Integrand size = 15, antiderivative size = 276 \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=-\frac {b^7}{2 (b c-a d)^8 (a+b x)^2}+\frac {8 b^7 d}{(b c-a d)^9 (a+b x)}+\frac {d^2}{7 (b c-a d)^3 (c+d x)^7}+\frac {b d^2}{2 (b c-a d)^4 (c+d x)^6}+\frac {6 b^2 d^2}{5 (b c-a d)^5 (c+d x)^5}+\frac {5 b^3 d^2}{2 (b c-a d)^6 (c+d x)^4}+\frac {5 b^4 d^2}{(b c-a d)^7 (c+d x)^3}+\frac {21 b^5 d^2}{2 (b c-a d)^8 (c+d x)^2}+\frac {28 b^6 d^2}{(b c-a d)^9 (c+d x)}+\frac {36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac {36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}} \]
-1/2*b^7/(-a*d+b*c)^8/(b*x+a)^2+8*b^7*d/(-a*d+b*c)^9/(b*x+a)+1/7*d^2/(-a*d +b*c)^3/(d*x+c)^7+1/2*b*d^2/(-a*d+b*c)^4/(d*x+c)^6+6/5*b^2*d^2/(-a*d+b*c)^ 5/(d*x+c)^5+5/2*b^3*d^2/(-a*d+b*c)^6/(d*x+c)^4+5*b^4*d^2/(-a*d+b*c)^7/(d*x +c)^3+21/2*b^5*d^2/(-a*d+b*c)^8/(d*x+c)^2+28*b^6*d^2/(-a*d+b*c)^9/(d*x+c)+ 36*b^7*d^2*ln(b*x+a)/(-a*d+b*c)^10-36*b^7*d^2*ln(d*x+c)/(-a*d+b*c)^10
Time = 0.11 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=\frac {-\frac {35 b^7 (b c-a d)^2}{(a+b x)^2}+\frac {560 b^7 d (b c-a d)}{a+b x}+\frac {10 d^2 (b c-a d)^7}{(c+d x)^7}+\frac {35 b d^2 (b c-a d)^6}{(c+d x)^6}+\frac {84 b^2 d^2 (b c-a d)^5}{(c+d x)^5}+\frac {175 b^3 d^2 (b c-a d)^4}{(c+d x)^4}+\frac {350 b^4 d^2 (b c-a d)^3}{(c+d x)^3}+\frac {735 b^5 d^2 (b c-a d)^2}{(c+d x)^2}+\frac {1960 b^6 d^2 (b c-a d)}{c+d x}+2520 b^7 d^2 \log (a+b x)-2520 b^7 d^2 \log (c+d x)}{70 (b c-a d)^{10}} \]
((-35*b^7*(b*c - a*d)^2)/(a + b*x)^2 + (560*b^7*d*(b*c - a*d))/(a + b*x) + (10*d^2*(b*c - a*d)^7)/(c + d*x)^7 + (35*b*d^2*(b*c - a*d)^6)/(c + d*x)^6 + (84*b^2*d^2*(b*c - a*d)^5)/(c + d*x)^5 + (175*b^3*d^2*(b*c - a*d)^4)/(c + d*x)^4 + (350*b^4*d^2*(b*c - a*d)^3)/(c + d*x)^3 + (735*b^5*d^2*(b*c - a*d)^2)/(c + d*x)^2 + (1960*b^6*d^2*(b*c - a*d))/(c + d*x) + 2520*b^7*d^2* Log[a + b*x] - 2520*b^7*d^2*Log[c + d*x])/(70*(b*c - a*d)^10)
Time = 0.58 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {54, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (\frac {36 b^8 d^2}{(a+b x) (b c-a d)^{10}}-\frac {8 b^8 d}{(a+b x)^2 (b c-a d)^9}+\frac {b^8}{(a+b x)^3 (b c-a d)^8}-\frac {36 b^7 d^3}{(c+d x) (b c-a d)^{10}}-\frac {28 b^6 d^3}{(c+d x)^2 (b c-a d)^9}-\frac {21 b^5 d^3}{(c+d x)^3 (b c-a d)^8}-\frac {15 b^4 d^3}{(c+d x)^4 (b c-a d)^7}-\frac {10 b^3 d^3}{(c+d x)^5 (b c-a d)^6}-\frac {6 b^2 d^3}{(c+d x)^6 (b c-a d)^5}-\frac {3 b d^3}{(c+d x)^7 (b c-a d)^4}-\frac {d^3}{(c+d x)^8 (b c-a d)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac {36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}}+\frac {8 b^7 d}{(a+b x) (b c-a d)^9}-\frac {b^7}{2 (a+b x)^2 (b c-a d)^8}+\frac {28 b^6 d^2}{(c+d x) (b c-a d)^9}+\frac {21 b^5 d^2}{2 (c+d x)^2 (b c-a d)^8}+\frac {5 b^4 d^2}{(c+d x)^3 (b c-a d)^7}+\frac {5 b^3 d^2}{2 (c+d x)^4 (b c-a d)^6}+\frac {6 b^2 d^2}{5 (c+d x)^5 (b c-a d)^5}+\frac {b d^2}{2 (c+d x)^6 (b c-a d)^4}+\frac {d^2}{7 (c+d x)^7 (b c-a d)^3}\) |
-1/2*b^7/((b*c - a*d)^8*(a + b*x)^2) + (8*b^7*d)/((b*c - a*d)^9*(a + b*x)) + d^2/(7*(b*c - a*d)^3*(c + d*x)^7) + (b*d^2)/(2*(b*c - a*d)^4*(c + d*x)^ 6) + (6*b^2*d^2)/(5*(b*c - a*d)^5*(c + d*x)^5) + (5*b^3*d^2)/(2*(b*c - a*d )^6*(c + d*x)^4) + (5*b^4*d^2)/((b*c - a*d)^7*(c + d*x)^3) + (21*b^5*d^2)/ (2*(b*c - a*d)^8*(c + d*x)^2) + (28*b^6*d^2)/((b*c - a*d)^9*(c + d*x)) + ( 36*b^7*d^2*Log[a + b*x])/(b*c - a*d)^10 - (36*b^7*d^2*Log[c + d*x])/(b*c - a*d)^10
3.14.74.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Time = 0.39 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {d^{2}}{7 \left (a d -b c \right )^{3} \left (d x +c \right )^{7}}-\frac {36 d^{2} b^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{10}}-\frac {28 d^{2} b^{6}}{\left (a d -b c \right )^{9} \left (d x +c \right )}+\frac {21 d^{2} b^{5}}{2 \left (a d -b c \right )^{8} \left (d x +c \right )^{2}}-\frac {5 d^{2} b^{4}}{\left (a d -b c \right )^{7} \left (d x +c \right )^{3}}+\frac {5 d^{2} b^{3}}{2 \left (a d -b c \right )^{6} \left (d x +c \right )^{4}}-\frac {6 d^{2} b^{2}}{5 \left (a d -b c \right )^{5} \left (d x +c \right )^{5}}+\frac {d^{2} b}{2 \left (a d -b c \right )^{4} \left (d x +c \right )^{6}}-\frac {b^{7}}{2 \left (a d -b c \right )^{8} \left (b x +a \right )^{2}}+\frac {36 d^{2} b^{7} \ln \left (b x +a \right )}{\left (a d -b c \right )^{10}}-\frac {8 b^{7} d}{\left (a d -b c \right )^{9} \left (b x +a \right )}\) | \(265\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1919\) |
| risch | \(\text {Expression too large to display}\) | \(1963\) |
| norman | \(\text {Expression too large to display}\) | \(2049\) |
-1/7*d^2/(a*d-b*c)^3/(d*x+c)^7-36*d^2/(a*d-b*c)^10*b^7*ln(d*x+c)-28*d^2/(a *d-b*c)^9*b^6/(d*x+c)+21/2*d^2/(a*d-b*c)^8*b^5/(d*x+c)^2-5*d^2/(a*d-b*c)^7 *b^4/(d*x+c)^3+5/2*d^2/(a*d-b*c)^6*b^3/(d*x+c)^4-6/5*d^2/(a*d-b*c)^5*b^2/( d*x+c)^5+1/2*d^2/(a*d-b*c)^4*b/(d*x+c)^6-1/2*b^7/(a*d-b*c)^8/(b*x+a)^2+36* d^2/(a*d-b*c)^10*b^7*ln(b*x+a)-8*b^7/(a*d-b*c)^9*d/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 3016 vs. \(2 (264) = 528\).
