Integrand size = 15, antiderivative size = 194 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}+\frac {d^7 \log (a+b x)}{b^8} \]
-1/7*(-a*d+b*c)^7/b^8/(b*x+a)^7-7/6*d*(-a*d+b*c)^6/b^8/(b*x+a)^6-21/5*d^2* (-a*d+b*c)^5/b^8/(b*x+a)^5-35/4*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^4-35/3*d^4*(- a*d+b*c)^3/b^8/(b*x+a)^3-21/2*d^5*(-a*d+b*c)^2/b^8/(b*x+a)^2-7*d^6*(-a*d+b *c)/b^8/(b*x+a)+d^7*ln(b*x+a)/b^8
Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.59 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {(b c-a d) \left (1089 a^6 d^6+3 a^5 b d^5 (223 c+2401 d x)+3 a^4 b^2 d^4 \left (153 c^2+1421 c d x+6713 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+2793 c^2 d x+11319 c d^2 x^2+30625 d^3 x^3\right )+a^2 b^4 d^2 \left (214 c^4+1813 c^3 d x+6909 c^2 d^2 x^2+15925 c d^3 x^3+26950 d^4 x^4\right )+a b^5 d \left (130 c^5+1078 c^4 d x+3969 c^3 d^2 x^2+8575 c^2 d^3 x^3+12250 c d^4 x^4+13230 d^5 x^5\right )+b^6 \left (60 c^6+490 c^5 d x+1764 c^4 d^2 x^2+3675 c^3 d^3 x^3+4900 c^2 d^4 x^4+4410 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8} \]
-1/420*((b*c - a*d)*(1089*a^6*d^6 + 3*a^5*b*d^5*(223*c + 2401*d*x) + 3*a^4 *b^2*d^4*(153*c^2 + 1421*c*d*x + 6713*d^2*x^2) + a^3*b^3*d^3*(319*c^3 + 27 93*c^2*d*x + 11319*c*d^2*x^2 + 30625*d^3*x^3) + a^2*b^4*d^2*(214*c^4 + 181 3*c^3*d*x + 6909*c^2*d^2*x^2 + 15925*c*d^3*x^3 + 26950*d^4*x^4) + a*b^5*d* (130*c^5 + 1078*c^4*d*x + 3969*c^3*d^2*x^2 + 8575*c^2*d^3*x^3 + 12250*c*d^ 4*x^4 + 13230*d^5*x^5) + b^6*(60*c^6 + 490*c^5*d*x + 1764*c^4*d^2*x^2 + 36 75*c^3*d^3*x^3 + 4900*c^2*d^4*x^4 + 4410*c*d^5*x^5 + 2940*d^6*x^6)))/(b^8* (a + b*x)^7) + (d^7*Log[a + b*x])/b^8
Time = 0.37 (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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^8} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^2}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^3}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^4}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^5}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^6}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^7}+\frac {(b c-a d)^7}{b^7 (a+b x)^8}+\frac {d^7}{b^7 (a+b x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8}\) |
-1/7*(b*c - a*d)^7/(b^8*(a + b*x)^7) - (7*d*(b*c - a*d)^6)/(6*b^8*(a + b*x )^6) - (21*d^2*(b*c - a*d)^5)/(5*b^8*(a + b*x)^5) - (35*d^3*(b*c - a*d)^4) /(4*b^8*(a + b*x)^4) - (35*d^4*(b*c - a*d)^3)/(3*b^8*(a + b*x)^3) - (21*d^ 5*(b*c - a*d)^2)/(2*b^8*(a + b*x)^2) - (7*d^6*(b*c - a*d))/(b^8*(a + b*x)) + (d^7*Log[a + b*x])/b^8
3.13.90.3.1 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. \(446\) vs. \(2(182)=364\).
