Integrand size = 15, antiderivative size = 209 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^8 x}{d^8}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}-\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9} \]
b^8*x/d^8-1/7*(-a*d+b*c)^8/d^9/(d*x+c)^7+4/3*b*(-a*d+b*c)^7/d^9/(d*x+c)^6- 28/5*b^2*(-a*d+b*c)^6/d^9/(d*x+c)^5+14*b^3*(-a*d+b*c)^5/d^9/(d*x+c)^4-70/3 *b^4*(-a*d+b*c)^4/d^9/(d*x+c)^3+28*b^5*(-a*d+b*c)^3/d^9/(d*x+c)^2-28*b^6*( -a*d+b*c)^2/d^9/(d*x+c)-8*b^7*(-a*d+b*c)*ln(d*x+c)/d^9
Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(209)=418\).
Time = 0.12 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.27 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=-\frac {15 a^8 d^8+20 a^7 b d^7 (c+7 d x)+28 a^6 b^2 d^6 \left (c^2+7 c d x+21 d^2 x^2\right )+42 a^5 b^3 d^5 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+70 a^4 b^4 d^4 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+140 a^3 b^5 d^3 \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+420 a^2 b^6 d^2 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )-2 a b^7 c d \left (1089 c^6+7203 c^5 d x+20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+13230 c d^5 x^5+2940 d^6 x^6\right )+b^8 \left (1443 c^8+9261 c^7 d x+24843 c^6 d^2 x^2+35525 c^5 d^3 x^3+28175 c^4 d^4 x^4+11025 c^3 d^5 x^5+735 c^2 d^6 x^6-735 c d^7 x^7-105 d^8 x^8\right )+840 b^7 (b c-a d) (c+d x)^7 \log (c+d x)}{105 d^9 (c+d x)^7} \]
-1/105*(15*a^8*d^8 + 20*a^7*b*d^7*(c + 7*d*x) + 28*a^6*b^2*d^6*(c^2 + 7*c* d*x + 21*d^2*x^2) + 42*a^5*b^3*d^5*(c^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^ 3*x^3) + 70*a^4*b^4*d^4*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3 + 35*d^4*x^4) + 140*a^3*b^5*d^3*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2* d^3*x^3 + 35*c*d^4*x^4 + 21*d^5*x^5) + 420*a^2*b^6*d^2*(c^6 + 7*c^5*d*x + 21*c^4*d^2*x^2 + 35*c^3*d^3*x^3 + 35*c^2*d^4*x^4 + 21*c*d^5*x^5 + 7*d^6*x^ 6) - 2*a*b^7*c*d*(1089*c^6 + 7203*c^5*d*x + 20139*c^4*d^2*x^2 + 30625*c^3* d^3*x^3 + 26950*c^2*d^4*x^4 + 13230*c*d^5*x^5 + 2940*d^6*x^6) + b^8*(1443* c^8 + 9261*c^7*d*x + 24843*c^6*d^2*x^2 + 35525*c^5*d^3*x^3 + 28175*c^4*d^4 *x^4 + 11025*c^3*d^5*x^5 + 735*c^2*d^6*x^6 - 735*c*d^7*x^7 - 105*d^8*x^8) + 840*b^7*(b*c - a*d)*(c + d*x)^7*Log[c + d*x])/(d^9*(c + d*x)^7)
Time = 0.48 (sec) , antiderivative size = 209, 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 = {49, 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.
