5} d^{5}-5 a^{4} b c \,d^{4}-5 a \,b^{4} c^{4} d +2 b^{5} c^{5}\right ) x^{3}}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a c \left (3 a^{5} d^{5}-7 a^{4} b c \,d^{4}-5 a^{3} b^{2} c^{2} d^{3}-5 a^{2} b^{3} c^{3} d^{2}-7 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x}{b^{3} d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (3 a^{6} d^{6}-a^{5} b c \,d^{5}-20 a^{4} b^{2} c^{2} d^{4}-20 a^{2} b^{4} c^{4} d^{2}-a \,b^{5} c^{5} d +3 b^{6} c^{6}\right ) x^{2}}{2 d^{3} b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a^{2} c^{2} \left (3 a^{4} d^{4}-9 a^{3} b c \,d^{3}-9 a \,b^{3} c^{3} d +3 b^{4} c^{4}\right )}{2 d^{3} b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {a^{3} \left (a^{2} d^{2}-5 a b c d +10 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) b^{3}}-\frac {c^{3} \left (10 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(674\) |
| risch | \(\frac {\frac {\left (2 a^{5} d^{5}-5 a^{4} b c \,d^{4}-5 a \,b^{4} c^{4} d +2 b^{5} c^{5}\right ) x^{3}}{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 (3 a^{6} d^{6}-a^{5} b c \,d^{5}-20 a^{4} b^{2} c^{2} d^{4}-20 a^{2} b^{4} c^{4} d^{2}-a \,b^{5} c^{5} d +3 b^{6} c^{6}\right ) x^{2}}{2 d^{3} b^{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 {a c \left (3 a^{5} d^{5}-7 a^{4} b c \,d^{4}-5 a^{3} b^{2} c^{2} d^{3}-5 a^{2} b^{3} c^{3} d^{2}-7 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x}{b^{3} d^{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 c^{2} a^{2} \left (a^{4} d^{4}-3 a^{3} b c \,d^{3}-3 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{2 d^{3} b^{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 )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {a^{5} \ln \left (-b x -a \right ) d^{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 ) b^{3}}-\frac {5 a^{4} \ln \left (-b x -a \right ) c d}{\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 ) b^{2}}+\frac {10 a^{3} \ln \left (-b x -a \right ) c^{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 ) b}-\frac {10 c^{3} \ln \left (d x +c \right ) a^{2}}{d \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 {5 c^{4} \ln \left (d x +c \right ) a b}{d^{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 )}-\frac {c^{5} \ln \left (d x +c \right ) b^{2}}{d^{3} \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 )}\) | \(973\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1216\) |
-c^4*(5*a*d-2*b*c)/d^3/(a*d-b*c)^4/(d*x+c)+1/2*c^5/d^3/(a*d-b*c)^3/(d*x+c) ^2-c^3*(10*a^2*d^2-5*a*b*c*d+b^2*c^2)/(a*d-b*c)^5/d^3*ln(d*x+c)-1/2*a^5/b^ 3/(a*d-b*c)^3/(b*x+a)^2+a^3*(a^2*d^2-5*a*b*c*d+10*b^2*c^2)/(a*d-b*c)^5/b^3 *ln(b*x+a)+a^4*(2*a*d-5*b*c)/b^3/(a*d-b*c)^4/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1267 vs. \(2 (209) = 418\).