Time = 0.22 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.30
| method | result | size |
| risch | \(\frac {\frac {7 d^{6} \left (a d -b c \right ) x^{6}}{b^{2}}+\frac {21 d^{5} \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x^{5}}{2 b^{3}}+\frac {35 d^{4} \left (11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{4}}{6 b^{4}}+\frac {35 d^{3} \left (25 a^{4} d^{4}-12 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{3}}{12 b^{5}}+\frac {7 d^{2} \left (137 a^{5} d^{5}-60 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-15 a \,b^{4} c^{4} d -12 b^{5} c^{5}\right ) x^{2}}{20 b^{6}}+\frac {7 d \left (147 a^{6} d^{6}-60 a^{5} b c \,d^{5}-30 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-15 a^{2} b^{4} c^{4} d^{2}-12 a \,b^{5} c^{5} d -10 b^{6} c^{6}\right ) x}{60 b^{7}}+\frac {1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}}{420 b^{8}}}{\left (b x +a \right )^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}\) | \(447\) |
| norman | \(\frac {\frac {1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}}{420 b^{8}}+\frac {7 \left (a \,d^{7}-b c \,d^{6}\right ) x^{6}}{b^{2}}+\frac {21 \left (3 a^{2} d^{7}-2 a b c \,d^{6}-b^{2} c^{2} d^{5}\right ) x^{5}}{2 b^{3}}+\frac {35 \left (11 a^{3} d^{7}-6 a^{2} b c \,d^{6}-3 a \,b^{2} c^{2} d^{5}-2 b^{3} c^{3} d^{4}\right ) x^{4}}{6 b^{4}}+\frac {35 \left (25 a^{4} d^{7}-12 a^{3} b c \,d^{6}-6 a^{2} b^{2} c^{2} d^{5}-4 a \,b^{3} c^{3} d^{4}-3 b^{4} c^{4} d^{3}\right ) x^{3}}{12 b^{5}}+\frac {7 \left (137 a^{5} d^{7}-60 a^{4} b c \,d^{6}-30 a^{3} b^{2} c^{2} d^{5}-20 a^{2} b^{3} c^{3} d^{4}-15 a \,b^{4} c^{4} d^{3}-12 b^{5} c^{5} d^{2}\right ) x^{2}}{20 b^{6}}+\frac {7 \left (147 a^{6} d^{7}-60 a^{5} b c \,d^{6}-30 a^{4} b^{2} c^{2} d^{5}-20 a^{3} b^{3} c^{3} d^{4}-15 a^{2} b^{4} c^{4} d^{3}-12 a \,b^{5} c^{5} d^{2}-10 b^{6} c^{6} d \right ) x}{60 b^{7}}}{\left (b x +a \right )^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}\) | \(459\) |
| default | \(\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 b^{8} \left (b x +a \right )^{3}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{6 b^{8} \left (b x +a \right )^{6}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{7 b^{8} \left (b x +a \right )^{7}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{5 b^{8} \left (b x +a \right )^{5}}+\frac {7 d^{6} \left (a d -b c \right )}{b^{8} \left (b x +a \right )}\) | \(462\) |
| parallelrisch | \(\frac {2940 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}-70 a \,b^{6} c^{6} d -84 a^{2} b^{5} c^{5} d^{2}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-60 b^{7} c^{7}+1089 a^{7} d^{7}+8820 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}-2940 x^{2} a^{2} b^{5} c^{3} d^{4}-2205 x^{2} a \,b^{6} c^{4} d^{3}-14700 x^{3} a^{3} b^{4} c \,d^{6}-7350 x^{3} a^{2} b^{5} c^{2} d^{5}-4900 x^{3} a \,b^{6} c^{3} d^{4}-14700 x^{4} a^{2} b^{5} c \,d^{6}-7350 x^{4} a \,b^{6} c^{2} d^{5}-8820 x^{5} a \,b^{6} c \,d^{6}-2940 x \,a^{5} b^{2} c \,d^{6}-1470 x \,a^{4} b^{3} c^{2} d^{5}-980 x \,a^{3} b^{4} c^{3} d^{4}-735 