Integrand size = 18, antiderivative size = 245 \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {3 (b c+a d) x}{b^4 d^4}+\frac {x^2}{2 b^3 d^3}+\frac {a^7}{2 b^5 (b c-a d)^3 (a+b x)^2}-\frac {a^6 (7 b c-4 a d)}{b^5 (b c-a d)^4 (a+b x)}-\frac {c^7}{2 d^5 (b c-a d)^3 (c+d x)^2}+\frac {c^6 (4 b c-7 a d)}{d^5 (b c-a d)^4 (c+d x)}-\frac {3 a^5 \left (7 b^2 c^2-7 a b c d+2 a^2 d^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}+\frac {3 c^5 \left (2 b^2 c^2-7 a b c d+7 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^5} \]
-3*(a*d+b*c)*x/b^4/d^4+1/2*x^2/b^3/d^3+1/2*a^7/b^5/(-a*d+b*c)^3/(b*x+a)^2- a^6*(-4*a*d+7*b*c)/b^5/(-a*d+b*c)^4/(b*x+a)-1/2*c^7/d^5/(-a*d+b*c)^3/(d*x+ c)^2+c^6*(-7*a*d+4*b*c)/d^5/(-a*d+b*c)^4/(d*x+c)-3*a^5*(2*a^2*d^2-7*a*b*c* d+7*b^2*c^2)*ln(b*x+a)/b^5/(-a*d+b*c)^5+3*c^5*(7*a^2*d^2-7*a*b*c*d+2*b^2*c ^2)*ln(d*x+c)/d^5/(-a*d+b*c)^5
Time = 0.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98 \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=\frac {1}{2} \left (-\frac {6 (b c+a d) x}{b^4 d^4}+\frac {x^2}{b^3 d^3}+\frac {a^7}{b^5 (b c-a d)^3 (a+b x)^2}+\frac {2 a^6 (-7 b c+4 a d)}{b^5 (b c-a d)^4 (a+b x)}+\frac {c^7}{d^5 (-b c+a d)^3 (c+d x)^2}+\frac {2 c^6 (4 b c-7 a d)}{d^5 (b c-a d)^4 (c+d x)}-\frac {6 a^5 \left (7 b^2 c^2-7 a b c d+2 a^2 d^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac {6 c^5 \left (2 b^2 c^2-7 a b c d+7 a^2 d^2\right ) \log (c+d x)}{d^5 (-b c+a d)^5}\right ) \]
((-6*(b*c + a*d)*x)/(b^4*d^4) + x^2/(b^3*d^3) + a^7/(b^5*(b*c - a*d)^3*(a + b*x)^2) + (2*a^6*(-7*b*c + 4*a*d))/(b^5*(b*c - a*d)^4*(a + b*x)) + c^7/( d^5*(-(b*c) + a*d)^3*(c + d*x)^2) + (2*c^6*(4*b*c - 7*a*d))/(d^5*(b*c - a* d)^4*(c + d*x)) - (6*a^5*(7*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*Log[a + b*x]) /(b^5*(b*c - a*d)^5) - (6*c^5*(2*b^2*c^2 - 7*a*b*c*d + 7*a^2*d^2)*Log[c + d*x])/(d^5*(-(b*c) + a*d)^5))/2
Time = 0.58 (sec) , antiderivative size = 245, 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.
