Integrand size = 15, antiderivative size = 231 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=-\frac {b^7}{(b c-a d)^8 (a+b x)}-\frac {d}{7 (b c-a d)^2 (c+d x)^7}-\frac {b d}{3 (b c-a d)^3 (c+d x)^6}-\frac {3 b^2 d}{5 (b c-a d)^4 (c+d x)^5}-\frac {b^3 d}{(b c-a d)^5 (c+d x)^4}-\frac {5 b^4 d}{3 (b c-a d)^6 (c+d x)^3}-\frac {3 b^5 d}{(b c-a d)^7 (c+d x)^2}-\frac {7 b^6 d}{(b c-a d)^8 (c+d x)}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9} \]
-b^7/(-a*d+b*c)^8/(b*x+a)-1/7*d/(-a*d+b*c)^2/(d*x+c)^7-1/3*b*d/(-a*d+b*c)^ 3/(d*x+c)^6-3/5*b^2*d/(-a*d+b*c)^4/(d*x+c)^5-b^3*d/(-a*d+b*c)^5/(d*x+c)^4- 5/3*b^4*d/(-a*d+b*c)^6/(d*x+c)^3-3*b^5*d/(-a*d+b*c)^7/(d*x+c)^2-7*b^6*d/(- a*d+b*c)^8/(d*x+c)-8*b^7*d*ln(b*x+a)/(-a*d+b*c)^9+8*b^7*d*ln(d*x+c)/(-a*d+ b*c)^9
Time = 0.14 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=-\frac {\frac {105 b^7 (b c-a d)}{a+b x}-\frac {15 d (-b c+a d)^7}{(c+d x)^7}+\frac {35 b d (b c-a d)^6}{(c+d x)^6}+\frac {63 b^2 d (b c-a d)^5}{(c+d x)^5}+\frac {105 b^3 d (b c-a d)^4}{(c+d x)^4}+\frac {175 b^4 d (b c-a d)^3}{(c+d x)^3}+\frac {315 b^5 d (b c-a d)^2}{(c+d x)^2}+\frac {735 b^6 d (b c-a d)}{c+d x}+840 b^7 d \log (a+b x)-840 b^7 d \log (c+d x)}{105 (b c-a d)^9} \]
-1/105*((105*b^7*(b*c - a*d))/(a + b*x) - (15*d*(-(b*c) + a*d)^7)/(c + d*x )^7 + (35*b*d*(b*c - a*d)^6)/(c + d*x)^6 + (63*b^2*d*(b*c - a*d)^5)/(c + d *x)^5 + (105*b^3*d*(b*c - a*d)^4)/(c + d*x)^4 + (175*b^4*d*(b*c - a*d)^3)/ (c + d*x)^3 + (315*b^5*d*(b*c - a*d)^2)/(c + d*x)^2 + (735*b^6*d*(b*c - a* d))/(c + d*x) + 840*b^7*d*Log[a + b*x] - 840*b^7*d*Log[c + d*x])/(b*c - a* d)^9
Time = 0.50 (sec) , antiderivative size = 231, 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.
\(\displaystyle \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (-\frac {8 b^8 d}{(a+b x) (b c-a d)^9}+\frac {b^8}{(a+b x)^2 (b c-a d)^8}+\frac {8 b^7 d^2}{(c+d x) (b c-a d)^9}+\frac {7 b^6 d^2}{(c+d x)^2 (b c-a d)^8}+\frac {6 b^5 d^2}{(c+d x)^3 (b c-a d)^7}+\frac {5 b^4 d^2}{(c+d x)^4 (b c-a d)^6}+\frac {4 b^3 d^2}{(c+d x)^5 (b c-a d)^5}+\frac {3 b^2 d^2}{(c+d x)^6 (b c-a d)^4}+\frac {2 b d^2}{(c+d x)^7 (b c-a d)^3}+\frac {d^2}{(c+d x)^8 (b c-a d)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^7}{(a+b x) (b c-a d)^8}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9}-\frac {7 b^6 d}{(c+d x) (b c-a d)^8}-\frac {3 b^5 d}{(c+d x)^2 (b c-a d)^7}-\frac {5 b^4 d}{3 (c+d x)^3 (b c-a d)^6}-\frac {b^3 d}{(c+d x)^4 (b c-a d)^5}-\frac {3 b^2 d}{5 (c+d x)^5 (b c-a d)^4}-\frac {b d}{3 (c+d x)^6 (b c-a d)^3}-\frac {d}{7 (c+d x)^7 (b c-a d)^2}\) |
-(b^7/((b*c - a*d)^8*(a + b*x))) - d/(7*(b*c - a*d)^2*(c + d*x)^7) - (b*d) /(3*(b*c - a*d)^3*(c + d*x)^6) - (3*b^2*d)/(5*(b*c - a*d)^4*(c + d*x)^5) - (b^3*d)/((b*c - a*d)^5*(c + d*x)^4) - (5*b^4*d)/(3*(b*c - a*d)^6*(c + d*x )^3) - (3*b^5*d)/((b*c - a*d)^7*(c + d*x)^2) - (7*b^6*d)/((b*c - a*d)^8*(c + d*x)) - (8*b^7*d*Log[a + b*x])/(b*c - a*d)^9 + (8*b^7*d*Log[c + d*x])/( b*c - a*d)^9