x \,a^{2} b^{5} c^{4} d^{3}-588 x a \,b^{6} c^{5} d^{2}-8820 x^{2} a^{4} b^{3} c \,d^{6}-4410 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (b x +a \right ) x^{7} b^{7} d^{7}+420 \ln \left (b x +a \right ) a^{7} d^{7}+14700 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}+8820 \ln \left (b x +a \right ) x^{5} a^{2} b^{5} d^{7}+2940 \ln \left (b x +a \right ) x^{6} a \,b^{6} d^{7}+13230 x^{5} a^{2} b^{5} d^{7}-4410 x^{5} b^{7} c^{2} d^{5}+2940 x^{6} a \,b^{6} d^{7}+7203 x \,a^{6} b \,d^{7}-490 x \,b^{7} c^{6} d +20139 x^{2} a^{5} b^{2} d^{7}-1764 x^{2} b^{7} c^{5} d^{2}+30625 x^{3} a^{4} b^{3} d^{7}-3675 x^{3} b^{7} c^{4} d^{3}+26950 x^{4} a^{3} b^{4} d^{7}-4900 x^{4} b^{7} c^{3} d^{4}-2940 x^{6} b^{7} c \,d^{6}+14700 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}}{420 b^{8} \left (b x +a \right )^{7}}\) | \(632\) |
(7*d^6*(a*d-b*c)/b^2*x^6+21/2*d^5*(3*a^2*d^2-2*a*b*c*d-b^2*c^2)/b^3*x^5+35 /6*d^4*(11*a^3*d^3-6*a^2*b*c*d^2-3*a*b^2*c^2*d-2*b^3*c^3)/b^4*x^4+35/12*d^ 3*(25*a^4*d^4-12*a^3*b*c*d^3-6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d-3*b^4*c^4)/b^ 5*x^3+7/20*d^2*(137*a^5*d^5-60*a^4*b*c*d^4-30*a^3*b^2*c^2*d^3-20*a^2*b^3*c ^3*d^2-15*a*b^4*c^4*d-12*b^5*c^5)/b^6*x^2+7/60*d*(147*a^6*d^6-60*a^5*b*c*d ^5-30*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^2-12*a*b^5*c^5*d -10*b^6*c^6)/b^7*x+1/420*(1089*a^7*d^7-420*a^6*b*c*d^6-210*a^5*b^2*c^2*d^5 -140*a^4*b^3*c^3*d^4-105*a^3*b^4*c^4*d^3-84*a^2*b^5*c^5*d^2-70*a*b^6*c^6*d -60*b^7*c^7)/b^8)/(b*x+a)^7+d^7*ln(b*x+a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (182) = 364\).
Time = 0.23 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.22 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x - 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, a b^{6} d^{7} x^{6} + 21 \, a^{2} b^{5} d^{7} x^{5} + 35 \, a^{3} b^{4} d^{7} x^{4} + 35 \, a^{4} b^{3} d^{7} x^{3} + 21 \, a^{5} b^{2} d^{7} x^{2} + 7 \, a^{6} b d^{7} x + a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} \]
-1/420*(60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4 *d^3 + 140*a^4*b^3*c^3*d^4 + 210*a^5*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089* a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(b^7*c^2*d^5 + 2*a*b^6*c *d^6 - 3*a^2*b^5*d^7)*x^5 + 2450*(2*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 6*a^2* b^5*c*d^6 - 11*a^3*b^4*d^7)*x^4 + 1225*(3*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 12*a^3*b^4*c*d^6 - 25*a^4*b^3*d^7)*x^3 + 147*(12*b^7*c ^5*d^2 + 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 + 60*a ^4*b^3*c*d^6 - 137*a^5*b^2*d^7)*x^2 + 49*(10*b^7*c^6*d + 12*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 + 60*a^5*b^ 2*c*d^6 - 147*a^6*b*d^7)*x - 420*(b^7*d^7*x^7 + 7*a*b^6*d^7*x^6 + 21*a^2*b ^5*d^7*x^5 + 35*a^3*b^4*d^7*x^4 + 35*a^4*b^3*d^7*x^3 + 21*a^5*b^2*d^7*x^2 + 7*a^6*b*d^7*x + a^7*d^7)*log(b*x + a))/(b^15*x^7 + 7*a*b^14*x^6 + 21*a^2 *b^13*x^5 + 35*a^3*b^12*x^4 + 35*a^4*b^11*x^3 + 21*a^5*b^10*x^2 + 7*a^6*b^ 9*x + a^7*b^8)
Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (182) = 364\).
