Integrand size = 15, antiderivative size = 194 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {(b c-a d)^7}{7 d^8 (c+d x)^7}-\frac {7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac {21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac {35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac {35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac {21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac {7 b^6 (b c-a d)}{d^8 (c+d x)}+\frac {b^7 \log (c+d x)}{d^8} \]
1/7*(-a*d+b*c)^7/d^8/(d*x+c)^7-7/6*b*(-a*d+b*c)^6/d^8/(d*x+c)^6+21/5*b^2*( -a*d+b*c)^5/d^8/(d*x+c)^5-35/4*b^3*(-a*d+b*c)^4/d^8/(d*x+c)^4+35/3*b^4*(-a *d+b*c)^3/d^8/(d*x+c)^3-21/2*b^5*(-a*d+b*c)^2/d^8/(d*x+c)^2+7*b^6*(-a*d+b* c)/d^8/(d*x+c)+b^7*ln(d*x+c)/d^8
Time = 0.09 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {(b c-a d) \left (60 a^6 d^6+10 a^5 b d^5 (13 c+49 d x)+2 a^4 b^2 d^4 \left (107 c^2+539 c d x+882 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+1813 c^2 d x+3969 c d^2 x^2+3675 d^3 x^3\right )+a^2 b^4 d^2 \left (459 c^4+2793 c^3 d x+6909 c^2 d^2 x^2+8575 c d^3 x^3+4900 d^4 x^4\right )+a b^5 d \left (669 c^5+4263 c^4 d x+11319 c^3 d^2 x^2+15925 c^2 d^3 x^3+12250 c d^4 x^4+4410 d^5 x^5\right )+b^6 \left (1089 c^6+7203 c^5 d x+20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+13230 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 d^8 (c+d x)^7}+\frac {b^7 \log (c+d x)}{d^8} \]
((b*c - a*d)*(60*a^6*d^6 + 10*a^5*b*d^5*(13*c + 49*d*x) + 2*a^4*b^2*d^4*(1 07*c^2 + 539*c*d*x + 882*d^2*x^2) + a^3*b^3*d^3*(319*c^3 + 1813*c^2*d*x + 3969*c*d^2*x^2 + 3675*d^3*x^3) + a^2*b^4*d^2*(459*c^4 + 2793*c^3*d*x + 690 9*c^2*d^2*x^2 + 8575*c*d^3*x^3 + 4900*d^4*x^4) + a*b^5*d*(669*c^5 + 4263*c ^4*d*x + 11319*c^3*d^2*x^2 + 15925*c^2*d^3*x^3 + 12250*c*d^4*x^4 + 4410*d^ 5*x^5) + b^6*(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)))/(420*d^8*(c + d*x)^7) + (b^7*Log[c + d*x])/d^8
Time = 0.42 (sec) , antiderivative size = 194, 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.
| \(\displaystyle \int \frac {(a+b x)^7}{(c+d x)^8} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {7 b^6 (b c-a d)}{d^7 (c+d x)^2}+\frac {21 b^5 (b c-a d)^2}{d^7 (c+d x)^3}-\frac {35 b^4 (b c-a d)^3}{d^7 (c+d x)^4}+\frac {35 b^3 (b c-a d)^4}{d^7 (c+d x)^5}-\frac {21 b^2 (b c-a d)^5}{d^7 (c+d x)^6}+\frac {7 b (b c-a d)^6}{d^7 (c+d x)^7}+\frac {(a d-b c)^7}{d^7 (c+d x)^8}+\frac {b^7}{d^7 (c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 b^6 (b c-a d)}{d^8 (c+d x)}-\frac {21 b^5 (b c-a d)^2}{2 d^8 (c+d x)^2}+\frac {35 b^4 (b c-a d)^3}{3 d^8 (c+d x)^3}-\frac {35 b^3 (b c-a d)^4}{4 d^8 (c+d x)^4}+\frac {21 b^2 (b c-a d)^5}{5 d^8 (c+d x)^5}-\frac {7 b (b c-a d)^6}{6 d^8 (c+d x)^6}+\frac {(b c-a d)^7}{7 d^8 (c+d x)^7}+\frac {b^7 \log (c+d x)}{d^8}\) |
(b*c - a*d)^7/(7*d^8*(c + d*x)^7) - (7*b*(b*c - a*d)^6)/(6*d^8*(c + d*x)^6 ) + (21*b^2*(b*c - a*d)^5)/(5*d^8*(c + d*x)^5) - (35*b^3*(b*c - a*d)^4)/(4 *d^8*(c + d*x)^4) + (35*b^4*(b*c - a*d)^3)/(3*d^8*(c + d*x)^3) - (21*b^5*( b*c - a*d)^2)/(2*d^8*(c + d*x)^2) + (7*b^6*(b*c - a*d))/(d^8*(c + d*x)) + (b^7*Log[c + d*x])/d^8
3.14.64.3.1 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. \(445\) vs. \(2(182)=364\).