3.14.73.3.1 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 = 0.37 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {d}{7 \left (a d -b c \right )^{2} \left (d x +c \right )^{7}}-\frac {8 d \,b^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{9}}-\frac {7 d \,b^{6}}{\left (a d -b c \right )^{8} \left (d x +c \right )}+\frac {3 d \,b^{5}}{\left (a d -b c \right )^{7} \left (d x +c \right )^{2}}-\frac {5 d \,b^{4}}{3 \left (a d -b c \right )^{6} \left (d x +c \right )^{3}}+\frac {d \,b^{3}}{\left (a d -b c \right )^{5} \left (d x +c \right )^{4}}-\frac {3 d \,b^{2}}{5 \left (a d -b c \right )^{4} \left (d x +c \right )^{5}}+\frac {d b}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{6}}-\frac {b^{7}}{\left (a d -b c \right )^{8} \left (b x +a \right )}+\frac {8 d \,b^{7} \ln \left (b x +a \right )}{\left (a d -b c \right )^{9}}\) | \(223\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1391\) |
risch | \(\text {Expression too large to display}\) | \(1572\) |
norman | \(\text {Expression too large to display}\) | \(1649\) |
-1/7*d/(a*d-b*c)^2/(d*x+c)^7-8*d/(a*d-b*c)^9*b^7*ln(d*x+c)-7*d/(a*d-b*c)^8 *b^6/(d*x+c)+3*d/(a*d-b*c)^7*b^5/(d*x+c)^2-5/3*d/(a*d-b*c)^6*b^4/(d*x+c)^3 +d/(a*d-b*c)^5*b^3/(d*x+c)^4-3/5*d/(a*d-b*c)^4*b^2/(d*x+c)^5+1/3*d/(a*d-b* c)^3*b/(d*x+c)^6-b^7/(a*d-b*c)^8/(b*x+a)+8*d/(a*d-b*c)^9*b^7*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 2264 vs. \(2 (223) = 446\).
Time = 0.29 (sec) , antiderivative size = 2264, normalized size of antiderivative = 9.80 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=\text {Too large to display} \]
-1/105*(105*b^8*c^8 + 1338*a*b^7*c^7*d - 2940*a^2*b^6*c^6*d^2 + 2940*a^3*b ^5*c^5*d^3 - 2450*a^4*b^4*c^4*d^4 + 1470*a^5*b^3*c^3*d^5 - 588*a^6*b^2*c^2 *d^6 + 140*a^7*b*c*d^7 - 15*a^8*d^8 + 840*(b^8*c*d^7 - a*b^7*d^8)*x^7 + 42 0*(13*b^8*c^2*d^6 - 12*a*b^7*c*d^7 - a^2*b^6*d^8)*x^6 + 140*(107*b^8*c^3*d ^5 - 87*a*b^7*c^2*d^6 - 21*a^2*b^6*c*d^7 + a^3*b^5*d^8)*x^5 + 70*(319*b^8* c^4*d^4 - 206*a*b^7*c^3*d^5 - 126*a^2*b^6*c^2*d^6 + 14*a^3*b^5*c*d^7 - a^4 *b^4*d^8)*x^4 + 14*(1377*b^8*c^5*d^3 - 505*a*b^7*c^4*d^4 - 1050*a^2*b^6*c^ 3*d^5 + 210*a^3*b^5*c^2*d^6 - 35*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 14*( 669*b^8*c^6*d^2 + 117*a*b^7*c^5*d^3 - 1050*a^2*b^6*c^4*d^4 + 350*a^3*b^5*c ^3*d^5 - 105*a^4*b^4*c^2*d^6 + 21*a^5*b^3*c*d^7 - 2*a^6*b^2*d^8)*x^2 + 2*( 1089*b^8*c^7*d + 1743*a*b^7*c^6*d^2 - 4410*a^2*b^6*c^5*d^3 + 2450*a^3*b^5* c^4*d^4 - 1225*a^4*b^4*c^3*d^5 + 441*a^5*b^3*c^2*d^6 - 98*a^6*b^2*c*d^7 + 10*a^7*b*d^8)*x + 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^ 8)*x^7 + 7*(3*b^8*c^2*d^6 + a*b^7*c*d^7)*x^6 + 7*(5*b^8*c^3*d^5 + 3*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 + 7*(3*b^8*c^5*d^3 + 5 *a*b^7*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(b*x + a) - 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^ 8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 + a*b^7*c*d^7)*x^6 + 7*(5*b^8* c^3*d^5 + 3*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 + 7* (3*b^8*c^5*d^3 + 5*a*b^7*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^...