Time = 0.31 (sec) , antiderivative size = 3016, normalized size of antiderivative = 10.93 \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=\text {Too large to display} \]
-1/70*(35*b^9*c^9 - 630*a*b^8*c^8*d - 2754*a^2*b^7*c^7*d^2 + 5880*a^3*b^6* c^6*d^3 - 4410*a^4*b^5*c^5*d^4 + 2940*a^5*b^4*c^4*d^5 - 1470*a^6*b^3*c^3*d ^6 + 504*a^7*b^2*c^2*d^7 - 105*a^8*b*c*d^8 + 10*a^9*d^9 - 2520*(b^9*c*d^8 - a*b^8*d^9)*x^8 - 1260*(13*b^9*c^2*d^7 - 10*a*b^8*c*d^8 - 3*a^2*b^7*d^9)* x^7 - 420*(107*b^9*c^3*d^6 - 48*a*b^8*c^2*d^7 - 57*a^2*b^7*c*d^8 - 2*a^3*b ^6*d^9)*x^6 - 210*(319*b^9*c^4*d^5 + 8*a*b^8*c^3*d^6 - 300*a^2*b^7*c^2*d^7 - 28*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^5 - 42*(1377*b^9*c^5*d^4 + 1090*a*b^8 *c^4*d^5 - 2080*a^2*b^7*c^3*d^6 - 420*a^3*b^6*c^2*d^7 + 35*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 - 42*(669*b^9*c^6*d^3 + 1494*a*b^8*c^5*d^4 - 1555*a^2* b^7*c^4*d^5 - 700*a^3*b^6*c^3*d^6 + 105*a^4*b^5*c^2*d^7 - 14*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^3 - 6*(1089*b^9*c^7*d^2 + 6426*a*b^8*c^6*d^3 - 3591*a^2* b^7*c^5*d^4 - 4900*a^3*b^6*c^4*d^5 + 1225*a^4*b^5*c^3*d^6 - 294*a^5*b^4*c^ 2*d^7 + 49*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 - 3*(105*b^9*c^8*d + 3516*a* b^8*c^7*d^2 + 546*a^2*b^7*c^6*d^3 - 5880*a^3*b^6*c^5*d^4 + 2450*a^4*b^5*c^ 4*d^5 - 980*a^5*b^4*c^3*d^6 + 294*a^6*b^3*c^2*d^7 - 56*a^7*b^2*c*d^8 + 5*a ^8*b*d^9)*x - 2520*(b^9*d^9*x^9 + a^2*b^7*c^7*d^2 + (7*b^9*c*d^8 + 2*a*b^8 *d^9)*x^8 + (21*b^9*c^2*d^7 + 14*a*b^8*c*d^8 + a^2*b^7*d^9)*x^7 + 7*(5*b^9 *c^3*d^6 + 6*a*b^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 7*(5*b^9*c^4*d^5 + 10*a* b^8*c^3*d^6 + 3*a^2*b^7*c^2*d^7)*x^5 + 7*(3*b^9*c^5*d^4 + 10*a*b^8*c^4*d^5 + 5*a^2*b^7*c^3*d^6)*x^4 + 7*(b^9*c^6*d^3 + 6*a*b^8*c^5*d^4 + 5*a^2*b^...
Timed out. \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2399 vs. \(2 (264) = 528\).
