Integrand size = 15, antiderivative size = 202 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx=\frac {1}{7 (b c-a d) (c+d x)^7}+\frac {b}{6 (b c-a d)^2 (c+d x)^6}+\frac {b^2}{5 (b c-a d)^3 (c+d x)^5}+\frac {b^3}{4 (b c-a d)^4 (c+d x)^4}+\frac {b^4}{3 (b c-a d)^5 (c+d x)^3}+\frac {b^5}{2 (b c-a d)^6 (c+d x)^2}+\frac {b^6}{(b c-a d)^7 (c+d x)}+\frac {b^7 \log (a+b x)}{(b c-a d)^8}-\frac {b^7 \log (c+d x)}{(b c-a d)^8} \]
1/7/(-a*d+b*c)/(d*x+c)^7+1/6*b/(-a*d+b*c)^2/(d*x+c)^6+1/5*b^2/(-a*d+b*c)^3 /(d*x+c)^5+1/4*b^3/(-a*d+b*c)^4/(d*x+c)^4+1/3*b^4/(-a*d+b*c)^5/(d*x+c)^3+1 /2*b^5/(-a*d+b*c)^6/(d*x+c)^2+b^6/(-a*d+b*c)^7/(d*x+c)+b^7*ln(b*x+a)/(-a*d +b*c)^8-b^7*ln(d*x+c)/(-a*d+b*c)^8
Time = 0.05 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx=\frac {60 (b c-a d)^7+70 b (b c-a d)^6 (c+d x)+84 b^2 (b c-a d)^5 (c+d x)^2+105 b^3 (b c-a d)^4 (c+d x)^3+140 b^4 (b c-a d)^3 (c+d x)^4+210 b^5 (b c-a d)^2 (c+d x)^5+420 b^6 (b c-a d) (c+d x)^6+420 b^7 (c+d x)^7 \log (a+b x)-420 b^7 (c+d x)^7 \log (c+d x)}{420 (b c-a d)^8 (c+d x)^7} \]
(60*(b*c - a*d)^7 + 70*b*(b*c - a*d)^6*(c + d*x) + 84*b^2*(b*c - a*d)^5*(c + d*x)^2 + 105*b^3*(b*c - a*d)^4*(c + d*x)^3 + 140*b^4*(b*c - a*d)^3*(c + d*x)^4 + 210*b^5*(b*c - a*d)^2*(c + d*x)^5 + 420*b^6*(b*c - a*d)*(c + d*x )^6 + 420*b^7*(c + d*x)^7*Log[a + b*x] - 420*b^7*(c + d*x)^7*Log[c + d*x]) /(420*(b*c - a*d)^8*(c + d*x)^7)
Time = 0.40 (sec) , antiderivative size = 202, 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.
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x)^8} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (\frac {b^8}{(a+b x) (b c-a d)^8}-\frac {b^7 d}{(c+d x) (b c-a d)^8}-\frac {b^6 d}{(c+d x)^2 (b c-a d)^7}-\frac {b^5 d}{(c+d x)^3 (b c-a d)^6}-\frac {b^4 d}{(c+d x)^4 (b c-a d)^5}-\frac {b^3 d}{(c+d x)^5 (b c-a d)^4}-\frac {b^2 d}{(c+d x)^6 (b c-a d)^3}-\frac {b d}{(c+d x)^7 (b c-a d)^2}-\frac {d}{(c+d x)^8 (b c-a d)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^7 \log (a+b x)}{(b c-a d)^8}-\frac {b^7 \log (c+d x)}{(b c-a d)^8}+\frac {b^6}{(c+d x) (b c-a d)^7}+\frac {b^5}{2 (c+d x)^2 (b c-a d)^6}+\frac {b^4}{3 (c+d x)^3 (b c-a d)^5}+\frac {b^3}{4 (c+d x)^4 (b c-a d)^4}+\frac {b^2}{5 (c+d x)^5 (b c-a d)^3}+\frac {b}{6 (c+d x)^6 (b c-a d)^2}+\frac {1}{7 (c+d x)^7 (b c-a d)}\) |
1/(7*(b*c - a*d)*(c + d*x)^7) + b/(6*(b*c - a*d)^2*(c + d*x)^6) + b^2/(5*( b*c - a*d)^3*(c + d*x)^5) + b^3/(4*(b*c - a*d)^4*(c + d*x)^4) + b^4/(3*(b* c - a*d)^5*(c + d*x)^3) + b^5/(2*(b*c - a*d)^6*(c + d*x)^2) + b^6/((b*c - a*d)^7*(c + d*x)) + (b^7*Log[a + b*x])/(b*c - a*d)^8 - (b^7*Log[c + d*x])/ (b*c - a*d)^8