Time = 0.23 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.75 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=-\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} + \frac {d^{7} \log \left (b x + a\right )}{b^{8}} \]
-1/420*(60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4 *d^3 + 140*a^4*b^3*c^3*d^4 + 210*a^5*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089* a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(b^7*c^2*d^5 + 2*a*b^6*c *d^6 - 3*a^2*b^5*d^7)*x^5 + 2450*(2*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 6*a^2* b^5*c*d^6 - 11*a^3*b^4*d^7)*x^4 + 1225*(3*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 12*a^3*b^4*c*d^6 - 25*a^4*b^3*d^7)*x^3 + 147*(12*b^7*c ^5*d^2 + 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 + 60*a ^4*b^3*c*d^6 - 137*a^5*b^2*d^7)*x^2 + 49*(10*b^7*c^6*d + 12*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 + 60*a^5*b^ 2*c*d^6 - 147*a^6*b*d^7)*x)/(b^15*x^7 + 7*a*b^14*x^6 + 21*a^2*b^13*x^5 + 3 5*a^3*b^12*x^4 + 35*a^4*b^11*x^3 + 21*a^5*b^10*x^2 + 7*a^6*b^9*x + a^7*b^8 ) + d^7*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (182) = 364\).
Time = 0.29 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.40 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\frac {d^{7} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {2940 \, {\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 4410 \, {\left (b^{6} c^{2} d^{5} + 2 \, a b^{5} c d^{6} - 3 \, a^{2} b^{4} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{6} c^{3} d^{4} + 3 \, a b^{5} c^{2} d^{5} + 6 \, a^{2} b^{4} c d^{6} - 11 \, a^{3} b^{3} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{6} c^{4} d^{3} + 4 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} + 12 \, a^{3} b^{3} c d^{6} - 25 \, a^{4} b^{2} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{6} c^{5} d^{2} + 15 \, a b^{5} c^{4} d^{3} + 20 \, a^{2} b^{4} c^{3} d^{4} + 30 \, a^{3} b^{3} c^{2} d^{5} + 60 \, a^{4} b^{2} c d^{6} - 137 \, a^{5} b d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{6} c^{6} d + 12 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} + 20 \, a^{3} b^{3} c^{3} d^{4} + 30 \, a^{4} b^{2} c^{2} d^{5} + 60 \, a^{5} b c d^{6} - 147 \, a^{6} d^{7}\right )} x + \frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7}}{b}}{420 \, {\left (b x + a\right )}^{7} b^{7}} \]
d^7*log(abs(b*x + a))/b^8 - 1/420*(2940*(b^6*c*d^6 - a*b^5*d^7)*x^6 + 4410 *(b^6*c^2*d^5 + 2*a*b^5*c*d^6 - 3*a^2*b^4*d^7)*x^5 + 2450*(2*b^6*c^3*d^4 + 3*a*b^5*c^2*d^5 + 6*a^2*b^4*c*d^6 - 11*a^3*b^3*d^7)*x^4 + 1225*(3*b^6*c^4 *d^3 + 4*a*b^5*c^3*d^4 + 6*a^2*b^4*c^2*d^5 + 12*a^3*b^3*c*d^6 - 25*a^4*b^2 *d^7)*x^3 + 147*(12*b^6*c^5*d^2 + 15*a*b^5*c^4*d^3 + 20*a^2*b^4*c^3*d^4 + 30*a^3*b^3*c^2*d^5 + 60*a^4*b^2*c*d^6 - 137*a^5*b*d^7)*x^2 + 49*(10*b^6*c^ 6*d + 12*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^3*d^4 + 30*a^4* b^2*c^2*d^5 + 60*a^5*b*c*d^6 - 147*a^6*d^7)*x + (60*b^7*c^7 + 70*a*b^6*c^6 *d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 + 210* a^5*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089*a^7*d^7)/b)/((b*x + a)^7*b^7)