Time = 0.42 (sec) , antiderivative size = 2399, normalized size of antiderivative = 8.69 \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=\text {Too large to display} \]
36*b^7*d^2*log(b*x + a)/(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6 *b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9 + a^10*d^10) - 36*b^7*d^2*log(d*x + c)/(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2 *b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5 *d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10 *a^9*b*c*d^9 + a^10*d^10) + 1/70*(2520*b^8*d^8*x^8 - 35*b^8*c^8 + 595*a*b^ 7*c^7*d + 3349*a^2*b^6*c^6*d^2 - 2531*a^3*b^5*c^5*d^3 + 1879*a^4*b^4*c^4*d ^4 - 1061*a^5*b^3*c^3*d^5 + 409*a^6*b^2*c^2*d^6 - 95*a^7*b*c*d^7 + 10*a^8* d^8 + 1260*(13*b^8*c*d^7 + 3*a*b^7*d^8)*x^7 + 420*(107*b^8*c^2*d^6 + 59*a* b^7*c*d^7 + 2*a^2*b^6*d^8)*x^6 + 210*(319*b^8*c^3*d^5 + 327*a*b^7*c^2*d^6 + 27*a^2*b^6*c*d^7 - a^3*b^5*d^8)*x^5 + 42*(1377*b^8*c^4*d^4 + 2467*a*b^7* c^3*d^5 + 387*a^2*b^6*c^2*d^6 - 33*a^3*b^5*c*d^7 + 2*a^4*b^4*d^8)*x^4 + 42 *(669*b^8*c^5*d^3 + 2163*a*b^7*c^4*d^4 + 608*a^2*b^6*c^3*d^5 - 92*a^3*b^5* c^2*d^6 + 13*a^4*b^4*c*d^7 - a^5*b^3*d^8)*x^3 + 6*(1089*b^8*c^6*d^2 + 7515 *a*b^7*c^5*d^3 + 3924*a^2*b^6*c^4*d^4 - 976*a^3*b^5*c^3*d^5 + 249*a^4*b^4* c^2*d^6 - 45*a^5*b^3*c*d^7 + 4*a^6*b^2*d^8)*x^2 + 3*(105*b^8*c^7*d + 3621* a*b^7*c^6*d^2 + 4167*a^2*b^6*c^5*d^3 - 1713*a^3*b^5*c^4*d^4 + 737*a^4*b^4* c^3*d^5 - 243*a^5*b^3*c^2*d^6 + 51*a^6*b^2*c*d^7 - 5*a^7*b*d^8)*x)/(a^2*b^ 9*c^16 - 9*a^3*b^8*c^15*d + 36*a^4*b^7*c^14*d^2 - 84*a^5*b^6*c^13*d^3 +...
Leaf count of result is larger than twice the leaf count of optimal. 1029 vs. \(2 (264) = 528\).
Time = 0.31 (sec) , antiderivative size = 1029, normalized size of antiderivative = 3.73 \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=\frac {36 \, b^{8} d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{11} c^{10} - 10 \, a b^{10} c^{9} d + 45 \, a^{2} b^{9} c^{8} d^{2} - 120 \, a^{3} b^{8} c^{7} d^{3} + 210 \, a^{4} b^{7} c^{6} d^{4} - 252 \, a^{5} b^{6} c^{5} d^{5} + 210 \, a^{6} b^{5} c^{4} d^{6} - 120 \, a^{7} b^{4} c^{3} d^{7} + 45 \, a^{8} b^{3} c^{2} d^{8} - 10 \, a^{9} b^{2} c d^{9} + a^{10} b d^{10}} - \frac {36 \, b^{7} d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{10} c^{10} d - 10 \, a b^{9} c^{9} d^{2} + 45 \, a^{2} b^{8} c^{8} d^{3} - 120 \, a^{3} b^{7} c^{7} d^{4} + 210 \, a^{4} b^{6} c^{6} d^{5} - 252 \, a^{5} b^{5} c^{5} d^{6} + 210 \, a^{6} b^{4} c^{4} d^{7} - 120 \, a^{7} b^{3} c^{3} d^{8} + 45 \, a^{8} b^{2} c^{2} d^{9} - 10 \, a^{9} b