Time = 0.29 (sec) , antiderivative size = 1267, normalized size of antiderivative = 5.95 \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]
1/2*(3*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 9*a^4*b^3*c^5*d^2 - 9*a^5*b^2*c^4* d^3 + 12*a^6*b*c^3*d^4 - 3*a^7*c^2*d^5 + 2*(2*b^7*c^6*d - 7*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 7*a^5*b^2*c*d^6 - 2*a^6*b*d^7)*x ^3 + (3*b^7*c^7 - 4*a*b^6*c^6*d - 19*a^2*b^5*c^5*d^2 + 20*a^3*b^4*c^4*d^3 - 20*a^4*b^3*c^3*d^4 + 19*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 - 3*a^7*d^7)*x^2 + 2*(3*a*b^6*c^7 - 10*a^2*b^5*c^6*d + 2*a^3*b^4*c^5*d^2 - 2*a^5*b^2*c^3*d ^4 + 10*a^6*b*c^2*d^5 - 3*a^7*c*d^6)*x - 2*(10*a^5*b^2*c^4*d^3 - 5*a^6*b*c ^3*d^4 + a^7*c^2*d^5 + (10*a^3*b^4*c^2*d^5 - 5*a^4*b^3*c*d^6 + a^5*b^2*d^7 )*x^4 + 2*(10*a^3*b^4*c^3*d^4 + 5*a^4*b^3*c^2*d^5 - 4*a^5*b^2*c*d^6 + a^6* b*d^7)*x^3 + (10*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 - a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(10*a^4*b^3*c^4*d^3 + 5*a^5*b^2*c^3*d^4 - 4*a^6*b*c^2*d^5 + a^7*c*d^6)*x)*log(b*x + a) + 2*(a^2*b^5*c^7 - 5*a^3*b^4 *c^6*d + 10*a^4*b^3*c^5*d^2 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5* c^3*d^4)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3 *b^4*c^3*d^4)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 35*a^3*b^ 4*c^4*d^3 + 10*a^4*b^3*c^3*d^4)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a ^3*b^4*c^5*d^2 + 10*a^4*b^3*c^4*d^3)*x)*log(d*x + c))/(a^2*b^8*c^7*d^3 - 5 *a^3*b^7*c^6*d^4 + 10*a^4*b^6*c^5*d^5 - 10*a^5*b^5*c^4*d^6 + 5*a^6*b^4*c^3 *d^7 - a^7*b^3*c^2*d^8 + (b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3* d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^4 + 2*(b^1...
Timed out. \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (209) = 418\).
Time = 0.23 (sec) , antiderivative size = 813, normalized size of antiderivative = 3.82 \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {{\left (10 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{8} c^{5} - 5 \, a b^{7} c^{4} d + 10 \, a^{2} b^{6} c^{3} d^{2} - 10 \, a^{3} b^{5} c^{2} d^{3} + 5 \, a^{4} b^{4} c d^{4} - a^{5} b^{3} d^{5}} + \frac {{\left (b^{2} c^{5} - 5 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} d^{3} - 5 \, a b^{4} c^{4} d^{4} + 10 \, a^{2} b^{3} c^{3} d^{5} - 10 \, a^{3} b^{2} c^{2} d^{6} + 5 \, a^{4} b c d^{7} - a^{5} d^{8}} + \frac {3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \, {\left (2 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{4} b^{2} c d^{5} + 2 \, a^{5} b d^{6}\right )} x^{3} + {\left (3 \, b^{6} c^{6} - a b^{5} c^{5} d - 20 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{4} b^{2} c^{2} d^{4} - a^{5} b c d^{5} + 3 \, a^{6} d^{6}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} - 5 \, a^{4} b^{2} c^{3} d^{3} - 7 \, a^{5} b c^{2} d^{4} + 3 \, a^{6} c d^{5}\right )} x}{2 \, {\left (a^{2} b^{7} c^{6} d^{3} - 4 \, a^{3} b^{6} c^{5} d^{4} + 6 \, a^{4} b^{5} c^{4} d^{5} - 4 \, a^{5} b^{4} c^{3} d^{6} + a^{6} b^{3} c^{2} d^{7} + {\left (b^{9} c^{4} d^{5} - 4 \, a b^{8} c^{3} d^{6} + 6 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )} x^{4} + 2 \, {\left (b^{9} c^{5} d^{4} - 3 \, a b^{8} c^{4} d^{5} + 2 \, a^{2} b^{7} c^{3} d^{6} + 2 \, a^{3} b^{6} c^{2} d^{7} - 3 \, a^{4} b^{5} c d^{8} + a^{5} b^{4} d^{9}\right )} x^{3} + {\left (b^{9} c^{6} d^{3} - 9 \, a^{2} b^{7} c^{4} d^{5} + 16 \, a^{3} b^{6} c^{3} d^{6} - 9 \, a^{4} b^{5} c^{2} d^{7} + a^{6} b^{3} d^{9}\right )} x^{2} + 2 \, {\left (a b^{8} c^{6} d^{3} - 3 \, a^{2} b^{7} c^{5} d^{4} + 2 \, a^{3} b^{6} c^{4} d^{5} + 2 \, a^{4} b^{5} c^{3} d^{6} - 3 \, a^{5} b^{4} c^{2} d^{7} + a^{6} b^{3} c d^{8}\right )} x\right )}} \]
-(10*a^3*b^2*c^2 - 5*a^4*b*c*d + a^5*d^2)*log(b*x + a)/(b^8*c^5 - 5*a*b^7* c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^ 3*d^5) + (b^2*c^5 - 5*a*b*c^4*d + 10*a^2*c^3*d^2)*log(d*x + c)/(b^5*c^5*d^ 3 - 5*a*b^4*c^4*d^4 + 10*a^2*b^3*c^3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c* d^7 - a^5*d^8) + 1/2*(3*a^2*b^4*c^6 - 9*a^3*b^3*c^5*d - 9*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + 2*(2*b^6*c^5*d - 5*a*b^5*c^4*d^2 - 5*a^4*b^2*c*d^5 + 2*a^5 *b*d^6)*x^3 + (3*b^6*c^6 - a*b^5*c^5*d - 20*a^2*b^4*c^4*d^2 - 20*a^4*b^2*c ^2*d^4 - a^5*b*c*d^5 + 3*a^6*d^6)*x^2 + 2*(3*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 + 3*a^6*c*d^5)*x) /(a^2*b^7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c^4*d^5 - 4*a^5*b^4*c^3* d^6 + a^6*b^3*c^2*d^7 + (b^9*c^4*d^5 - 4*a*b^8*c^3*d^6 + 6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c^5*d^4 - 3*a*b^8*c^4*d^5 + 2*a^2*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^9)*x^ 3 + (b^9*c^6*d^3 - 9*a^2*b^7*c^4*d^5 + 16*a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2* d^7 + a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*a^2*b^7*c^5*d^4 + 2*a^3*b^6* c^4*d^5 + 2*a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*x)
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (209) = 418\).
Time = 0.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.27 \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {{\left (10 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8} c^{5} - 5 \, a b^{7} c^{4} d + 10 \, a^{2} b^{6} c^{3} d^{2} - 10 \, a^{3} b^{5} c^{2} d^{3} + 5 \, a^{4} b^{4} c d^{4} - a^{5} b^{3} d^{5}} + \frac {{\left (b^{2} c^{5} - 5 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d^{3} - 5 \, a b^{4} c^{4} d^{4} + 10 \, a^{2} b^{3} c^{3} d^{5} - 10 \, a^{3} b^{2} c^{2} d^{6} + 5 \, a^{4} b c d^{7} - a^{5} d^{8}} + \frac {3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \, {\left (2 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{4} b^{2} c d^{5} + 2 \, a^{5} b d^{6}\right )} x^{3} + {\left (3 \, b^{6} c^{6} - a b^{5} c^{5} d - 20 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{4} b^{2} c^{2} d^{4} - a^{5} b c d^{5} + 3 \, a^{6} d^{6}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} - 5 \, a^{4} b^{2} c^{3} d^{3} - 7 \, a^{5} b c^{2} d^{4} + 3 \, a^{6} c d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} b^{3} d^{3}} \]