| \(\displaystyle \int \frac {(a+b x)^8}{(c+d x)^8} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {8 b^7 (b c-a d)}{d^8 (c+d x)}+\frac {28 b^6 (b c-a d)^2}{d^8 (c+d x)^2}-\frac {56 b^5 (b c-a d)^3}{d^8 (c+d x)^3}+\frac {70 b^4 (b c-a d)^4}{d^8 (c+d x)^4}-\frac {56 b^3 (b c-a d)^5}{d^8 (c+d x)^5}+\frac {28 b^2 (b c-a d)^6}{d^8 (c+d x)^6}-\frac {8 b (b c-a d)^7}{d^8 (c+d x)^7}+\frac {(a d-b c)^8}{d^8 (c+d x)^8}+\frac {b^8}{d^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {b^8 x}{d^8}\) |
(b^8*x)/d^8 - (b*c - a*d)^8/(7*d^9*(c + d*x)^7) + (4*b*(b*c - a*d)^7)/(3*d ^9*(c + d*x)^6) - (28*b^2*(b*c - a*d)^6)/(5*d^9*(c + d*x)^5) + (14*b^3*(b* c - a*d)^5)/(d^9*(c + d*x)^4) - (70*b^4*(b*c - a*d)^4)/(3*d^9*(c + d*x)^3) + (28*b^5*(b*c - a*d)^3)/(d^9*(c + d*x)^2) - (28*b^6*(b*c - a*d)^2)/(d^9* (c + d*x)) - (8*b^7*(b*c - a*d)*Log[c + d*x])/d^9
3.14.63.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(569\) vs. \(2(201)=402\).
Time = 0.23 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.73
| method | result | size |
| risch | \(\frac {b^{8} x}{d^{8}}+\frac {\left (-28 a^{2} b^{6} d^{7}+56 a \,b^{7} c \,d^{6}-28 b^{8} c^{2} d^{5}\right ) x^{6}-28 b^{5} d^{4} \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) x^{5}-\frac {70 b^{4} d^{3} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-22 a \,b^{3} c^{3} d +13 b^{4} c^{4}\right ) x^{4}}{3}-\frac {14 b^{3} d^{2} \left (3 a^{5} d^{5}+5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+30 a^{2} b^{3} c^{3} d^{2}-125 a \,b^{4} c^{4} d +77 b^{5} c^{5}\right ) x^{3}}{3}-\frac {14 b^{2} d \left (2 a^{6} d^{6}+3 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+30 a^{2} b^{4} c^{4} d^{2}-137 a \,b^{5} c^{5} d +87 b^{6} c^{6}\right ) x^{2}}{5}-\frac {2 b \left (10 a^{7} d^{7}+14 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}-1029 a \,b^{6} c^{6} d +669 b^{7} c^{7}\right ) x}{15}-\frac {15 a^{8} d^{8}+20 a^{7} b c \,d^{7}+28 a^{6} b^{2} d^{6} c^{2}+42 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} d^{4} c^{4}+140 a^{3} b^{5} c^{5} d^{3}+420 a^{2} b^{6} d^{2} c^{6}-2178 a \,b^{7} c^{7} d +1443 b^{8} c^{8}}{105 d}}{d^{8} \left (d x +c \right )^{7}}+\frac {8 b^{7} \ln \left (d x +c \right ) a}{d^{8}}-\frac {8 b^{8} \ln \left (d x +c \right ) c}{d^{9}}\) | \(570\) |