| \(\displaystyle \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^7}{b^4 (a+b x)^3 (b c-a d)^3}-\frac {a^6 (4 a d-7 b c)}{b^4 (a+b x)^2 (b c-a d)^4}-\frac {3 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right )}{d^4 (c+d x) (a d-b c)^5}-\frac {3 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right )}{b^4 (a+b x) (b c-a d)^5}-\frac {3 (a d+b c)}{b^4 d^4}-\frac {c^7}{d^4 (c+d x)^3 (a d-b c)^3}-\frac {c^6 (4 b c-7 a d)}{d^4 (c+d x)^2 (a d-b c)^4}+\frac {x}{b^3 d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^7}{2 b^5 (a+b x)^2 (b c-a d)^3}-\frac {a^6 (7 b c-4 a d)}{b^5 (a+b x) (b c-a d)^4}+\frac {3 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^5}-\frac {3 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac {3 x (a d+b c)}{b^4 d^4}-\frac {c^7}{2 d^5 (c+d x)^2 (b c-a d)^3}+\frac {c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}+\frac {x^2}{2 b^3 d^3}\) |
(-3*(b*c + a*d)*x)/(b^4*d^4) + x^2/(2*b^3*d^3) + a^7/(2*b^5*(b*c - a*d)^3* (a + b*x)^2) - (a^6*(7*b*c - 4*a*d))/(b^5*(b*c - a*d)^4*(a + b*x)) - c^7/( 2*d^5*(b*c - a*d)^3*(c + d*x)^2) + (c^6*(4*b*c - 7*a*d))/(d^5*(b*c - a*d)^ 4*(c + d*x)) - (3*a^5*(7*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2)*Log[a + b*x])/(b ^5*(b*c - a*d)^5) + (3*c^5*(2*b^2*c^2 - 7*a*b*c*d + 7*a^2*d^2)*Log[c + d*x ])/(d^5*(b*c - a*d)^5)
3.4.9.3.1 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 = 239, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {-\frac {1}{2} b d \,x^{2}+3 a d x +3 b c x}{b^{4} d^{4}}-\frac {c^{6} \left (7 a d -4 b c \right )}{d^{5} \left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {c^{7}}{2 d^{5} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {3 c^{5} \left (7 a^{2} d^{2}-7 a b c d +2 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{5}}-\frac {a^{7}}{2 b^{5} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {3 a^{5} \left (2 a^{2} d^{2}-7 a b c d +7 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )^{5}}+\frac {a^{6} \left (4 a d -7 b c \right )}{b^{5} \left (a d -b c \right )^{4} \left (b x +a \right )}\) | \(239\) |
| norman | \(\frac {\frac {\left (12 a^{7} d^{7}-22 a^{6} b c \,d^{6}-3 a^{5} b^{2} c^{2} d^{5}+10 a^{4} b^{3} c^{3} d^{4}+10 a^{3} b^{4} c^{4} d^{3}-3 a^{2} b^{5} c^{5} d^{2}-22 a \,b^{6} c^{6} d +12 b^{7} c^{7}\right ) x^{3}}{d^{4} b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {c a \left (18 a^{7} d^{7}-25 a^{6} b c \,d^{6}-25 a^{5} b^{2} c^{2} d^{5}+23 a^{4} b^{3} c^{3} d^{4}+23 a^{3} b^{4} c^{4} d^{3}-25 a^{2} b^{5} c^{5} d^{2}-25 a \,b^{6} c^{6} d +18 b^{7} c^{7}\right ) x}{d^{5} b^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {x^{6}}{2 b d}-\frac {2 \left (a d +b c \right ) x^{5}}{b^{2} d^{2}}+\frac {\left (18 a^{8} d^{8}+11 a^{7} b c \,d^{7}-91 a^{6} b^{2} d^{6} c^{2}+4 a^{5} b^{3} c^{3} d^{5}+80 a^{4} b^{4} d^{4} c^{4}+4 a^{3} b^{5} c^{5} d^{3}-91 a^{2} b^{6} d^{2} c^{6}+11 a \,b^{7} c^{7} d +18 b^{8} c^{8}\right ) x^{2}}{2 d^{5} b^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a^{2} c^{2} \left (18 a^{6} d^{6}-37 a^{5} b c \,d^{5}-3 a^{4} b^{2} c^{2} d^{4}+32 a^{3} b^{3} c^{3} d^{3}-3 a^{2} b^{4} c^{4} d^{2}-37 a \,b^{5} c^{5} d +18 b^{6} c^{6}\right )}{2 d^{5} b^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {3 a^{5} \left (2 a^{2} d^{2}-7 a b c d +7 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) b^{5}}-\frac {3 c^{5} \left (7 a^{2} d^{2}-7 a b c d +2 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(874\) |
| risch | \(\text {Expression too large to display}\) | \(1007\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1407\) |
-1/b^4/d^4*(-1/2*b*d*x^2+3*a*d*x+3*b*c*x)-1/d^5*c^6*(7*a*d-4*b*c)/(a*d-b*c )^4/(d*x+c)+1/2/d^5*c^7/(a*d-b*c)^3/(d*x+c)^2-3/d^5*c^5*(7*a^2*d^2-7*a*b*c *d+2*b^2*c^2)/(a*d-b*c)^5*ln(d*x+c)-1/2/b^5*a^7/(a*d-b*c)^3/(b*x+a)^2+3/b^ 5*a^5*(2*a^2*d^2-7*a*b*c*d+7*b^2*c^2)/(a*d-b*c)^5*ln(b*x+a)+1/b^5*a^6*(4*a *d-7*b*c)/(a*d-b*c)^4/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1567 vs. \(2 (239) = 478\).