Time = 0.37 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.38 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\frac {d^7\,\ln \left (a+b\,x\right )}{b^8}-\frac {x\,\left (-\frac {343\,a^6\,b\,d^7}{20}+7\,a^5\,b^2\,c\,d^6+\frac {7\,a^4\,b^3\,c^2\,d^5}{2}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{4}+\frac {7\,a\,b^6\,c^5\,d^2}{5}+\frac {7\,b^7\,c^6\,d}{6}\right )-x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (-\frac {875\,a^4\,b^3\,d^7}{12}+35\,a^3\,b^4\,c\,d^6+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+\frac {35\,a\,b^6\,c^3\,d^4}{3}+\frac {35\,b^7\,c^4\,d^3}{4}\right )+x^5\,\left (-\frac {63\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6+\frac {21\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (-\frac {959\,a^5\,b^2\,d^7}{20}+21\,a^4\,b^3\,c\,d^6+\frac {21\,a^3\,b^4\,c^2\,d^5}{2}+7\,a^2\,b^5\,c^3\,d^4+\frac {21\,a\,b^6\,c^4\,d^3}{4}+\frac {21\,b^7\,c^5\,d^2}{5}\right )-\frac {363\,a^7\,d^7}{140}+\frac {b^7\,c^7}{7}+x^4\,\left (-\frac {385\,a^3\,b^4\,d^7}{6}+35\,a^2\,b^5\,c\,d^6+\frac {35\,a\,b^6\,c^2\,d^5}{2}+\frac {35\,b^7\,c^3\,d^4}{3}\right )+\frac {a^2\,b^5\,c^5\,d^2}{5}+\frac {a^3\,b^4\,c^4\,d^3}{4}+\frac {a^4\,b^3\,c^3\,d^4}{3}+\frac {a^5\,b^2\,c^2\,d^5}{2}+\frac {a\,b^6\,c^6\,d}{6}+a^6\,b\,c\,d^6}{b^8\,{\left (a+b\,x\right )}^7} \]
(d^7*log(a + b*x))/b^8 - (x*((7*b^7*c^6*d)/6 - (343*a^6*b*d^7)/20 + (7*a*b ^6*c^5*d^2)/5 + 7*a^5*b^2*c*d^6 + (7*a^2*b^5*c^4*d^3)/4 + (7*a^3*b^4*c^3*d ^4)/3 + (7*a^4*b^3*c^2*d^5)/2) - x^6*(7*a*b^6*d^7 - 7*b^7*c*d^6) + x^3*((3 5*b^7*c^4*d^3)/4 - (875*a^4*b^3*d^7)/12 + (35*a*b^6*c^3*d^4)/3 + 35*a^3*b^ 4*c*d^6 + (35*a^2*b^5*c^2*d^5)/2) + x^5*((21*b^7*c^2*d^5)/2 - (63*a^2*b^5* d^7)/2 + 21*a*b^6*c*d^6) + x^2*((21*b^7*c^5*d^2)/5 - (959*a^5*b^2*d^7)/20 + (21*a*b^6*c^4*d^3)/4 + 21*a^4*b^3*c*d^6 + 7*a^2*b^5*c^3*d^4 + (21*a^3*b^ 4*c^2*d^5)/2) - (363*a^7*d^7)/140 + (b^7*c^7)/7 + x^4*((35*b^7*c^3*d^4)/3 - (385*a^3*b^4*d^7)/6 + (35*a*b^6*c^2*d^5)/2 + 35*a^2*b^5*c*d^6) + (a^2*b^ 5*c^5*d^2)/5 + (a^3*b^4*c^4*d^3)/4 + (a^4*b^3*c^3*d^4)/3 + (a^5*b^2*c^2*d^ 5)/2 + (a*b^6*c^6*d)/6 + a^6*b*c*d^6)/(b^8*(a + b*x)^7)