| default | \(\frac {b^{8} x}{d^{8}}-\frac {14 b^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{9} \left (d x +c \right )^{4}}-\frac {28 b^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{9} \left (d x +c \right )^{2}}+\frac {8 b^{7} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{9}}-\frac {4 b \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{3 d^{9} \left (d x +c \right )^{6}}-\frac {28 b^{2} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{5 d^{9} \left (d x +c \right )^{5}}-\frac {a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} d^{6} c^{2}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} d^{4} c^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} d^{2} c^{6}-8 a \,b^{7} c^{7} d +b^{8} c^{8}}{7 d^{9} \left (d x +c \right )^{7}}-\frac {70 b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 d^{9} \left (d x +c \right )^{3}}-\frac {28 b^{6} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{9} \left (d x +c \right )}\) | \(576\) |
| norman | \(\frac {\frac {b^{8} x^{8}}{d}-\frac {15 a^{8} d^{8}+20 a^{7} b c \,d^{7}+28 a^{6} b^{2} d^{6} c^{2}+42 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} d^{4} c^{4}+140 a^{3} b^{5} c^{5} d^{3}+420 a^{2} b^{6} d^{2} c^{6}-2178 a \,b^{7} c^{7} d +2178 b^{8} c^{8}}{105 d^{9}}-\frac {7 \left (4 a^{2} b^{6} d^{2}-8 a \,b^{7} c d +8 b^{8} c^{2}\right ) x^{6}}{d^{3}}-\frac {7 \left (4 a^{3} b^{5} d^{3}+12 a^{2} b^{6} c \,d^{2}-36 a \,b^{7} c^{2} d +36 b^{8} c^{3}\right ) x^{5}}{d^{4}}-\frac {35 \left (2 a^{4} b^{4} d^{4}+4 a^{3} b^{5} c \,d^{3}+12 a^{2} b^{6} c^{2} d^{2}-44 a \,b^{7} c^{3} d +44 b^{8} c^{4}\right ) x^{4}}{3 d^{5}}-\frac {7 \left (6 a^{5} b^{3} d^{5}+10 a^{4} b^{4} c \,d^{4}+20 a^{3} b^{5} c^{2} d^{3}+60 a^{2} b^{6} c^{3} d^{2}-250 a \,b^{7} c^{4} d +250 b^{8} c^{5}\right ) x^{3}}{3 d^{6}}-\frac {7 \left (4 a^{6} b^{2} d^{6}+6 a^{5} b^{3} c \,d^{5}+10 a^{4} b^{4} c^{2} d^{4}+20 a^{3} b^{5} c^{3} d^{3}+60 a^{2} b^{6} c^{4} d^{2}-274 a \,b^{7} c^{5} d +274 b^{8} c^{6}\right ) x^{2}}{5 d^{7}}-\frac {\left (20 a^{7} b \,d^{7}+28 a^{6} b^{2} c \,d^{6}+42 a^{5} b^{3} c^{2} d^{5}+70 a^{4} b^{4} c^{3} d^{4}+140 a^{3} b^{5} c^{4} d^{3}+420 a^{2} b^{6} c^{5} d^{2}-2058 a \,b^{7} c^{6} d +2058 b^{8} c^{7}\right ) x}{15 d^{8}}}{\left (d x +c \right )^{7}}+\frac {8 b^{7} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{9}}\) | \(577\) |
| parallelrisch | \(\frac {-8820 x^{2} a^{2} b^{6} c^{4} d^{4}-14700 x^{4} a^{2} b^{6} c^{2} d^{6}+53900 x^{4} a \,b^{7} c^{3} d^{5}-8820 x^{5} a^{2} b^{6} c \,d^{7}+26460 x^{5} a \,b^{7} c^{2} d^{6}+5880 x^{6} a \,b^{7} c \,d^{7}+840 \ln \left (d x +c \right ) x^{7} a \,b^{7} d^{8}-840 \ln \left (d x +c \right ) x^{7} b^{8} c \,d^{7}-5880 \ln \left (d x +c \right ) x^{6} b^{8} c^{2} d^{6}-17640 \ln \left (d x +c \right ) x^{5} b^{8} c^{3} d^{5}+40278 x^{2} a \,b^{7} c^{5} d^{3}-2450 x^{3} a^{4} b^{4} c \,d^{7}-4900 x^{3} a^{3} b^{5} c^{2} d^{6}-14700 