Time = 0.32 (sec) , antiderivative size = 1567, normalized size of antiderivative = 6.40 \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]
1/2*(7*a^2*b^7*c^9 - 20*a^3*b^6*c^8*d + 13*a^4*b^5*c^7*d^2 - 13*a^7*b^2*c^ 4*d^5 + 20*a^8*b*c^3*d^6 - 7*a^9*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^6 - 4*(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^5 - (11*b^9*c^7*d^2 - 35*a*b^8*c^ 6*d^3 + 21*a^2*b^7*c^5*d^4 + 35*a^3*b^6*c^4*d^5 - 35*a^4*b^5*c^3*d^6 - 21* a^5*b^4*c^2*d^7 + 35*a^6*b^3*c*d^8 - 11*a^7*b^2*d^9)*x^4 + 2*(b^9*c^8*d - 10*a*b^8*c^7*d^2 + 33*a^2*b^7*c^6*d^3 - 43*a^3*b^6*c^5*d^4 + 43*a^5*b^4*c^ 3*d^6 - 33*a^6*b^3*c^2*d^7 + 10*a^7*b^2*c*d^8 - a^8*b*d^9)*x^3 + (7*b^9*c^ 9 - 16*a*b^8*c^8*d + 6*a^2*b^7*c^7*d^2 + 11*a^3*b^6*c^6*d^3 - 50*a^4*b^5*c ^5*d^4 + 50*a^5*b^4*c^4*d^5 - 11*a^6*b^3*c^3*d^6 - 6*a^7*b^2*c^2*d^7 + 16* a^8*b*c*d^8 - 7*a^9*d^9)*x^2 + 2*(7*a*b^8*c^9 - 19*a^2*b^7*c^8*d + 14*a^3* b^6*c^7*d^2 - 8*a^4*b^5*c^6*d^3 + 8*a^6*b^3*c^4*d^5 - 14*a^7*b^2*c^3*d^6 + 19*a^8*b*c^2*d^7 - 7*a^9*c*d^8)*x - 6*(7*a^7*b^2*c^4*d^5 - 7*a^8*b*c^3*d^ 6 + 2*a^9*c^2*d^7 + (7*a^5*b^4*c^2*d^7 - 7*a^6*b^3*c*d^8 + 2*a^7*b^2*d^9)* x^4 + 2*(7*a^5*b^4*c^3*d^6 - 5*a^7*b^2*c*d^8 + 2*a^8*b*d^9)*x^3 + (7*a^5*b ^4*c^4*d^5 + 21*a^6*b^3*c^3*d^6 - 19*a^7*b^2*c^2*d^7 + a^8*b*c*d^8 + 2*a^9 *d^9)*x^2 + 2*(7*a^6*b^3*c^4*d^5 - 5*a^8*b*c^2*d^7 + 2*a^9*c*d^8)*x)*log(b *x + a) + 6*(2*a^2*b^7*c^9 - 7*a^3*b^6*c^8*d + 7*a^4*b^5*c^7*d^2 + (2*b^9* c^7*d^2 - 7*a*b^8*c^6*d^3 + 7*a^2*b^7*c^5*d^4)*x^4 + 2*(2*b^9*c^8*d - 5...
Timed out. \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (239) = 478\).