3.14.72.3.1 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 = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {1}{7 \left (a d -b c \right ) \left (d x +c \right )^{7}}-\frac {b^{2}}{5 \left (a d -b c \right )^{3} \left (d x +c \right )^{5}}-\frac {b^{4}}{3 \left (a d -b c \right )^{5} \left (d x +c \right )^{3}}-\frac {b^{6}}{\left (a d -b c \right )^{7} \left (d x +c \right )}+\frac {b}{6 \left (a d -b c \right )^{2} \left (d x +c \right )^{6}}+\frac {b^{3}}{4 \left (a d -b c \right )^{4} \left (d x +c \right )^{4}}+\frac {b^{5}}{2 \left (a d -b c \right )^{6} \left (d x +c \right )^{2}}-\frac {b^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{8}}+\frac {b^{7} \ln \left (b x +a \right )}{\left (a d -b c \right )^{8}}\) | \(192\) |
| parallelrisch | \(\frac {490 a^{6} b c \,d^{13}-1764 a^{5} b^{2} c^{2} d^{12}+3675 a^{4} b^{3} c^{3} d^{11}-4900 a^{3} b^{4} c^{4} d^{10}+4410 a^{2} b^{5} c^{5} d^{9}-2940 a \,b^{6} c^{6} d^{8}-420 x^{6} a \,b^{6} d^{14}+420 x^{6} b^{7} c \,d^{13}+210 x^{5} a^{2} b^{5} d^{14}+2730 x^{5} b^{7} c^{2} d^{12}-140 x^{4} a^{3} b^{4} d^{14}+7490 x^{4} b^{7} c^{3} d^{11}+105 x^{3} a^{4} b^{3} d^{14}+11165 x^{3} b^{7} c^{4} d^{10}-84 x^{2} a^{5} b^{2} d^{14}+9639 x^{2} b^{7} c^{5} d^{9}+70 x \,a^{6} b \,d^{14}+4683 x \,b^{7} c^{6} d^{8}+420 \ln \left (b x +a \right ) x^{7} b^{7} d^{14}-420 \ln \left (d x +c \right ) x^{7} b^{7} d^{14}+420 \ln \left (b x +a \right ) b^{7} c^{7} d^{7}-420 \ln \left (d x +c \right ) b^{7} c^{7} d^{7}-60 a^{7} d^{14}+735 x^{2} a^{4} b^{3} c \,d^{13}-2940 x^{2} a^{3} b^{4} c^{2} d^{12}+7350 x^{2} a^{2} b^{5} c^{3} d^{11}-14700 x^{2} a \,b^{6} c^{4} d^{10}-588 x \,a^{5} b^{2} c \,d^{13}+2205 x \,a^{4} b^{3} c^{2} d^{12}-4900 x \,a^{3} b^{4} c^{3} d^{11}+7350 x \,a^{2} b^{5} c^{4} d^{10}-8820 x a \,b^{6} c^{5} d^{9}-2940 \ln \left (d x +c \right ) x^{6} b^{7} c \,d^{13}+8820 \ln \left (b x +a \right ) x^{5} b^{7} c^{2} d^{12}-8820 \ln \left (d x +c \right ) x^{5} b^{7} c^{2} d^{12}+14700 \ln \left (b x +a \right ) x^{4} b^{7} c^{3} d^{11}-14700 \ln \left (d x +c \right ) x^{4} b^{7} c^{3} d^{11}+14700 \ln \left (b x +a \right ) x^{3} b^{7} c^{4} d^{10}-14700 \ln \left (d x +c \right ) x^{3} b^{7} c^{4} d^{10}+8820 \ln \left (b x +a \right ) x^{2} b^{7} c^{5} d^{9}-8820 \ln \left (d x +c \right ) x^{2} b^{7} c^{5} d^{9}+2940 \ln \left (b x +a \right ) x \,b^{7} c^{6} d^{8}-2940 \ln \left (d x +c \right ) x \,b^{7} c^{6} d^{8}+1089 b^{7} c^{7} d^{7}+2940 \ln \left (b x +a \right ) x^{6} b^{7} c \,d^{13}-2940 x^{5} a \,b^{6} c \,d^{13}+1470 x^{4} a^{2} b^{5} c \,d^{13}-8820 x^{4} a \,b^{6} c^{2} d^{12}-980 x^{3} a^{3} b^{4} c \,d^{13}+4410 x^{3} a^{2} b^{5} c^{2} d^{12}-14700 x^{3} a \,b^{6} c^{3} d^{11}}{420 \left (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}\right ) \left (d x +c \right )^{7} d^{7}}\) | \(901\) |
| risch | \(\text {Expression too large to display}\) | \(1232\) |
| norman | \(\text {Expression too large to display}\) | \(1274\) |
-1/7/(a*d-b*c)/(d*x+c)^7-1/5*b^2/(a*d-b*c)^3/(d*x+c)^5-1/3*b^4/(a*d-b*c)^5 /(d*x+c)^3-b^6/(a*d-b*c)^7/(d*x+c)+1/6*b/(a*d-b*c)^2/(d*x+c)^6+1/4*b^3/(a* d-b*c)^4/(d*x+c)^4+1/2*b^5/(a*d-b*c)^6/(d*x+c)^2-b^7/(a*d-b*c)^8*ln(d*x+c) +b^7/(a*d-b*c)^8*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1589 vs. \(2 (190) = 380\).
Time = 0.27 (sec) , antiderivative size = 1589, normalized size of antiderivative = 7.87 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx=\text {Too large to display} \]
1/420*(1089*b^7*c^7 - 2940*a*b^6*c^6*d + 4410*a^2*b^5*c^5*d^2 - 4900*a^3*b ^4*c^4*d^3 + 3675*a^4*b^3*c^3*d^4 - 1764*a^5*b^2*c^2*d^5 + 490*a^6*b*c*d^6 - 60*a^7*d^7 + 420*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 210*(13*b^7*c^2*d^5 - 14 *a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 70*(107*b^7*c^3*d^4 - 126*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 35*(319*b^7*c^4*d^3 - 420*a*b^6* c^3*d^4 + 126*a^2*b^5*c^2*d^5 - 28*a^3*b^4*c*d^6 + 3*a^4*b^3*d^7)*x^3 + 21 *(459*b^7*c^5*d^2 - 700*a*b^6*c^4*d^3 + 350*a^2*b^5*c^3*d^4 - 140*a^3*b^4* c^2*d^5 + 35*a^4*b^3*c*d^6 - 4*a^5*b^2*d^7)*x^2 + 7*(669*b^7*c^6*d - 1260* a*b^6*c^5*d^2 + 1050*a^2*b^5*c^4*d^3 - 700*a^3*b^4*c^3*d^4 + 315*a^4*b^3*c ^2*d^5 - 84*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(b*x + a) - 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))/(b^ 8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a ^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d ^7 + a^8*c^7*d^8 + (b^8*c^8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 5 6*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^ 2*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15)*x^7 + 7*(b^8*c^9*d^6 - 8*a*b^7*c^8 *d^7 + 28*a^2*b^6*c^7*d^8 - 56*a^3*b^5*c^6*d^9 + 70*a^4*b^4*c^5*d^10 - ...
Leaf count of result is larger than twice the leaf count of optimal. 1776 vs. \(2 (170) = 340\).