x^{3} a^{2} b^{6} c^{3} d^{5}+61250 x^{3} a \,b^{7} c^{4} d^{4}-4900 x^{4} a^{3} b^{5} c \,d^{7}-196 x \,a^{6} b^{2} c \,d^{7}-294 x \,a^{5} b^{3} c^{2} d^{6}-490 x \,a^{4} b^{4} c^{3} d^{5}-29400 \ln \left (d x +c \right ) x^{4} b^{8} c^{4} d^{4}-29400 \ln \left (d x +c \right ) x^{3} b^{8} c^{5} d^{3}-17640 \ln \left (d x +c \right ) x^{2} b^{8} c^{6} d^{2}-5880 \ln \left (d x +c \right ) x \,b^{8} c^{7} d +840 \ln \left (d x +c \right ) a \,b^{7} c^{7} d -980 x \,a^{3} b^{5} c^{4} d^{4}-2940 x \,a^{2} b^{6} c^{5} d^{3}+14406 x a \,b^{7} c^{6} d^{2}-882 x^{2} a^{5} b^{3} c \,d^{7}-1470 x^{2} a^{4} b^{4} c^{2} d^{6}-2940 x^{2} a^{3} b^{5} c^{3} d^{5}+29400 \ln \left (d x +c \right ) x^{3} a \,b^{7} c^{4} d^{4}+17640 \ln \left (d x +c \right ) x^{2} a \,b^{7} c^{5} d^{3}+5880 \ln \left (d x +c \right ) x a \,b^{7} c^{6} d^{2}+5880 \ln \left (d x +c \right ) x^{6} a \,b^{7} c \,d^{7}+17640 \ln \left (d x +c \right ) x^{5} a \,b^{7} c^{2} d^{6}+29400 \ln \left (d x +c \right ) x^{4} a \,b^{7} c^{3} d^{5}-14406 b^{8} c^{7} d x -140 a^{7} b \,d^{8} x -40278 b^{8} c^{6} d^{2} x^{2}-588 a^{6} b^{2} d^{8} x^{2}-61250 b^{8} c^{5} d^{3} x^{3}-1470 a^{5} b^{3} d^{8} x^{3}-53900 b^{8} c^{4} d^{4} x^{4}-2450 a^{4} b^{4} d^{8} x^{4}-26460 b^{8} c^{3} d^{5} x^{5}-2940 a^{3} b^{5} d^{8} x^{5}-5880 b^{8} c^{2} d^{6} x^{6}-2940 a^{2} b^{6} d^{8} x^{6}+105 x^{8} b^{8} d^{8}-420 a^{2} b^{6} d^{2} c^{6}-28 a^{6} b^{2} d^{6} c^{2}-70 a^{4} b^{4} d^{4} c^{4}-140 a^{3} b^{5} c^{5} d^{3}-840 \ln \left (d x +c \right ) b^{8} c^{8}-42 a^{5} b^{3} c^{3} d^{5}-15 a^{8} d^{8}-2178 b^{8} c^{8}-20 a^{7} b c \,d^{7}+2178 a \,b^{7} c^{7} d}{105 d^{9} \left (d x +c \right )^{7}}\) | \(916\) |
b^8*x/d^8+((-28*a^2*b^6*d^7+56*a*b^7*c*d^6-28*b^8*c^2*d^5)*x^6-28*b^5*d^4* (a^3*d^3+3*a^2*b*c*d^2-9*a*b^2*c^2*d+5*b^3*c^3)*x^5-70/3*b^4*d^3*(a^4*d^4+ 2*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-22*a*b^3*c^3*d+13*b^4*c^4)*x^4-14/3*b^3*d^ 2*(3*a^5*d^5+5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3+30*a^2*b^3*c^3*d^2-125*a*b^4 *c^4*d+77*b^5*c^5)*x^3-14/5*b^2*d*(2*a^6*d^6+3*a^5*b*c*d^5+5*a^4*b^2*c^2*d ^4+10*a^3*b^3*c^3*d^3+30*a^2*b^4*c^4*d^2-137*a*b^5*c^5*d+87*b^6*c^6)*x^2-2 /15*b*(10*a^7*d^7+14*a^6*b*c*d^6+21*a^5*b^2*c^2*d^5+35*a^4*b^3*c^3*d^4+70* a^3*b^4*c^4*d^3+210*a^2*b^5*c^5*d^2-1029*a*b^6*c^6*d+669*b^7*c^7)*x-1/105/ d*(15*a^8*d^8+20*a^7*b*c*d^7+28*a^6*b^2*c^2*d^6+42*a^5*b^3*c^3*d^5+70*a^4* b^4*c^4*d^4+140*a^3*b^5*c^5*d^3+420*a^2*b^6*c^6*d^2-2178*a*b^7*c^7*d+1443* b^8*c^8))/d^8/(d*x+c)^7+8*b^7/d^8*ln(d*x+c)*a-8*b^8/d^9*ln(d*x+c)*c
Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (201) = 402\).