Time = 0.24 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.43 \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {3 \, {\left (7 \, a^{5} b^{2} c^{2} - 7 \, a^{6} b c d + 2 \, a^{7} d^{2}\right )} \log \left (b x + a\right )}{b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}} + \frac {3 \, {\left (2 \, b^{2} c^{7} - 7 \, a b c^{6} d + 7 \, a^{2} c^{5} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}} + \frac {7 \, a^{2} b^{6} c^{8} - 13 \, a^{3} b^{5} c^{7} d - 13 \, a^{7} b c^{3} d^{5} + 7 \, a^{8} c^{2} d^{6} + 2 \, {\left (4 \, b^{8} c^{7} d - 7 \, a b^{7} c^{6} d^{2} - 7 \, a^{6} b^{2} c d^{7} + 4 \, a^{7} b d^{8}\right )} x^{3} + {\left (7 \, b^{8} c^{8} + 3 \, a b^{7} c^{7} d - 28 \, a^{2} b^{6} c^{6} d^{2} - 28 \, a^{6} b^{2} c^{2} d^{6} + 3 \, a^{7} b c d^{7} + 7 \, a^{8} d^{8}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} c^{8} - 9 \, a^{2} b^{6} c^{7} d - 7 \, a^{3} b^{5} c^{6} d^{2} - 7 \, a^{6} b^{2} c^{3} d^{5} - 9 \, a^{7} b c^{2} d^{6} + 7 \, a^{8} c d^{7}\right )} x}{2 \, {\left (a^{2} b^{9} c^{6} d^{5} - 4 \, a^{3} b^{8} c^{5} d^{6} + 6 \, a^{4} b^{7} c^{4} d^{7} - 4 \, a^{5} b^{6} c^{3} d^{8} + a^{6} b^{5} c^{2} d^{9} + {\left (b^{11} c^{4} d^{7} - 4 \, a b^{10} c^{3} d^{8} + 6 \, a^{2} b^{9} c^{2} d^{9} - 4 \, a^{3} b^{8} c d^{10} + a^{4} b^{7} d^{11}\right )} x^{4} + 2 \, {\left (b^{11} c^{5} d^{6} - 3 \, a b^{10} c^{4} d^{7} + 2 \, a^{2} b^{9} c^{3} d^{8} + 2 \, a^{3} b^{8} c^{2} d^{9} - 3 \, a^{4} b^{7} c d^{10} + a^{5} b^{6} d^{11}\right )} x^{3} + {\left (b^{11} c^{6} d^{5} - 9 \, a^{2} b^{9} c^{4} d^{7} + 16 \, a^{3} b^{8} c^{3} d^{8} - 9 \, a^{4} b^{7} c^{2} d^{9} + a^{6} b^{5} d^{11}\right )} x^{2} + 2 \, {\left (a b^{10} c^{6} d^{5} - 3 \, a^{2} b^{9} c^{5} d^{6} + 2 \, a^{3} b^{8} c^{4} d^{7} + 2 \, a^{4} b^{7} c^{3} d^{8} - 3 \, a^{5} b^{6} c^{2} d^{9} + a^{6} b^{5} c d^{10}\right )} x\right )}} + \frac {b d x^{2} - 6 \, {\left (b c + a d\right )} x}{2 \, b^{4} d^{4}} \]
-3*(7*a^5*b^2*c^2 - 7*a^6*b*c*d + 2*a^7*d^2)*log(b*x + a)/(b^10*c^5 - 5*a* b^9*c^4*d + 10*a^2*b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^ 5*b^5*d^5) + 3*(2*b^2*c^7 - 7*a*b*c^6*d + 7*a^2*c^5*d^2)*log(d*x + c)/(b^5 *c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d^8 + 5*a ^4*b*c*d^9 - a^5*d^10) + 1/2*(7*a^2*b^6*c^8 - 13*a^3*b^5*c^7*d - 13*a^7*b* c^3*d^5 + 7*a^8*c^2*d^6 + 2*(4*b^8*c^7*d - 7*a*b^7*c^6*d^2 - 7*a^6*b^2*c*d ^7 + 4*a^7*b*d^8)*x^3 + (7*b^8*c^8 + 3*a*b^7*c^7*d - 28*a^2*b^6*c^6*d^2 - 28*a^6*b^2*c^2*d^6 + 3*a^7*b*c*d^7 + 7*a^8*d^8)*x^2 + 2*(7*a*b^7*c^8 - 9*a ^2*b^6*c^7*d - 7*a^3*b^5*c^6*d^2 - 7*a^6*b^2*c^3*d^5 - 9*a^7*b*c^2*d^6 + 7 *a^8*c*d^7)*x)/(a^2*b^9*c^6*d^5 - 4*a^3*b^8*c^5*d^6 + 6*a^4*b^7*c^4*d^7 - 4*a^5*b^6*c^3*d^8 + a^6*b^5*c^2*d^9 + (b^11*c^4*d^7 - 4*a*b^10*c^3*d^8 + 6 *a^2*b^9*c^2*d^9 - 4*a^3*b^8*c*d^10 + a^4*b^7*d^11)*x^4 + 2*(b^11*c^5*d^6 - 3*a*b^10*c^4*d^7 + 2*a^2*b^9*c^3*d^8 + 2*a^3*b^8*c^2*d^9 - 3*a^4*b^7*c*d ^10 + a^5*b^6*d^11)*x^3 + (b^11*c^6*d^5 - 9*a^2*b^9*c^4*d^7 + 16*a^3*b^8*c ^3*d^8 - 9*a^4*b^7*c^2*d^9 + a^6*b^5*d^11)*x^2 + 2*(a*b^10*c^6*d^5 - 3*a^2 *b^9*c^5*d^6 + 2*a^3*b^8*c^4*d^7 + 2*a^4*b^7*c^3*d^8 - 3*a^5*b^6*c^2*d^9 + a^6*b^5*c*d^10)*x) + 1/2*(b*d*x^2 - 6*(b*c + a*d)*x)/(b^4*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (239) = 478\).