c d^{10} + a^{10} d^{11}} - \frac {35 \, b^{9} c^{9} - 630 \, a b^{8} c^{8} d - 2754 \, a^{2} b^{7} c^{7} d^{2} + 5880 \, a^{3} b^{6} c^{6} d^{3} - 4410 \, a^{4} b^{5} c^{5} d^{4} + 2940 \, a^{5} b^{4} c^{4} d^{5} - 1470 \, a^{6} b^{3} c^{3} d^{6} + 504 \, a^{7} b^{2} c^{2} d^{7} - 105 \, a^{8} b c d^{8} + 10 \, a^{9} d^{9} - 2520 \, {\left (b^{9} c d^{8} - a b^{8} d^{9}\right )} x^{8} - 1260 \, {\left (13 \, b^{9} c^{2} d^{7} - 10 \, a b^{8} c d^{8} - 3 \, a^{2} b^{7} d^{9}\right )} x^{7} - 420 \, {\left (107 \, b^{9} c^{3} d^{6} - 48 \, a b^{8} c^{2} d^{7} - 57 \, a^{2} b^{7} c d^{8} - 2 \, a^{3} b^{6} d^{9}\right )} x^{6} - 210 \, {\left (319 \, b^{9} c^{4} d^{5} + 8 \, a b^{8} c^{3} d^{6} - 300 \, a^{2} b^{7} c^{2} d^{7} - 28 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )} x^{5} - 42 \, {\left (1377 \, b^{9} c^{5} d^{4} + 1090 \, a b^{8} c^{4} d^{5} - 2080 \, a^{2} b^{7} c^{3} d^{6} - 420 \, a^{3} b^{6} c^{2} d^{7} + 35 \, a^{4} b^{5} c d^{8} - 2 \, a^{5} b^{4} d^{9}\right )} x^{4} - 42 \, {\left (669 \, b^{9} c^{6} d^{3} + 1494 \, a b^{8} c^{5} d^{4} - 1555 \, a^{2} b^{7} c^{4} d^{5} - 700 \, a^{3} b^{6} c^{3} d^{6} + 105 \, a^{4} b^{5} c^{2} d^{7} - 14 \, a^{5} b^{4} c d^{8} + a^{6} b^{3} d^{9}\right )} x^{3} - 6 \, {\left (1089 \, b^{9} c^{7} d^{2} + 6426 \, a b^{8} c^{6} d^{3} - 3591 \, a^{2} b^{7} c^{5} d^{4} - 4900 \, a^{3} b^{6} c^{4} d^{5} + 1225 \, a^{4} b^{5} c^{3} d^{6} - 294 \, a^{5} b^{4} c^{2} d^{7} + 49 \, a^{6} b^{3} c d^{8} - 4 \, a^{7} b^{2} d^{9}\right )} x^{2} - 3 \, {\left (105 \, b^{9} c^{8} d + 3516 \, a b^{8} c^{7} d^{2} + 546 \, a^{2} b^{7} c^{6} d^{3} - 5880 \, a^{3} b^{6} c^{5} d^{4} + 2450 \, a^{4} b^{5} c^{4} d^{5} - 980 \, a^{5} b^{4} c^{3} d^{6} + 294 \, a^{6} b^{3} c^{2} d^{7} - 56 \, a^{7} b^{2} c d^{8} + 5 \, a^{8} b d^{9}\right )} x}{70 \, {\left (b c - a d\right )}^{10} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{7}} \]
36*b^8*d^2*log(abs(b*x + a))/(b^11*c^10 - 10*a*b^10*c^9*d + 45*a^2*b^9*c^8 *d^2 - 120*a^3*b^8*c^7*d^3 + 210*a^4*b^7*c^6*d^4 - 252*a^5*b^6*c^5*d^5 + 2 10*a^6*b^5*c^4*d^6 - 120*a^7*b^4*c^3*d^7 + 45*a^8*b^3*c^2*d^8 - 10*a^9*b^2 *c*d^9 + a^10*b*d^10) - 36*b^7*d^3*log(abs(d*x + c))/(b^10*c^10*d - 10*a*b ^9*c^9*d^2 + 45*a^2*b^8*c^8*d^3 - 120*a^3*b^7*c^7*d^4 + 210*a^4*b^6*c^6*d^ 5 - 252*a^5*b^5*c^5*d^6 + 210*a^6*b^4*c^4*d^7 - 120*a^7*b^3*c^3*d^8 + 45*a ^8*b^2*c^2*d^9 - 10*a^9*b*c*d^10 + a^10*d^11) - 1/70*(35*b^9*c^9 - 630*a*b ^8*c^8*d - 2754*a^2*b^7*c^7*d^2 + 5880*a^3*b^6*c^6*d^3 - 4410*a^4*b^5*c^5* d^4 + 2940*a^5*b^4*c^4*d^5 - 1470*a^6*b^3*c^3*d^6 + 