Time = 0.22 (sec) , antiderivative size = 852, normalized size of antiderivative = 4.08 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx =\text {Too large to display} \]
1/105*(105*b^8*d^8*x^8 + 735*b^8*c*d^7*x^7 - 1443*b^8*c^8 + 2178*a*b^7*c^7 *d - 420*a^2*b^6*c^6*d^2 - 140*a^3*b^5*c^5*d^3 - 70*a^4*b^4*c^4*d^4 - 42*a ^5*b^3*c^3*d^5 - 28*a^6*b^2*c^2*d^6 - 20*a^7*b*c*d^7 - 15*a^8*d^8 - 735*(b ^8*c^2*d^6 - 8*a*b^7*c*d^7 + 4*a^2*b^6*d^8)*x^6 - 735*(15*b^8*c^3*d^5 - 36 *a*b^7*c^2*d^6 + 12*a^2*b^6*c*d^7 + 4*a^3*b^5*d^8)*x^5 - 1225*(23*b^8*c^4* d^4 - 44*a*b^7*c^3*d^5 + 12*a^2*b^6*c^2*d^6 + 4*a^3*b^5*c*d^7 + 2*a^4*b^4* d^8)*x^4 - 245*(145*b^8*c^5*d^3 - 250*a*b^7*c^4*d^4 + 60*a^2*b^6*c^3*d^5 + 20*a^3*b^5*c^2*d^6 + 10*a^4*b^4*c*d^7 + 6*a^5*b^3*d^8)*x^3 - 147*(169*b^8 *c^6*d^2 - 274*a*b^7*c^5*d^3 + 60*a^2*b^6*c^4*d^4 + 20*a^3*b^5*c^3*d^5 + 1 0*a^4*b^4*c^2*d^6 + 6*a^5*b^3*c*d^7 + 4*a^6*b^2*d^8)*x^2 - 7*(1323*b^8*c^7 *d - 2058*a*b^7*c^6*d^2 + 420*a^2*b^6*c^5*d^3 + 140*a^3*b^5*c^4*d^4 + 70*a ^4*b^4*c^3*d^5 + 42*a^5*b^3*c^2*d^6 + 28*a^6*b^2*c*d^7 + 20*a^7*b*d^8)*x - 840*(b^8*c^8 - a*b^7*c^7*d + (b^8*c*d^7 - a*b^7*d^8)*x^7 + 7*(b^8*c^2*d^6 - a*b^7*c*d^7)*x^6 + 21*(b^8*c^3*d^5 - a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d ^4 - a*b^7*c^3*d^5)*x^4 + 35*(b^8*c^5*d^3 - a*b^7*c^4*d^4)*x^3 + 21*(b^8*c ^6*d^2 - a*b^7*c^5*d^3)*x^2 + 7*(b^8*c^7*d - a*b^7*c^6*d^2)*x)*log(d*x + c ))/(d^16*x^7 + 7*c*d^15*x^6 + 21*c^2*d^14*x^5 + 35*c^3*d^13*x^4 + 35*c^4*d ^12*x^3 + 21*c^5*d^11*x^2 + 7*c^6*d^10*x + c^7*d^9)
Timed out. \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (201) = 402\).
Time = 0.24 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^{8} x}{d^{8}} - \frac {1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \, {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \, {\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \, {\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \, {\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \, {\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \, {\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \, {\left (d^{16} x^{7} + 7 \, c d^{15} x^{6} + 21 \, c^{2} d^{14} x^{5} + 35 \, c^{3} d^{13} x^{4} + 35 \, c^{4} d^{12} x^{3} + 21 \, c^{5} d^{11} x^{2} + 7 \, c^{6} d^{10} x + c^{7} d^{9}\right )}} - \frac {8 \, {\left (b^{8} c - a b^{7} d\right )} \log \left (d x + c\right )}{d^{9}} \]