Time = 0.33 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.15 \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {3 \, {\left (7 \, a^{5} b^{2} c^{2} - 7 \, a^{6} b c d + 2 \, a^{7} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}} + \frac {3 \, {\left (2 \, b^{2} c^{7} - 7 \, a b c^{6} d + 7 \, a^{2} c^{5} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}} + \frac {b^{3} d^{3} x^{2} - 6 \, b^{3} c d^{2} x - 6 \, a b^{2} d^{3} x}{2 \, b^{6} d^{6}} + \frac {7 \, a^{2} b^{6} c^{8} - 13 \, a^{3} b^{5} c^{7} d - 13 \, a^{7} b c^{3} d^{5} + 7 \, a^{8} c^{2} d^{6} + 2 \, {\left (4 \, b^{8} c^{7} d - 7 \, a b^{7} c^{6} d^{2} - 7 \, a^{6} b^{2} c d^{7} + 4 \, a^{7} b d^{8}\right )} x^{3} + {\left (7 \, b^{8} c^{8} + 3 \, a b^{7} c^{7} d - 28 \, a^{2} b^{6} c^{6} d^{2} - 28 \, a^{6} b^{2} c^{2} d^{6} + 3 \, a^{7} b c d^{7} + 7 \, a^{8} d^{8}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} c^{8} - 9 \, a^{2} b^{6} c^{7} d - 7 \, a^{3} b^{5} c^{6} d^{2} - 7 \, a^{6} b^{2} c^{3} d^{5} - 9 \, a^{7} b c^{2} d^{6} + 7 \, a^{8} c d^{7}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} b^{5} d^{5}} \]
-3*(7*a^5*b^2*c^2 - 7*a^6*b*c*d + 2*a^7*d^2)*log(abs(b*x + a))/(b^10*c^5 - 5*a*b^9*c^4*d + 10*a^2*b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5) + 3*(2*b^2*c^7 - 7*a*b*c^6*d + 7*a^2*c^5*d^2)*log(abs(d*x + c))/(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2 *d^8 + 5*a^4*b*c*d^9 - a^5*d^10) + 1/2*(b^3*d^3*x^2 - 6*b^3*c*d^2*x - 6*a* b^2*d^3*x)/(b^6*d^6) + 1/2*(7*a^2*b^6*c^8 - 13*a^3*b^5*c^7*d - 13*a^7*b*c^ 3*d^5 + 7*a^8*c^2*d^6 + 2*(4*b^8*c^7*d - 7*a*b^7*c^6*d^2 - 7*a^6*b^2*c*d^7 + 4*a^7*b*d^8)*x^3 + (7*b^8*c^8 + 3*a*b^7*c^7*d - 28*a^2*b^6*c^6*d^2 - 28 *a^6*b^2*c^2*d^6 + 3*a^7*b*c*d^7 + 7*a^8*d^8)*x^2 + 2*(7*a*b^7*c^8 - 9*a^2 *b^6*c^7*d - 7*a^3*b^5*c^6*d^2 - 7*a^6*b^2*c^3*d^5 - 9*a^7*b*c^2*d^6 + 7*a ^8*c*d^7)*x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*b^5*d^5)
Time = 1.13 (sec) , antiderivative size = 880, normalized size of antiderivative = 3.59 \[ \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {x^3\,\left (a\,d+b\,c\right )\,\left (4\,a^6\,d^6-11\,a^5\,b\,c\,d^5+11\,a^4\,b^2\,c^2\,d^4-11\,a^3\,b^3\,c^3\,d^3+11\,a^2\,b^4\,c^4\,d^2-11\,a\,b^5\,c^5\,d+4\,b^6\,c^6\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {7\,a^8\,c^2\,d^6-13\,a^7\,b\,c^3\,d^5-13\,a^3\,b^5\,c^7\,d+7\,a^2\,b^6\,c^8}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x^2\,\left (7\,a^8\,d^8+3\,a^7\,b\,c\,d^7-28\,a^6\,b^2\,c^2\,d^6-28\,a^2\,b^6\,c^6\,d^2+3\,a\,b^7\,c^7\,d+7\,b^8\,c^8\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (7\,a^7\,c\,d^6-16\,a^6\,b\,c^2\,d^5+9\,a^5\,b^2\,c^3\,d^4-9\,a^4\,b^3\,c^4\,d^3+9\,a^3\,b^4\,c^5\,d^2-16\,a^2\,b^5\,c^6\,d+7\,a\,b^6\,c^7\right )}{b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x^3\,\left (2\,c\,b^6\,d^5+2\,a\,b^5\,d^6\right )+x\,\left (2\,a^2\,b^4\,c\,d^5+2\,a\,b^5\,c^2\,d^4\right )+x^2\,\left (a^2\,b^4\,d^6+4\,a\,b^5\,c\,d^5+b^6\,c^2\,d^4\right )+b^6\,d^6\,x^4+a^2\,b^4\,c^2\,d^4}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a^7\,d^2-21\,a^6\,b\,c\,d+21\,a^5\,b^2\,c^2\right )}{-a^5\,b^5\,d^5+5\,a^4\,b^6\,c\,d^4-10\,a^3\,b^7\,c^2\,d^3+10\,a^2\,b^8\,c^3\,d^2-5\,a\,b^9\,c^4\,d+b^{10}\,c^5}+\frac {x^2}{2\,b^3\,d^3}-\frac {\ln \left (c+d\,x\right )\,\left (21\,a^2\,c^5\,d^2-21\,a\,b\,c^6\,d+6\,b^2\,c^7\right )}{a^5\,d^{10}-5\,a^4\,b\,c\,d^9+10\,a^3\,b^2\,c^2\,d^8-10\,a^2\,b^3\,c^3\,d^7+5\,a\,b^4\,c^4\,d^6-b^5\,c^5\,d^5}-\frac {3\,x\,\left (a\,d+b\,c\right )}{b^4\,d^4} \]
((x^3*(a*d + b*c)*(4*a^6*d^6 + 4*b^6*c^6 + 11*a^2*b^4*c^4*d^2 - 11*a^3*b^3 *c^3*d^3 + 11*a^4*b^2*c^2*d^4 - 11*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(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) + (7*a^2*b ^6*c^8 + 7*a^8*c^2*d^6 - 13*a^3*b^5*c^7*d - 13*a^7*b*c^3*d^5)/(2*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)) + (x^2 *(7*a^8*d^8 + 7*b^8*c^8 - 28*a^2*b^6*c^6*d^2 - 28*a^6*b^2*c^2*d^6 + 3*a*b^ 7*c^7*d + 3*a^7*b*c*d^7))/(2*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)) + (x*(a*d + b*c)*(7*a*b^6*c^7 + 7*a^7*c*d^ 6 - 16*a^2*b^5*c^6*d - 16*a^6*b*c^2*d^5 + 9*a^3*b^4*c^5*d^2 - 9*a^4*b^3*c^ 4*d^3 + 9*a^5*b^2*c^3*d^4))/(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)))/(x^3*(2*a*b^5*d^6 + 2*b^6*c*d^5) + x*(2*a *b^5*c^2*d^4 + 2*a^2*b^4*c*d^5) + x^2*(a^2*b^4*d^6 + b^6*c^2*d^4 + 4*a*b^5 *c*d^5) + b^6*d^6*x^4 + a^2*b^4*c^2*d^4) - (log(a + b*x)*(6*a^7*d^2 + 21*a ^5*b^2*c^2 - 21*a^6*b*c*d))/(b^10*c^5 - a^5*b^5*d^5 + 5*a^4*b^6*c*d^4 + 10 *a^2*b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 - 5*a*b^9*c^4*d) + x^2/(2*b^3*d^3) - (log(c + d*x)*(6*b^2*c^7 + 21*a^2*c^5*d^2 - 21*a*b*c^6*d))/(a^5*d^10 - b^ 5*c^5*d^5 + 5*a*b^4*c^4*d^6 - 10*a^2*b^3*c^3*d^7 + 10*a^3*b^2*c^2*d^8 - 5* a^4*b*c*d^9) - (3*x*(a*d + b*c))/(b^4*d^4)