Time = 0.23 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.30
| method | result | size |
| risch | \(\frac {-\frac {7 b^{6} \left (a d -b c \right ) x^{6}}{d^{2}}-\frac {21 b^{5} \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) x^{5}}{2 d^{3}}-\frac {35 b^{4} \left (2 a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -11 b^{3} c^{3}\right ) x^{4}}{6 d^{4}}-\frac {35 b^{3} \left (3 a^{4} d^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+12 a \,b^{3} c^{3} d -25 b^{4} c^{4}\right ) x^{3}}{12 d^{5}}-\frac {7 b^{2} \left (12 a^{5} d^{5}+15 a^{4} b c \,d^{4}+20 a^{3} b^{2} c^{2} d^{3}+30 a^{2} b^{3} c^{3} d^{2}+60 a \,b^{4} c^{4} d -137 b^{5} c^{5}\right ) x^{2}}{20 d^{6}}-\frac {7 b \left (10 a^{6} d^{6}+12 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+20 a^{3} b^{3} c^{3} d^{3}+30 a^{2} b^{4} c^{4} d^{2}+60 a \,b^{5} c^{5} d -147 b^{6} c^{6}\right ) x}{60 d^{7}}-\frac {60 a^{7} d^{7}+70 a^{6} b c \,d^{6}+84 a^{5} b^{2} c^{2} d^{5}+105 a^{4} b^{3} c^{3} d^{4}+140 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}+420 a \,b^{6} c^{6} d -1089 b^{7} c^{7}}{420 d^{8}}}{\left (d x +c \right )^{7}}+\frac {b^{7} \ln \left (d x +c \right )}{d^{8}}\) | \(446\) |
| norman | \(\frac {-\frac {60 a^{7} d^{7}+70 a^{6} b c \,d^{6}+84 a^{5} b^{2} c^{2} d^{5}+105 a^{4} b^{3} c^{3} d^{4}+140 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}+420 a \,b^{6} c^{6} d -1089 b^{7} c^{7}}{420 d^{8}}-\frac {7 \left (a \,b^{6} d -b^{7} c \right ) x^{6}}{d^{2}}-\frac {21 \left (a^{2} b^{5} d^{2}+2 a \,b^{6} c d -3 b^{7} c^{2}\right ) x^{5}}{2 d^{3}}-\frac {35 \left (2 a^{3} b^{4} d^{3}+3 a^{2} b^{5} c \,d^{2}+6 a \,b^{6} c^{2} d -11 b^{7} c^{3}\right ) x^{4}}{6 d^{4}}-\frac {35 \left (3 a^{4} b^{3} d^{4}+4 a^{3} b^{4} c \,d^{3}+6 a^{2} b^{5} c^{2} d^{2}+12 a \,b^{6} c^{3} d -25 b^{7} c^{4}\right ) x^{3}}{12 d^{5}}-\frac {7 \left (12 a^{5} b^{2} d^{5}+15 a^{4} b^{3} c \,d^{4}+20 a^{3} b^{4} c^{2} d^{3}+30 a^{2} b^{5} c^{3} d^{2}+60 a \,b^{6} c^{4} d -137 b^{7} c^{5}\right ) x^{2}}{20 d^{6}}-\frac {7 \left (10 a^{6} b \,d^{6}+12 a^{5} b^{2} c \,d^{5}+15 a^{4} b^{3} c^{2} d^{4}+20 a^{3} b^{4} c^{3} d^{3}+30 a^{2} b^{5} c^{4} d^{2}+60 a \,b^{6} c^{5} d -147 b^{7} c^{6}\right ) x}{60 d^{7}}}{\left (d x +c \right )^{7}}+\frac {b^{7} \ln \left (d x +c \right )}{d^{8}}\) | \(458\) |