504*a^7*b^2*c^2*d^7 - 105*a^8*b*c*d^8 + 10*a^9*d^9 - 2520*(b^9*c*d^8 - a*b^8*d^9)*x^8 - 1260*(13 *b^9*c^2*d^7 - 10*a*b^8*c*d^8 - 3*a^2*b^7*d^9)*x^7 - 420*(107*b^9*c^3*d^6 - 48*a*b^8*c^2*d^7 - 57*a^2*b^7*c*d^8 - 2*a^3*b^6*d^9)*x^6 - 210*(319*b^9* c^4*d^5 + 8*a*b^8*c^3*d^6 - 300*a^2*b^7*c^2*d^7 - 28*a^3*b^6*c*d^8 + a^4*b ^5*d^9)*x^5 - 42*(1377*b^9*c^5*d^4 + 1090*a*b^8*c^4*d^5 - 2080*a^2*b^7*c^3 *d^6 - 420*a^3*b^6*c^2*d^7 + 35*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 - 42*(6 69*b^9*c^6*d^3 + 1494*a*b^8*c^5*d^4 - 1555*a^2*b^7*c^4*d^5 - 700*a^3*b^6*c ^3*d^6 + 105*a^4*b^5*c^2*d^7 - 14*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^3 - 6*(10 89*b^9*c^7*d^2 + 6426*a*b^8*c^6*d^3 - 3591*a^2*b^7*c^5*d^4 - 4900*a^3*b^6* c^4*d^5 + 1225*a^4*b^5*c^3*d^6 - 294*a^5*b^4*c^2*d^7 + 49*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 - 3*(105*b^9*c^8*d + 3516*a*b^8*c^7*d^2 + 546*a^2*b^...
Time = 2.33 (sec) , antiderivative size = 2224, normalized size of antiderivative = 8.06 \[ \int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx=\text {Too large to display} \]
(72*b^7*d^2*atanh((a^10*d^10 - b^10*c^10 - 27*a^2*b^8*c^8*d^2 + 48*a^3*b^7 *c^7*d^3 - 42*a^4*b^6*c^6*d^4 + 42*a^6*b^4*c^4*d^6 - 48*a^7*b^3*c^3*d^7 + 27*a^8*b^2*c^2*d^8 + 8*a*b^9*c^9*d - 8*a^9*b*c*d^9)/(a*d - b*c)^10 + (2*b* d*x*(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4 *b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d ^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8))/(a*d - b*c)^10))/(a*d - b*c)^10 - ((1 0*a^8*d^8 - 35*b^8*c^8 + 3349*a^2*b^6*c^6*d^2 - 2531*a^3*b^5*c^5*d^3 + 187 9*a^4*b^4*c^4*d^4 - 1061*a^5*b^3*c^3*d^5 + 409*a^6*b^2*c^2*d^6 + 595*a*b^7 *c^7*d - 95*a^7*b*c*d^7)/(70*(a^9*d^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84* a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c ^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8*d - 9*a^8*b*c*d^8)) + (3*b^2*x^2 *(4*a^6*d^8 + 1089*b^6*c^6*d^2 + 7515*a*b^5*c^5*d^3 + 3924*a^2*b^4*c^4*d^4 - 976*a^3*b^3*c^3*d^5 + 249*a^4*b^2*c^2*d^6 - 45*a^5*b*c*d^7))/(35*(a^9*d ^9 - b^9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d ^4 + 126*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b ^8*c^8*d - 9*a^8*b*c*d^8)) + (3*b^4*x^4*(2*a^4*d^8 + 1377*b^4*c^4*d^4 + 24 67*a*b^3*c^3*d^5 + 387*a^2*b^2*c^2*d^6 - 33*a^3*b*c*d^7))/(5*(a^9*d^9 - b^ 9*c^9 - 36*a^2*b^7*c^7*d^2 + 84*a^3*b^6*c^6*d^3 - 126*a^4*b^5*c^5*d^4 + 12 6*a^5*b^4*c^4*d^5 - 84*a^6*b^3*c^3*d^6 + 36*a^7*b^2*c^2*d^7 + 9*a*b^8*c^8* d - 9*a^8*b*c*d^8)) + (3*b*x*(105*b^7*c^7*d - 5*a^7*d^8 + 3621*a*b^6*c^...