Time = 8.34 (sec) , antiderivative size = 1776, normalized size of antiderivative = 8.79 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx=\text {Too large to display} \]
-b**7*log(x + (-a**9*b**7*d**9/(a*d - b*c)**8 + 9*a**8*b**8*c*d**8/(a*d - b*c)**8 - 36*a**7*b**9*c**2*d**7/(a*d - b*c)**8 + 84*a**6*b**10*c**3*d**6/ (a*d - b*c)**8 - 126*a**5*b**11*c**4*d**5/(a*d - b*c)**8 + 126*a**4*b**12* c**5*d**4/(a*d - b*c)**8 - 84*a**3*b**13*c**6*d**3/(a*d - b*c)**8 + 36*a** 2*b**14*c**7*d**2/(a*d - b*c)**8 - 9*a*b**15*c**8*d/(a*d - b*c)**8 + a*b** 7*d + b**16*c**9/(a*d - b*c)**8 + b**8*c)/(2*b**8*d))/(a*d - b*c)**8 + b** 7*log(x + (a**9*b**7*d**9/(a*d - b*c)**8 - 9*a**8*b**8*c*d**8/(a*d - b*c)* *8 + 36*a**7*b**9*c**2*d**7/(a*d - b*c)**8 - 84*a**6*b**10*c**3*d**6/(a*d - b*c)**8 + 126*a**5*b**11*c**4*d**5/(a*d - b*c)**8 - 126*a**4*b**12*c**5* d**4/(a*d - b*c)**8 + 84*a**3*b**13*c**6*d**3/(a*d - b*c)**8 - 36*a**2*b** 14*c**7*d**2/(a*d - b*c)**8 + 9*a*b**15*c**8*d/(a*d - b*c)**8 + a*b**7*d - b**16*c**9/(a*d - b*c)**8 + b**8*c)/(2*b**8*d))/(a*d - b*c)**8 + (-60*a** 6*d**6 + 430*a**5*b*c*d**5 - 1334*a**4*b**2*c**2*d**4 + 2341*a**3*b**3*c** 3*d**3 - 2559*a**2*b**4*c**4*d**2 + 1851*a*b**5*c**5*d - 1089*b**6*c**6 - 420*b**6*d**6*x**6 + x**5*(210*a*b**5*d**6 - 2730*b**6*c*d**5) + x**4*(-14 0*a**2*b**4*d**6 + 1330*a*b**5*c*d**5 - 7490*b**6*c**2*d**4) + x**3*(105*a **3*b**3*d**6 - 875*a**2*b**4*c*d**5 + 3535*a*b**5*c**2*d**4 - 11165*b**6* c**3*d**3) + x**2*(-84*a**4*b**2*d**6 + 651*a**3*b**3*c*d**5 - 2289*a**2*b **4*c**2*d**4 + 5061*a*b**5*c**3*d**3 - 9639*b**6*c**4*d**2) + x*(70*a**5* b*d**6 - 518*a**4*b**2*c*d**5 + 1687*a**3*b**3*c**2*d**4 - 3213*a**2*b*...
Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (190) = 380\).