Time = 0.00 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.29 \[ \int \frac {(c+d x)^7}{(a+b x)^8} \, dx=\frac {420 \,\mathrm {log}\left (b x +a \right ) a^{8} d^{7}-60 a \,b^{7} c^{7}+669 a^{8} d^{7}+420 b^{8} c \,d^{6} x^{7}+8820 \,\mathrm {log}\left (b x +a \right ) a^{6} b^{2} d^{7} x^{2}+14700 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{4} d^{7} x^{4}-210 a^{6} b^{2} c^{2} d^{5}-140 a^{5} b^{3} c^{3} d^{4}-105 a^{4} b^{4} c^{4} d^{3}-84 a^{3} b^{5} c^{5} d^{2}+14700 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{3} d^{7} x^{3}+2940 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{6} d^{7} x^{6}-1470 a^{5} b^{3} c^{2} d^{5} x -980 a^{4} b^{4} c^{3} d^{4} x -4410 a^{4} b^{4} c^{2} d^{5} x^{2}-735 a^{3} b^{5} c^{4} d^{3} x -2940 a^{3} b^{5} c^{3} d^{4} x^{2}-7350 a^{3} b^{5} c^{2} d^{5} x^{3}-588 a^{2} b^{6} c^{5} d^{2} x -2205 a^{2} b^{6} c^{4} d^{3} x^{2}-4900 a^{2} b^{6} c^{3} d^{4} x^{3}-7350 a^{2} b^{6} c^{2} d^{5} x^{4}-490 a \,b^{7} c^{6} d x -1764 a \,b^{7} c^{5} d^{2} x^{2}-3675 a \,b^{7} c^{4} d^{3} x^{3}-4900 a \,b^{7} c^{3} d^{4} x^{4}-4410 a \,b^{7} c^{2} d^{5} x^{5}+8820 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{5} d^{7} x^{5}-70 a^{2} b^{6} c^{6} d +420 \,\mathrm {log}\left (b x +a \right ) a \,b^{7} d^{7} x^{7}+2940 \,\mathrm {log}\left (b x +a \right ) a^{7} b \,d^{7} x +4263 a^{7} b \,d^{7} x +11319 a^{6} b^{2} d^{7} x^{2}+15925 a^{5} b^{3} d^{7} x^{3}+12250 a^{4} b^{4} d^{7} x^{4}+4410 a^{3} b^{5} d^{7} x^{5}-420 a \,b^{7} d^{7} x^{7}}{420 a \,b^{8} \left (b^{7} x^{7}+7 a \,b^{6} x^{6}+21 a^{2} b^{5} x^{5}+35 a^{3} b^{4} x^{4}+35 a^{4} b^{3} x^{3}+21 a^{5} b^{2} x^{2}+7 a^{6} b x +a^{7}\right )} \]
int((c**7 + 7*c**6*d*x + 21*c**5*d**2*x**2 + 35*c**4*d**3*x**3 + 35*c**3*d **4*x**4 + 21*c**2*d**5*x**5 + 7*c*d**6*x**6 + d**7*x**7)/(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),x)
(420*log(a + b*x)*a**8*d**7 + 2940*log(a + b*x)*a**7*b*d**7*x + 8820*log(a + b*x)*a**6*b**2*d**7*x**2 + 14700*log(a + b*x)*a**5*b**3*d**7*x**3 + 147 00*log(a + b*x)*a**4*b**4*d**7*x**4 + 8820*log(a + b*x)*a**3*b**5*d**7*x** 5 + 2940*log(a + b*x)*a**2*b**6*d**7*x**6 + 420*log(a + b*x)*a*b**7*d**7*x **7 + 669*a**8*d**7 + 4263*a**7*b*d**7*x - 210*a**6*b**2*c**2*d**5 + 11319 *a**6*b**2*d**7*x**2 - 140*a**5*b**3*c**3*d**4 - 1470*a**5*b**3*c**2*d**5* x + 15925*a**5*b**3*d**7*x**3 - 105*a**4*b**4*c**4*d**3 - 980*a**4*b**4*c* *3*d**4*x - 4410*a**4*b**4*c**2*d**5*x**2 + 12250*a**4*b**4*d**7*x**4 - 84 *a**3*b**5*c**5*d**2 - 735*a**3*b**5*c**4*d**3*x - 2940*a**3*b**5*c**3*d** 4*x**2 - 7350*a**3*b**5*c**2*d**5*x**3 + 4410*a**3*b**5*d**7*x**5 - 70*a** 2*b**6*c**6*d - 588*a**2*b**6*c**5*d**2*x - 2205*a**2*b**6*c**4*d**3*x**2 - 4900*a**2*b**6*c**3*d**4*x**3 - 7350*a**2*b**6*c**2*d**5*x**4 - 60*a*b** 7*c**7 - 490*a*b**7*c**6*d*x - 1764*a*b**7*c**5*d**2*x**2 - 3675*a*b**7*c* *4*d**3*x**3 - 4900*a*b**7*c**3*d**4*x**4 - 4410*a*b**7*c**2*d**5*x**5 - 4 20*a*b**7*d**7*x**7 + 420*b**8*c*d**6*x**7)/(420*a*b**8*(a**7 + 7*a**6*b*x + 21*a**5*b**2*x**2 + 35*a**4*b**3*x**3 + 35*a**3*b**4*x**4 + 21*a**2*b** 5*x**5 + 7*a*b**6*x**6 + b**7*x**7))