| default | \(-\frac {21 b^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{8} \left (d x +c \right )^{2}}-\frac {21 b^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{5 d^{8} \left (d x +c \right )^{5}}+\frac {b^{7} \ln \left (d x +c \right )}{d^{8}}-\frac {7 b \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{6 d^{8} \left (d x +c \right )^{6}}-\frac {a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}{7 d^{8} \left (d x +c \right )^{7}}-\frac {35 b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 d^{8} \left (d x +c \right )^{3}}-\frac {7 b^{6} \left (a d -b c \right )}{d^{8} \left (d x +c \right )}-\frac {35 b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{4 d^{8} \left (d x +c \right )^{4}}\) | \(462\) |
| parallelrisch | \(\frac {2940 \ln \left (d x +c \right ) x^{6} b^{7} c \,d^{6}+8820 \ln \left (d x +c \right ) x^{5} b^{7} c^{2} d^{5}+14700 \ln \left (d x +c \right ) x^{4} b^{7} c^{3} d^{4}+14700 \ln \left (d x +c \right ) x^{3} b^{7} c^{4} d^{3}+8820 \ln \left (d x +c \right ) x^{2} b^{7} c^{5} d^{2}+2940 \ln \left (d x +c \right ) x \,b^{7} c^{6} d -420 a \,b^{6} c^{6} d -210 a^{2} b^{5} c^{5} d^{2}-105 a^{4} b^{3} c^{3} d^{4}-140 a^{3} b^{4} c^{4} d^{3}-70 a^{6} b c \,d^{6}-84 a^{5} b^{2} c^{2} d^{5}+1089 b^{7} c^{7}-60 a^{7} d^{7}-4410 x^{2} a^{2} b^{5} c^{3} d^{4}-8820 x^{2} a \,b^{6} c^{4} d^{3}-4900 x^{3} a^{3} b^{4} c \,d^{6}-7350 x^{3} a^{2} b^{5} c^{2} d^{5}-14700 x^{3} a \,b^{6} c^{3} d^{4}-7350 x^{4} a^{2} b^{5} c \,d^{6}-14700 x^{4} a \,b^{6} c^{2} d^{5}-8820 x^{5} a \,b^{6} c \,d^{6}-588 x \,a^{5} b^{2} c \,d^{6}-735 x \,a^{4} b^{3} c^{2} d^{5}-980 x \,a^{3} b^{4} c^{3} d^{4}-1470 x \,a^{2} b^{5} c^{4} d^{3}-2940 x a \,b^{6} c^{5} d^{2}-2205 x^{2} a^{4} b^{3} c \,d^{6}-2940 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (d x +c \right ) x^{7} b^{7} d^{7}+420 \ln \left (d x +c \right ) b^{7} c^{7}-4410 x^{5} a^{2} b^{5} d^{7}+13230 x^{5} b^{7} c^{2} d^{5}-2940 x^{6} a \,b^{6} d^{7}-490 x \,a^{6} b \,d^{7}+7203 x \,b^{7} c^{6} d -1764 x^{2} a^{5} b^{2} d^{7}+20139 x^{2} b^{7} c^{5} d^{2}-3675 x^{3} a^{4} b^{3} d^{7}+30625 x^{3} b^{7} c^{4} d^{3}-4900 x^{4} a^{3} b^{4} d^{7}+26950 x^{4} b^{7} c^{3} d^{4}+2940 x^{6} b^{7} c \,d^{6}}{420 d^{8} \left (d x +c \right )^{7}}\) | \(632\) |
(-7*b^6*(a*d-b*c)/d^2*x^6-21/2*b^5*(a^2*d^2+2*a*b*c*d-3*b^2*c^2)/d^3*x^5-3 5/6*b^4*(2*a^3*d^3+3*a^2*b*c*d^2+6*a*b^2*c^2*d-11*b^3*c^3)/d^4*x^4-35/12*b ^3*(3*a^4*d^4+4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+12*a*b^3*c^3*d-25*b^4*c^4)/d ^5*x^3-7/20*b^2*(12*a^5*d^5+15*a^4*b*c*d^4+20*a^3*b^2*c^2*d^3+30*a^2*b^3*c ^3*d^2+60*a*b^4*c^4*d-137*b^5*c^5)/d^6*x^2-7/60*b*(10*a^6*d^6+12*a^5*b*c*d ^5+15*a^4*b^2*c^2*d^4+20*a^3*b^3*c^3*d^3+30*a^2*b^4*c^4*d^2+60*a*b^5*c^5*d -147*b^6*c^6)/d^7*x-1/420*(60*a^7*d^7+70*a^6*b*c*d^6+84*a^5*b^2*c^2*d^5+10 5*a^4*b^3*c^3*d^4+140*a^3*b^4*c^4*d^3+210*a^2*b^5*c^5*d^2+420*a*b^6*c^6*d- 1089*b^7*c^7)/d^8)/(d*x+c)^7+b^7*ln(d*x+c)/d^8
Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (182) = 364\).