Time = 0.29 (sec) , antiderivative size = 1418, normalized size of antiderivative = 7.02 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx=\text {Too large to display} \]
b^7*log(b*x + a)/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^ 5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8) - b^7*log(d*x + c)/(b^8*c^8 - 8*a*b^7*c^7*d + 28 *a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^ 3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8) + 1/420*(420*b^6*d^6 *x^6 + 1089*b^6*c^6 - 1851*a*b^5*c^5*d + 2559*a^2*b^4*c^4*d^2 - 2341*a^3*b ^3*c^3*d^3 + 1334*a^4*b^2*c^2*d^4 - 430*a^5*b*c*d^5 + 60*a^6*d^6 + 210*(13 *b^6*c*d^5 - a*b^5*d^6)*x^5 + 70*(107*b^6*c^2*d^4 - 19*a*b^5*c*d^5 + 2*a^2 *b^4*d^6)*x^4 + 35*(319*b^6*c^3*d^3 - 101*a*b^5*c^2*d^4 + 25*a^2*b^4*c*d^5 - 3*a^3*b^3*d^6)*x^3 + 21*(459*b^6*c^4*d^2 - 241*a*b^5*c^3*d^3 + 109*a^2* b^4*c^2*d^4 - 31*a^3*b^3*c*d^5 + 4*a^4*b^2*d^6)*x^2 + 7*(669*b^6*c^5*d - 5 91*a*b^5*c^4*d^2 + 459*a^2*b^4*c^3*d^3 - 241*a^3*b^3*c^2*d^4 + 74*a^4*b^2* c*d^5 - 10*a^5*b*d^6)*x)/(b^7*c^14 - 7*a*b^6*c^13*d + 21*a^2*b^5*c^12*d^2 - 35*a^3*b^4*c^11*d^3 + 35*a^4*b^3*c^10*d^4 - 21*a^5*b^2*c^9*d^5 + 7*a^6*b *c^8*d^6 - a^7*c^7*d^7 + (b^7*c^7*d^7 - 7*a*b^6*c^6*d^8 + 21*a^2*b^5*c^5*d ^9 - 35*a^3*b^4*c^4*d^10 + 35*a^4*b^3*c^3*d^11 - 21*a^5*b^2*c^2*d^12 + 7*a ^6*b*c*d^13 - a^7*d^14)*x^7 + 7*(b^7*c^8*d^6 - 7*a*b^6*c^7*d^7 + 21*a^2*b^ 5*c^6*d^8 - 35*a^3*b^4*c^5*d^9 + 35*a^4*b^3*c^4*d^10 - 21*a^5*b^2*c^3*d^11 + 7*a^6*b*c^2*d^12 - a^7*c*d^13)*x^6 + 21*(b^7*c^9*d^5 - 7*a*b^6*c^8*d^6 + 21*a^2*b^5*c^7*d^7 - 35*a^3*b^4*c^6*d^8 + 35*a^4*b^3*c^5*d^9 - 21*a^5...
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (190) = 380\).
Time = 0.33 (sec) , antiderivative size = 703, normalized size of antiderivative = 3.48 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx=\frac {b^{8} \log \left ({\left | b x + a \right |}\right )}{b^{9} c^{8} - 8 \, a b^{8} c^{7} d + 28 \, a^{2} b^{7} c^{6} d^{2} - 56 \, a^{3} b^{6} c^{5} d^{3} + 70 \, a^{4} b^{5} c^{4} d^{4} - 56 \, a^{5} b^{4} c^{3} d^{5} + 28 \, a^{6} b^{3} c^{2} d^{6} - 8 \, a^{7} b^{2} c d^{7} + a^{8} b d^{8}} - \frac {b^{7} d \log \left ({\left | d x + c \right |}\right )}{b^{8} c^{8} d - 8 \, a b^{7} c^{7} d^{2} + 28 \, a^{2} b^{6} c^{6} d^{3} - 56 \, a^{3} b^{5} c^{5} d^{4} + 70 \, a^{4} b^{4} c^{4} d^{5} - 56 \, a^{5} b^{3} c^{3} d^{6} + 28 \, a^{6} b^{2} c^{2} d^{7} - 8 \, a^{7} b c d^{8} + a^{8} d^{9}} + \frac {1089 \, b^{7} c^{7} - 2940 \, a b^{6} c^{6} d + 4410 \, a^{2} b^{5} c^{5} d^{2} - 4900 \, a^{3} b^{4} c^{4} d^{3} + 3675 \, a^{4} b^{3} c^{3} d^{4} - 1764 \, a^{5} b^{2} c^{2} d^{5} + 490 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 420 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 210 \, {\left (13 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (107 \, b^{7} c^{3} d^{4} - 126 \, a b^{6} c^{2} d^{5} + 21 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 35 \, {\left (319 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 126 \, a^{2} b^{5} c^{2} d^{5} - 28 \, a^{3} b^{4} c d^{6} + 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 21 \, {\left (459 \, b^{7} c^{5} d^{2} - 700 \, a b^{6} c^{4} d^{3} + 350 \, a^{2} b^{5} c^{3} d^{4} - 140 \, a^{3} b^{4} c^{2} d^{5} + 35 \, a^{4} b^{3} c d^{6} - 4 \, a^{5} b^{2} d^{7}\right )} x^{2} + 7 \, {\left (669 \, b^{7} c^{6} d - 1260 \, a b^{6} c^{5} d^{2} + 1050 \, a^{2} b^{5} c^{4} d^{3} - 700 \, a^{3} b^{4} c^{3} d^{4} + 315 \, a^{4} b^{3} c^{2} d^{5} - 84 \, a^{5} b^{2} c d^{6} + 10 \, a^{6} b d^{7}\right )} x}{420 \, {\left (b c - a d\right )}^{8} {\left (d x + c\right )}^{7}} \]
b^8*log(abs(b*x + a))/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a ^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2* d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8) - b^7*d*log(abs(d*x + c))/(b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4 *d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8 + a^8*d^9) + 1/420*(1089*b^7*c^7 - 2940*a*b^6*c^6*d + 4410*a^2*b^5*c^5*d^2 - 4900*a^3 *b^4*c^4*d^3 + 3675*a^4*b^3*c^3*d^4 - 1764*a^5*b^2*c^2*d^5 + 490*a^6*b*c*d ^6 - 60*a^7*d^7 + 420*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 210*(13*b^7*c^2*d^5 - 14*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 70*(107*b^7*c^3*d^4 - 126*a*b^6*c^2*d^ 5 + 21*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 35*(319*b^7*c^4*d^3 - 420*a*b^ 6*c^3*d^4 + 126*a^2*b^5*c^2*d^5 - 28*a^3*b^4*c*d^6 + 3*a^4*b^3*d^7)*x^3 + 21*(459*b^7*c^5*d^2 - 700*a*b^6*c^4*d^3 + 350*a^2*b^5*c^3*d^4 - 140*a^3*b^ 4*c^2*d^5 + 35*a^4*b^3*c*d^6 - 4*a^5*b^2*d^7)*x^2 + 7*(669*b^7*c^6*d - 126 0*a*b^6*c^5*d^2 + 1050*a^2*b^5*c^4*d^3 - 700*a^3*b^4*c^3*d^4 + 315*a^4*b^3 *c^2*d^5 - 84*a^5*b^2*c*d^6 + 10*a^6*b*d^7)*x)/((b*c - a*d)^8*(d*x + c)^7)
Time = 1.02 (sec) , antiderivative size = 1299, normalized size of antiderivative = 6.43 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx =\text {Too large to display} \]
(2*b^7*atanh((a^8*d^8 - b^8*c^8 - 14*a^2*b^6*c^6*d^2 + 14*a^3*b^5*c^5*d^3 - 14*a^5*b^3*c^3*d^5 + 14*a^6*b^2*c^2*d^6 + 6*a*b^7*c^7*d - 6*a^7*b*c*d^7) /(a*d - b*c)^8 + (2*b*d*x*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3 *b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7 *a^6*b*c*d^6))/(a*d - b*c)^8))/(a*d - b*c)^8 - ((60*a^6*d^6 + 1089*b^6*c^6 + 2559*a^2*b^4*c^4*d^2 - 2341*a^3*b^3*c^3*d^3 + 1334*a^4*b^2*c^2*d^4 - 18 51*a*b^5*c^5*d - 430*a^5*b*c*d^5)/(420*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5 *d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a* b^6*c^6*d - 7*a^6*b*c*d^6)) - (b^3*x^3*(3*a^3*d^6 - 319*b^3*c^3*d^3 + 101* a*b^2*c^2*d^4 - 25*a^2*b*c*d^5))/(12*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d ^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^ 6*c^6*d - 7*a^6*b*c*d^6)) + (b^6*d^6*x^6)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5* c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7 *a*b^6*c^6*d - 7*a^6*b*c*d^6) - (b^5*x^5*(a*d^6 - 13*b*c*d^5))/(2*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) + (b^2*x^2*(4*a^4*d ^6 + 459*b^4*c^4*d^2 - 241*a*b^3*c^3*d^3 + 109*a^2*b^2*c^2*d^4 - 31*a^3*b* c*d^5))/(20*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) + (b^4*x^4*(2*a^2*d^6 + 107*b^2*c^2*d^4 - 19*a*b*c*d^5))/(6*(a^7*d^7 -...