-(10*a^3*b^2*c^2 - 5*a^4*b*c*d + a^5*d^2)*log(abs(b*x + a))/(b^8*c^5 - 5*a *b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a ^5*b^3*d^5) + (b^2*c^5 - 5*a*b*c^4*d + 10*a^2*c^3*d^2)*log(abs(d*x + c))/( b^5*c^5*d^3 - 5*a*b^4*c^4*d^4 + 10*a^2*b^3*c^3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c*d^7 - a^5*d^8) + 1/2*(3*a^2*b^4*c^6 - 9*a^3*b^3*c^5*d - 9*a^5*b* c^3*d^3 + 3*a^6*c^2*d^4 + 2*(2*b^6*c^5*d - 5*a*b^5*c^4*d^2 - 5*a^4*b^2*c*d ^5 + 2*a^5*b*d^6)*x^3 + (3*b^6*c^6 - a*b^5*c^5*d - 20*a^2*b^4*c^4*d^2 - 20 *a^4*b^2*c^2*d^4 - a^5*b*c*d^5 + 3*a^6*d^6)*x^2 + 2*(3*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 + 3*a^6 *c*d^5)*x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*b^3*d^3)
Time = 1.51 (sec) , antiderivative size = 759, normalized size of antiderivative = 3.56 \[ \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {3\,a^2\,c^2\,\left (a^4\,d^4-3\,a^3\,b\,c\,d^3-3\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,b^3\,d^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x^2\,\left (-3\,a^6\,d^6+a^5\,b\,c\,d^5+20\,a^4\,b^2\,c^2\,d^4+20\,a^2\,b^4\,c^4\,d^2+a\,b^5\,c^5\,d-3\,b^6\,c^6\right )}{2\,b^3\,d^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x^3\,\left (a\,d+b\,c\right )\,\left (2\,a^4\,d^4-7\,a^3\,b\,c\,d^3+7\,a^2\,b^2\,c^2\,d^2-7\,a\,b^3\,c^3\,d+2\,b^4\,c^4\right )}{b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a\,c\,x\,\left (a\,d+b\,c\right )\,\left (3\,a^4\,d^4-10\,a^3\,b\,c\,d^3+5\,a^2\,b^2\,c^2\,d^2-10\,a\,b^3\,c^3\,d+3\,b^4\,c^4\right )}{b^3\,d^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {\ln \left (a+b\,x\right )\,\left (a^5\,d^2-5\,a^4\,b\,c\,d+10\,a^3\,b^2\,c^2\right )}{-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^2-5\,a\,b\,c^4\,d+b^2\,c^5\right )}{a^5\,d^8-5\,a^4\,b\,c\,d^7+10\,a^3\,b^2\,c^2\,d^6-10\,a^2\,b^3\,c^3\,d^5+5\,a\,b^4\,c^4\,d^4-b^5\,c^5\,d^3} \]
((3*a^2*c^2*(a^4*d^4 + b^4*c^4 - 3*a*b^3*c^3*d - 3*a^3*b*c*d^3))/(2*b^3*d^ 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*(20*a^2*b^4*c^4*d^2 - 3*b^6*c^6 - 3*a^6*d^6 + 20*a^4*b^2*c^2*d^4 + a*b^5*c^5*d + a^5*b*c*d^5))/(2*b^3*d^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^3*(a*d + b*c)*(2*a^4*d^4 + 2*b ^4*c^4 + 7*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 7*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)) + (a* c*x*(a*d + b*c)*(3*a^4*d^4 + 3*b^4*c^4 + 5*a^2*b^2*c^2*d^2 - 10*a*b^3*c^3* d - 10*a^3*b*c*d^3))/(b^3*d^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*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 ) - (log(a + b*x)*(a^5*d^2 + 10*a^3*b^2*c^2 - 5*a^4*b*c*d))/(b^8*c^5 - a^5 *b^3*d^5 + 5*a^4*b^4*c*d^4 + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 - 5*a *b^7*c^4*d) - (log(c + d*x)*(b^2*c^5 + 10*a^2*c^3*d^2 - 5*a*b*c^4*d))/(a^5 *d^8 - b^5*c^5*d^3 + 5*a*b^4*c^4*d^4 - 10*a^2*b^3*c^3*d^5 + 10*a^3*b^2*c^2 *d^6 - 5*a^4*b*c*d^7)