Time = 0.22 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.22 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x + 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, b^{7} c d^{6} x^{6} + 21 \, b^{7} c^{2} d^{5} x^{5} + 35 \, b^{7} c^{3} d^{4} x^{4} + 35 \, b^{7} c^{4} d^{3} x^{3} + 21 \, b^{7} c^{5} d^{2} x^{2} + 7 \, b^{7} c^{6} d x + b^{7} c^{7}\right )} \log \left (d x + c\right )}{420 \, {\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} \]
1/420*(1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4* c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a ^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(3*b^7*c^2*d^5 - 2*a*b^6* c*d^6 - a^2*b^5*d^7)*x^5 + 2450*(11*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 - 3*a^2* b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 1225*(25*b^7*c^4*d^3 - 12*a*b^6*c^3*d^4 - 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 147*(137*b^7*c ^5*d^2 - 60*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 - 20*a^3*b^4*c^2*d^5 - 15*a ^4*b^3*c*d^6 - 12*a^5*b^2*d^7)*x^2 + 49*(147*b^7*c^6*d - 60*a*b^6*c^5*d^2 - 30*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*d^4 - 15*a^4*b^3*c^2*d^5 - 12*a^5*b^ 2*c*d^6 - 10*a^6*b*d^7)*x + 420*(b^7*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^ 2*d^5*x^5 + 35*b^7*c^3*d^4*x^4 + 35*b^7*c^4*d^3*x^3 + 21*b^7*c^5*d^2*x^2 + 7*b^7*c^6*d*x + b^7*c^7)*log(d*x + c))/(d^15*x^7 + 7*c*d^14*x^6 + 21*c^2* d^13*x^5 + 35*c^3*d^12*x^4 + 35*c^4*d^11*x^3 + 21*c^5*d^10*x^2 + 7*c^6*d^9 *x + c^7*d^8)
Timed out. \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (182) = 364\).
Time = 0.22 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} - 3 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{3} - 12 \, a b^{6} c^{3} d^{4} - 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d^{2} - 60 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} - 20 \, a^{3} b^{4} c^{2} d^{5} - 15 \, a^{4} b^{3} c d^{6} - 12 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} d - 60 \, a b^{6} c^{5} d^{2} - 30 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} - 15 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x}{420 \, {\left (d^{15} x^{7} + 7 \, c d^{14} x^{6} + 21 \, c^{2} d^{13} x^{5} + 35 \, c^{3} d^{12} x^{4} + 35 \, c^{4} d^{11} x^{3} + 21 \, c^{5} d^{10} x^{2} + 7 \, c^{6} d^{9} x + c^{7} d^{8}\right )}} + \frac {b^{7} \log \left (d x + c\right )}{d^{8}} \]
1/420*(1089*b^7*c^7 - 420*a*b^6*c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4* c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a ^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(3*b^7*c^2*d^5 - 2*a*b^6* c*d^6 - a^2*b^5*d^7)*x^5 + 2450*(11*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 - 3*a^2* b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 1225*(25*b^7*c^4*d^3 - 12*a*b^6*c^3*d^4 - 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 147*(137*b^7*c ^5*d^2 - 60*a*b^6*c^4*d^3 - 30*a^2*b^5*c^3*d^4 - 20*a^3*b^4*c^2*d^5 - 15*a ^4*b^3*c*d^6 - 12*a^5*b^2*d^7)*x^2 + 49*(147*b^7*c^6*d - 60*a*b^6*c^5*d^2 - 30*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*d^4 - 15*a^4*b^3*c^2*d^5 - 12*a^5*b^ 2*c*d^6 - 10*a^6*b*d^7)*x)/(d^15*x^7 + 7*c*d^14*x^6 + 21*c^2*d^13*x^5 + 35 *c^3*d^12*x^4 + 35*c^4*d^11*x^3 + 21*c^5*d^10*x^2 + 7*c^6*d^9*x + c^7*d^8) + b^7*log(d*x + c)/d^8
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (182) = 364\).