b^8*x/d^8 - 1/105*(1443*b^8*c^8 - 2178*a*b^7*c^7*d + 420*a^2*b^6*c^6*d^2 + 140*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 + 42*a^5*b^3*c^3*d^5 + 28*a^6*b^ 2*c^2*d^6 + 20*a^7*b*c*d^7 + 15*a^8*d^8 + 2940*(b^8*c^2*d^6 - 2*a*b^7*c*d^ 7 + a^2*b^6*d^8)*x^6 + 2940*(5*b^8*c^3*d^5 - 9*a*b^7*c^2*d^6 + 3*a^2*b^6*c *d^7 + a^3*b^5*d^8)*x^5 + 2450*(13*b^8*c^4*d^4 - 22*a*b^7*c^3*d^5 + 6*a^2* b^6*c^2*d^6 + 2*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^4 + 490*(77*b^8*c^5*d^3 - 1 25*a*b^7*c^4*d^4 + 30*a^2*b^6*c^3*d^5 + 10*a^3*b^5*c^2*d^6 + 5*a^4*b^4*c*d ^7 + 3*a^5*b^3*d^8)*x^3 + 294*(87*b^8*c^6*d^2 - 137*a*b^7*c^5*d^3 + 30*a^2 *b^6*c^4*d^4 + 10*a^3*b^5*c^3*d^5 + 5*a^4*b^4*c^2*d^6 + 3*a^5*b^3*c*d^7 + 2*a^6*b^2*d^8)*x^2 + 14*(669*b^8*c^7*d - 1029*a*b^7*c^6*d^2 + 210*a^2*b^6* c^5*d^3 + 70*a^3*b^5*c^4*d^4 + 35*a^4*b^4*c^3*d^5 + 21*a^5*b^3*c^2*d^6 + 1 4*a^6*b^2*c*d^7 + 10*a^7*b*d^8)*x)/(d^16*x^7 + 7*c*d^15*x^6 + 21*c^2*d^14* x^5 + 35*c^3*d^13*x^4 + 35*c^4*d^12*x^3 + 21*c^5*d^11*x^2 + 7*c^6*d^10*x + c^7*d^9) - 8*(b^8*c - a*b^7*d)*log(d*x + c)/d^9
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (201) = 402\).
Time = 0.29 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^{8} x}{d^{8}} - \frac {8 \, {\left (b^{8} c - a b^{7} d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{9}} - \frac {1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \, {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \, {\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \, {\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \, {\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \, {\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \, {\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \, {\left (d x + c\right )}^{7} d^{9}} \]
b^8*x/d^8 - 8*(b^8*c - a*b^7*d)*log(abs(d*x + c))/d^9 - 1/105*(1443*b^8*c^ 8 - 2178*a*b^7*c^7*d + 420*a^2*b^6*c^6*d^2 + 140*a^3*b^5*c^5*d^3 + 70*a^4* b^4*c^4*d^4 + 42*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 + 20*a^7*b*c*d^7 + 1 5*a^8*d^8 + 2940*(b^8*c^2*d^6 - 2*a*b^7*c*d^7 + a^2*b^6*d^8)*x^6 + 2940*(5 *b^8*c^3*d^5 - 9*a*b^7*c^2*d^6 + 3*a^2*b^6*c*d^7 + a^3*b^5*d^8)*x^5 + 2450 *(13*b^8*c^4*d^4 - 22*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 + 2*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^4 + 490*(77*b^8*c^5*d^3 - 125*a*b^7*c^4*d^4 + 30*a^2*b^6* c^3*d^5 + 10*a^3*b^5*c^2*d^6 + 5*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 294* (87*b^8*c^6*d^2 - 137*a*b^7*c^5*d^3 + 30*a^2*b^6*c^4*d^4 + 10*a^3*b^5*c^3* d^5 + 5*a^4*b^4*c^2*d^6 + 3*a^5*b^3*c*d^7 + 2*a^6*b^2*d^8)*x^2 + 14*(669*b ^8*c^7*d - 1029*a*b^7*c^6*d^2 + 210*a^2*b^6*c^5*d^3 + 70*a^3*b^5*c^4*d^4 + 35*a^4*b^4*c^3*d^5 + 21*a^5*b^3*c^2*d^6 + 14*a^6*b^2*c*d^7 + 10*a^7*b*d^8 )*x)/((d*x + c)^7*d^9)