Time = 0.01 (sec) , antiderivative size = 1877, normalized size of antiderivative = 9.29 \[ \int \frac {1}{(a+b x) (c+d x)^8} \, dx =\text {Too large to display} \]
int(1/(a*c**8 + 8*a*c**7*d*x + 28*a*c**6*d**2*x**2 + 56*a*c**5*d**3*x**3 + 70*a*c**4*d**4*x**4 + 56*a*c**3*d**5*x**5 + 28*a*c**2*d**6*x**6 + 8*a*c*d **7*x**7 + a*d**8*x**8 + b*c**8*x + 8*b*c**7*d*x**2 + 28*b*c**6*d**2*x**3 + 56*b*c**5*d**3*x**4 + 70*b*c**4*d**4*x**5 + 56*b*c**3*d**5*x**6 + 28*b*c **2*d**6*x**7 + 8*b*c*d**7*x**8 + b*d**8*x**9),x)
(420*log(a + b*x)*b**7*c**8 + 2940*log(a + b*x)*b**7*c**7*d*x + 8820*log(a + b*x)*b**7*c**6*d**2*x**2 + 14700*log(a + b*x)*b**7*c**5*d**3*x**3 + 147 00*log(a + b*x)*b**7*c**4*d**4*x**4 + 8820*log(a + b*x)*b**7*c**3*d**5*x** 5 + 2940*log(a + b*x)*b**7*c**2*d**6*x**6 + 420*log(a + b*x)*b**7*c*d**7*x **7 - 420*log(c + d*x)*b**7*c**8 - 2940*log(c + d*x)*b**7*c**7*d*x - 8820* log(c + d*x)*b**7*c**6*d**2*x**2 - 14700*log(c + d*x)*b**7*c**5*d**3*x**3 - 14700*log(c + d*x)*b**7*c**4*d**4*x**4 - 8820*log(c + d*x)*b**7*c**3*d** 5*x**5 - 2940*log(c + d*x)*b**7*c**2*d**6*x**6 - 420*log(c + d*x)*b**7*c*d **7*x**7 - 60*a**7*c*d**7 + 490*a**6*b*c**2*d**6 + 70*a**6*b*c*d**7*x - 17 64*a**5*b**2*c**3*d**5 - 588*a**5*b**2*c**2*d**6*x - 84*a**5*b**2*c*d**7*x **2 + 3675*a**4*b**3*c**4*d**4 + 2205*a**4*b**3*c**3*d**5*x + 735*a**4*b** 3*c**2*d**6*x**2 + 105*a**4*b**3*c*d**7*x**3 - 4900*a**3*b**4*c**5*d**3 - 4900*a**3*b**4*c**4*d**4*x - 2940*a**3*b**4*c**3*d**5*x**2 - 980*a**3*b**4 *c**2*d**6*x**3 - 140*a**3*b**4*c*d**7*x**4 + 4410*a**2*b**5*c**6*d**2 + 7 350*a**2*b**5*c**5*d**3*x + 7350*a**2*b**5*c**4*d**4*x**2 + 4410*a**2*b**5 *c**3*d**5*x**3 + 1470*a**2*b**5*c**2*d**6*x**4 + 210*a**2*b**5*c*d**7*x** 5 - 2880*a*b**6*c**7*d - 8400*a*b**6*c**6*d**2*x - 13440*a*b**6*c**5*d**3* x**2 - 12600*a*b**6*c**4*d**4*x**3 - 6720*a*b**6*c**3*d**5*x**4 - 1680*a*b **6*c**2*d**6*x**5 + 60*a*b**6*d**8*x**7 + 1029*b**7*c**8 + 4263*b**7*c**7 *d*x + 8379*b**7*c**6*d**2*x**2 + 9065*b**7*c**5*d**3*x**3 + 5390*b**7*...