Time = 0.31 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {b^{7} \log \left ({\left | d x + c \right |}\right )}{d^{8}} + \frac {2940 \, {\left (b^{7} c d^{5} - a b^{6} d^{6}\right )} x^{6} + 4410 \, {\left (3 \, b^{7} c^{2} d^{4} - 2 \, a b^{6} c d^{5} - a^{2} b^{5} d^{6}\right )} x^{5} + 2450 \, {\left (11 \, b^{7} c^{3} d^{3} - 6 \, a b^{6} c^{2} d^{4} - 3 \, a^{2} b^{5} c d^{5} - 2 \, a^{3} b^{4} d^{6}\right )} x^{4} + 1225 \, {\left (25 \, b^{7} c^{4} d^{2} - 12 \, a b^{6} c^{3} d^{3} - 6 \, a^{2} b^{5} c^{2} d^{4} - 4 \, a^{3} b^{4} c d^{5} - 3 \, a^{4} b^{3} d^{6}\right )} x^{3} + 147 \, {\left (137 \, b^{7} c^{5} d - 60 \, a b^{6} c^{4} d^{2} - 30 \, a^{2} b^{5} c^{3} d^{3} - 20 \, a^{3} b^{4} c^{2} d^{4} - 15 \, a^{4} b^{3} c d^{5} - 12 \, a^{5} b^{2} d^{6}\right )} x^{2} + 49 \, {\left (147 \, b^{7} c^{6} - 60 \, a b^{6} c^{5} d - 30 \, a^{2} b^{5} c^{4} d^{2} - 20 \, a^{3} b^{4} c^{3} d^{3} - 15 \, a^{4} b^{3} c^{2} d^{4} - 12 \, a^{5} b^{2} c d^{5} - 10 \, a^{6} b d^{6}\right )} x + \frac {1089 \, b^{7} c^{7} - 420 \, a b^{6} c^{6} d - 210 \, a^{2} b^{5} c^{5} d^{2} - 140 \, a^{3} b^{4} c^{4} d^{3} - 105 \, a^{4} b^{3} c^{3} d^{4} - 84 \, a^{5} b^{2} c^{2} d^{5} - 70 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7}}{d}}{420 \, {\left (d x + c\right )}^{7} d^{7}} \]
b^7*log(abs(d*x + c))/d^8 + 1/420*(2940*(b^7*c*d^5 - a*b^6*d^6)*x^6 + 4410 *(3*b^7*c^2*d^4 - 2*a*b^6*c*d^5 - a^2*b^5*d^6)*x^5 + 2450*(11*b^7*c^3*d^3 - 6*a*b^6*c^2*d^4 - 3*a^2*b^5*c*d^5 - 2*a^3*b^4*d^6)*x^4 + 1225*(25*b^7*c^ 4*d^2 - 12*a*b^6*c^3*d^3 - 6*a^2*b^5*c^2*d^4 - 4*a^3*b^4*c*d^5 - 3*a^4*b^3 *d^6)*x^3 + 147*(137*b^7*c^5*d - 60*a*b^6*c^4*d^2 - 30*a^2*b^5*c^3*d^3 - 2 0*a^3*b^4*c^2*d^4 - 15*a^4*b^3*c*d^5 - 12*a^5*b^2*d^6)*x^2 + 49*(147*b^7*c ^6 - 60*a*b^6*c^5*d - 30*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 - 15*a^4*b^3 *c^2*d^4 - 12*a^5*b^2*c*d^5 - 10*a^6*b*d^6)*x + (1089*b^7*c^7 - 420*a*b^6* c^6*d - 210*a^2*b^5*c^5*d^2 - 140*a^3*b^4*c^4*d^3 - 105*a^4*b^3*c^3*d^4 - 84*a^5*b^2*c^2*d^5 - 70*a^6*b*c*d^6 - 60*a^7*d^7)/d)/((d*x + c)^7*d^7)