Time = 0.47 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^8\,x}{d^8}-\frac {\ln \left (c+d\,x\right )\,\left (8\,b^8\,c-8\,a\,b^7\,d\right )}{d^9}-\frac {x^4\,\left (\frac {70\,a^4\,b^4\,d^7}{3}+\frac {140\,a^3\,b^5\,c\,d^6}{3}+140\,a^2\,b^6\,c^2\,d^5-\frac {1540\,a\,b^7\,c^3\,d^4}{3}+\frac {910\,b^8\,c^4\,d^3}{3}\right )+x^6\,\left (28\,a^2\,b^6\,d^7-56\,a\,b^7\,c\,d^6+28\,b^8\,c^2\,d^5\right )+\frac {15\,a^8\,d^8+20\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6+42\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4+140\,a^3\,b^5\,c^5\,d^3+420\,a^2\,b^6\,c^6\,d^2-2178\,a\,b^7\,c^7\,d+1443\,b^8\,c^8}{105\,d}+x\,\left (\frac {4\,a^7\,b\,d^7}{3}+\frac {28\,a^6\,b^2\,c\,d^6}{15}+\frac {14\,a^5\,b^3\,c^2\,d^5}{5}+\frac {14\,a^4\,b^4\,c^3\,d^4}{3}+\frac {28\,a^3\,b^5\,c^4\,d^3}{3}+28\,a^2\,b^6\,c^5\,d^2-\frac {686\,a\,b^7\,c^6\,d}{5}+\frac {446\,b^8\,c^7}{5}\right )+x^3\,\left (14\,a^5\,b^3\,d^7+\frac {70\,a^4\,b^4\,c\,d^6}{3}+\frac {140\,a^3\,b^5\,c^2\,d^5}{3}+140\,a^2\,b^6\,c^3\,d^4-\frac {1750\,a\,b^7\,c^4\,d^3}{3}+\frac {1078\,b^8\,c^5\,d^2}{3}\right )+x^2\,\left (\frac {28\,a^6\,b^2\,d^7}{5}+\frac {42\,a^5\,b^3\,c\,d^6}{5}+14\,a^4\,b^4\,c^2\,d^5+28\,a^3\,b^5\,c^3\,d^4+84\,a^2\,b^6\,c^4\,d^3-\frac {1918\,a\,b^7\,c^5\,d^2}{5}+\frac {1218\,b^8\,c^6\,d}{5}\right )+x^5\,\left (28\,a^3\,b^5\,d^7+84\,a^2\,b^6\,c\,d^6-252\,a\,b^7\,c^2\,d^5+140\,b^8\,c^3\,d^4\right )}{c^7\,d^8+7\,c^6\,d^9\,x+21\,c^5\,d^{10}\,x^2+35\,c^4\,d^{11}\,x^3+35\,c^3\,d^{12}\,x^4+21\,c^2\,d^{13}\,x^5+7\,c\,d^{14}\,x^6+d^{15}\,x^7} \]
(b^8*x)/d^8 - (log(c + d*x)*(8*b^8*c - 8*a*b^7*d))/d^9 - (x^4*((70*a^4*b^4 *d^7)/3 + (910*b^8*c^4*d^3)/3 - (1540*a*b^7*c^3*d^4)/3 + (140*a^3*b^5*c*d^ 6)/3 + 140*a^2*b^6*c^2*d^5) + x^6*(28*a^2*b^6*d^7 + 28*b^8*c^2*d^5 - 56*a* b^7*c*d^6) + (15*a^8*d^8 + 1443*b^8*c^8 + 420*a^2*b^6*c^6*d^2 + 140*a^3*b^ 5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 + 42*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 2178*a*b^7*c^7*d + 20*a^7*b*c*d^7)/(105*d) + x*((446*b^8*c^7)/5 + (4*a^7* b*d^7)/3 + (28*a^6*b^2*c*d^6)/15 + 28*a^2*b^6*c^5*d^2 + (28*a^3*b^5*c^4*d^ 3)/3 + (14*a^4*b^4*c^3*d^4)/3 + (14*a^5*b^3*c^2*d^5)/5 - (686*a*b^7*c^6*d) /5) + x^3*(14*a^5*b^3*d^7 + (1078*b^8*c^5*d^2)/3 - (1750*a*b^7*c^4*d^3)/3 + (70*a^4*b^4*c*d^6)/3 + 140*a^2*b^6*c^3*d^4 + (140*a^3*b^5*c^2*d^5)/3) + x^2*((1218*b^8*c^6*d)/5 + (28*a^6*b^2*d^7)/5 - (1918*a*b^7*c^5*d^2)/5 + (4 2*a^5*b^3*c*d^6)/5 + 84*a^2*b^6*c^4*d^3 + 28*a^3*b^5*c^3*d^4 + 14*a^4*b^4* c^2*d^5) + x^5*(28*a^3*b^5*d^7 + 140*b^8*c^3*d^4 - 252*a*b^7*c^2*d^5 + 84* a^2*b^6*c*d^6))/(c^7*d^8 + d^15*x^7 + 7*c^6*d^9*x + 7*c*d^14*x^6 + 21*c^5* d^10*x^2 + 35*c^4*d^11*x^3 + 35*c^3*d^12*x^4 + 21*c^2*d^13*x^5)
Time = 0.00 (sec) , antiderivative size = 911, normalized size of antiderivative = 4.36 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx =\text {Too large to display} \]
int((a**8 + 8*a**7*b*x + 28*a**6*b**2*x**2 + 56*a**5*b**3*x**3 + 70*a**4*b **4*x**4 + 56*a**3*b**5*x**5 + 28*a**2*b**6*x**6 + 8*a*b**7*x**7 + b**8*x* *8)/(c**8 + 8*c**7*d*x + 28*c**6*d**2*x**2 + 56*c**5*d**3*x**3 + 70*c**4*d **4*x**4 + 56*c**3*d**5*x**5 + 28*c**2*d**6*x**6 + 8*c*d**7*x**7 + d**8*x* *8),x)
(840*log(c + d*x)*a*b**7*c**8*d + 5880*log(c + d*x)*a*b**7*c**7*d**2*x + 1 7640*log(c + d*x)*a*b**7*c**6*d**3*x**2 + 29400*log(c + d*x)*a*b**7*c**5*d **4*x**3 + 29400*log(c + d*x)*a*b**7*c**4*d**5*x**4 + 17640*log(c + d*x)*a *b**7*c**3*d**6*x**5 + 5880*log(c + d*x)*a*b**7*c**2*d**7*x**6 + 840*log(c + d*x)*a*b**7*c*d**8*x**7 - 840*log(c + d*x)*b**8*c**9 - 5880*log(c + d*x )*b**8*c**8*d*x - 17640*log(c + d*x)*b**8*c**7*d**2*x**2 - 29400*log(c + d *x)*b**8*c**6*d**3*x**3 - 29400*log(c + d*x)*b**8*c**5*d**4*x**4 - 17640*l og(c + d*x)*b**8*c**4*d**5*x**5 - 5880*log(c + d*x)*b**8*c**3*d**6*x**6 - 840*log(c + d*x)*b**8*c**2*d**7*x**7 - 15*a**8*c*d**8 - 20*a**7*b*c**2*d** 7 - 140*a**7*b*c*d**8*x - 28*a**6*b**2*c**3*d**6 - 196*a**6*b**2*c**2*d**7 *x - 588*a**6*b**2*c*d**8*x**2 - 42*a**5*b**3*c**4*d**5 - 294*a**5*b**3*c* *3*d**6*x - 882*a**5*b**3*c**2*d**7*x**2 - 1470*a**5*b**3*c*d**8*x**3 - 70 *a**4*b**4*c**5*d**4 - 490*a**4*b**4*c**4*d**5*x - 1470*a**4*b**4*c**3*d** 6*x**2 - 2450*a**4*b**4*c**2*d**7*x**3 - 2450*a**4*b**4*c*d**8*x**4 - 140* a**3*b**5*c**6*d**3 - 980*a**3*b**5*c**5*d**4*x - 2940*a**3*b**5*c**4*d**5 *x**2 - 4900*a**3*b**5*c**3*d**6*x**3 - 4900*a**3*b**5*c**2*d**7*x**4 - 29 40*a**3*b**5*c*d**8*x**5 + 420*a**2*b**6*d**9*x**7 + 1338*a*b**7*c**8*d + 8526*a*b**7*c**7*d**2*x + 22638*a*b**7*c**6*d**3*x**2 + 31850*a*b**7*c**5* d**4*x**3 + 24500*a*b**7*c**4*d**5*x**4 + 8820*a*b**7*c**3*d**6*x**5 - 840 *a*b**7*c*d**8*x**7 - 1338*b**8*c**9 - 8526*b**8*c**8*d*x - 22638*b**8*...