Time = 0.50 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.37 \[ \int \frac {(a+b x)^7}{(c+d x)^8} \, dx=\frac {b^7\,\ln \left (c+d\,x\right )}{d^8}-\frac {x\,\left (\frac {7\,a^6\,b\,d^7}{6}+\frac {7\,a^5\,b^2\,c\,d^6}{5}+\frac {7\,a^4\,b^3\,c^2\,d^5}{4}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{2}+7\,a\,b^6\,c^5\,d^2-\frac {343\,b^7\,c^6\,d}{20}\right )+x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (\frac {35\,a^4\,b^3\,d^7}{4}+\frac {35\,a^3\,b^4\,c\,d^6}{3}+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+35\,a\,b^6\,c^3\,d^4-\frac {875\,b^7\,c^4\,d^3}{12}\right )+x^5\,\left (\frac {21\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6-\frac {63\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (\frac {21\,a^5\,b^2\,d^7}{5}+\frac {21\,a^4\,b^3\,c\,d^6}{4}+7\,a^3\,b^4\,c^2\,d^5+\frac {21\,a^2\,b^5\,c^3\,d^4}{2}+21\,a\,b^6\,c^4\,d^3-\frac {959\,b^7\,c^5\,d^2}{20}\right )+\frac {a^7\,d^7}{7}-\frac {363\,b^7\,c^7}{140}+x^4\,\left (\frac {35\,a^3\,b^4\,d^7}{3}+\frac {35\,a^2\,b^5\,c\,d^6}{2}+35\,a\,b^6\,c^2\,d^5-\frac {385\,b^7\,c^3\,d^4}{6}\right )+\frac {a^2\,b^5\,c^5\,d^2}{2}+\frac {a^3\,b^4\,c^4\,d^3}{3}+\frac {a^4\,b^3\,c^3\,d^4}{4}+\frac {a^5\,b^2\,c^2\,d^5}{5}+a\,b^6\,c^6\,d+\frac {a^6\,b\,c\,d^6}{6}}{d^8\,{\left (c+d\,x\right )}^7} \]
(b^7*log(c + d*x))/d^8 - (x*((7*a^6*b*d^7)/6 - (343*b^7*c^6*d)/20 + 7*a*b^ 6*c^5*d^2 + (7*a^5*b^2*c*d^6)/5 + (7*a^2*b^5*c^4*d^3)/2 + (7*a^3*b^4*c^3*d ^4)/3 + (7*a^4*b^3*c^2*d^5)/4) + x^6*(7*a*b^6*d^7 - 7*b^7*c*d^6) + x^3*((3 5*a^4*b^3*d^7)/4 - (875*b^7*c^4*d^3)/12 + 35*a*b^6*c^3*d^4 + (35*a^3*b^4*c *d^6)/3 + (35*a^2*b^5*c^2*d^5)/2) + x^5*((21*a^2*b^5*d^7)/2 - (63*b^7*c^2* d^5)/2 + 21*a*b^6*c*d^6) + x^2*((21*a^5*b^2*d^7)/5 - (959*b^7*c^5*d^2)/20 + 21*a*b^6*c^4*d^3 + (21*a^4*b^3*c*d^6)/4 + (21*a^2*b^5*c^3*d^4)/2 + 7*a^3 *b^4*c^2*d^5) + (a^7*d^7)/7 - (363*b^7*c^7)/140 + x^4*((35*a^3*b^4*d^7)/3 - (385*b^7*c^3*d^4)/6 + 35*a*b^6*c^2*d^5 + (35*a^2*b^5*c*d^6)/2) + (a^2*b^ 5*c^5*d^2)/2 + (a^3*b^4*c^4*d^3)/3 + (a^4*b^3*c^3*d^4)/4 + (a^5*b^2*c^2*d^ 5)/5 + a*b^6*c^6*d + (a^6*b*c*d^6)/6)/(d^8*(c + d*x)^7)