Leaf count of result is larger than twice the leaf count of optimal. 2336 vs. \(2 (209) = 418\).
Time = 27.86 (sec) , antiderivative size = 2336, normalized size of antiderivative = 10.11 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=\text {Too large to display} \]
-8*b**7*d*log(x + (-8*a**10*b**7*d**11/(a*d - b*c)**9 + 80*a**9*b**8*c*d** 10/(a*d - b*c)**9 - 360*a**8*b**9*c**2*d**9/(a*d - b*c)**9 + 960*a**7*b**1 0*c**3*d**8/(a*d - b*c)**9 - 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 + 20 16*a**5*b**12*c**5*d**6/(a*d - b*c)**9 - 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 + 960*a**3*b**14*c**7*d**4/(a*d - b*c)**9 - 360*a**2*b**15*c**8*d* *3/(a*d - b*c)**9 + 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**2 - 8*b**17*c**10*d/(a*d - b*c)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)** 9 + 8*b**7*d*log(x + (8*a**10*b**7*d**11/(a*d - b*c)**9 - 80*a**9*b**8*c*d **10/(a*d - b*c)**9 + 360*a**8*b**9*c**2*d**9/(a*d - b*c)**9 - 960*a**7*b* *10*c**3*d**8/(a*d - b*c)**9 + 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 - 2016*a**5*b**12*c**5*d**6/(a*d - b*c)**9 + 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 - 960*a**3*b**14*c**7*d**4/(a*d - b*c)**9 + 360*a**2*b**15*c**8* d**3/(a*d - b*c)**9 - 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**2 + 8*b**17*c**10*d/(a*d - b*c)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c) **9 + (-15*a**7*d**7 + 125*a**6*b*c*d**6 - 463*a**5*b**2*c**2*d**5 + 1007* a**4*b**3*c**3*d**4 - 1443*a**3*b**4*c**4*d**3 + 1497*a**2*b**5*c**5*d**2 - 1443*a*b**6*c**6*d - 105*b**7*c**7 - 840*b**7*d**7*x**7 + x**6*(-420*a*b **6*d**7 - 5460*b**7*c*d**6) + x**5*(140*a**2*b**5*d**7 - 2800*a*b**6*c*d* *6 - 14980*b**7*c**2*d**5) + x**4*(-70*a**3*b**4*d**7 + 910*a**2*b**5*c*d* *6 - 7910*a*b**6*c**2*d**5 - 22330*b**7*c**3*d**4) + x**3*(42*a**4*b**3...
Leaf count of result is larger than twice the leaf count of optimal. 1881 vs. \(2 (223) = 446\).
Time = 0.33 (sec) , antiderivative size = 1881, normalized size of antiderivative = 8.14 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=\text {Too large to display} \]
-8*b^7*d*log(b*x + a)/(b^9*c^9 - 9*a*b^8*c^8*d + 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^8*b*c*d^8 - a^9*d^9) + 8*b^7*d*log(d*x + c)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 12 6*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^8*b*c*d^8 - a^9*d^9) - 1/105*(840*b^7*d^7*x^7 + 105*b^7*c^7 + 1443*a*b^6*c^6*d - 1497*a^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^ 4*b^3*c^3*d^4 + 463*a^5*b^2*c^2*d^5 - 125*a^6*b*c*d^6 + 15*a^7*d^7 + 420*( 13*b^7*c*d^6 + a*b^6*d^7)*x^6 + 140*(107*b^7*c^2*d^5 + 20*a*b^6*c*d^6 - a^ 2*b^5*d^7)*x^5 + 70*(319*b^7*c^3*d^4 + 113*a*b^6*c^2*d^5 - 13*a^2*b^5*c*d^ 6 + a^3*b^4*d^7)*x^4 + 14*(1377*b^7*c^4*d^3 + 872*a*b^6*c^3*d^4 - 178*a^2* b^5*c^2*d^5 + 32*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 14*(669*b^7*c^5*d^2 + 786*a*b^6*c^4*d^3 - 264*a^2*b^5*c^3*d^4 + 86*a^3*b^4*c^2*d^5 - 19*a^4*b^ 3*c*d^6 + 2*a^5*b^2*d^7)*x^2 + 2*(1089*b^7*c^6*d + 2832*a*b^6*c^5*d^2 - 15 78*a^2*b^5*c^4*d^3 + 872*a^3*b^4*c^3*d^4 - 353*a^4*b^3*c^2*d^5 + 88*a^5*b^ 2*c*d^6 - 10*a^6*b*d^7)*x)/(a*b^8*c^15 - 8*a^2*b^7*c^14*d + 28*a^3*b^6*c^1 3*d^2 - 56*a^4*b^5*c^12*d^3 + 70*a^5*b^4*c^11*d^4 - 56*a^6*b^3*c^10*d^5 + 28*a^7*b^2*c^9*d^6 - 8*a^8*b*c^8*d^7 + a^9*c^7*d^8 + (b^9*c^8*d^7 - 8*a*b^ 8*c^7*d^8 + 28*a^2*b^7*c^6*d^9 - 56*a^3*b^6*c^5*d^10 + 70*a^4*b^5*c^4*d^11 - 56*a^5*b^4*c^3*d^12 + 28*a^6*b^3*c^2*d^13 - 8*a^7*b^2*c*d^14 + a^8*b...
Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (223) = 446\).
Time = 0.34 (sec) , antiderivative size = 714, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=-\frac {b^{15}}{{\left (b^{16} c^{8} - 8 \, a b^{15} c^{7} d + 28 \, a^{2} b^{14} c^{6} d^{2} - 56 \, a^{3} b^{13} c^{5} d^{3} + 70 \, a^{4} b^{12} c^{4} d^{4} - 56 \, a^{5} b^{11} c^{3} d^{5} + 28 \, a^{6} b^{10} c^{2} d^{6} - 8 \, a^{7} b^{9} c d^{7} + a^{8} b^{8} d^{8}\right )} {\left (b x + a\right )}} + \frac {8 \, b^{8} d \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{10} c^{9} - 9 \, a b^{9} c^{8} d + 36 \, a^{2} b^{8} c^{7} d^{2} - 84 \, a^{3} b^{7} c^{6} d^{3} + 126 \, a^{4} b^{6} c^{5} d^{4} - 126 \, a^{5} b^{5} c^{4} d^{5} + 84 \, a^{6} b^{4} c^{3} d^{6} - 36 \, a^{7} b^{3} c^{2} d^{7} + 9 \, a^{8} b^{2} c d^{8} - a^{9} b d^{9}} + \frac {1443 \, b^{7} d^{8} + \frac {9366 \, {\left (b^{9} c d^{7} - a b^{8} d^{8}\right )}}{{\left (b x + a\right )} b} + \frac {25578 \, {\left (b^{11} c^{2} d^{6} - 2 \, a b^{10} c d^{7} + a^{2} b^{9} d^{8}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {37730 \, {\left (b^{13} c^{3} d^{5} - 3 \, a b^{12} c^{2} d^{6} + 3 \, a^{2} b^{11} c d^{7} - a^{3} b^{10} d^{8}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {31850 \, {\left (b^{15} c^{4} d^{4} - 4 \, a b^{14} c^{3} d^{5} + 6 \, a^{2} b^{13} c^{2} d^{6} - 4 \, a^{3} b^{12} c d^{7} + a^{4} b^{11} d^{8}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {14700 \, {\left (b^{17} c^{5} d^{3} - 5 \, a b^{16} c^{4} d^{4} + 10 \, a^{2} b^{15} c^{3} d^{5} - 10 \, a^{3} b^{14} c^{2} d^{6} + 5 \, a^{4} b^{13} c d^{7} - a^{5} b^{12} d^{8}\right )}}{{\left (b x + a\right )}^{5} b^{5}} + \frac {2940 \, {\left (b^{19} c^{6} d^{2} - 6 \, a b^{18} c^{5} d^{3} + 15 \, a^{2} b^{17} c^{4} d^{4} - 20 \, a^{3} b^{16} c^{3} d^{5} + 15 \, a^{4} b^{15} c^{2} d^{6} - 6 \, a^{5} b^{14} c d^{7} + a^{6} b^{13} d^{8}\right )}}{{\left (b x + a\right )}^{6} b^{6}}}{105 \, {\left (b c - a d\right )}^{9} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{7}} \]
-b^15/((b^16*c^8 - 8*a*b^15*c^7*d + 28*a^2*b^14*c^6*d^2 - 56*a^3*b^13*c^5* d^3 + 70*a^4*b^12*c^4*d^4 - 56*a^5*b^11*c^3*d^5 + 28*a^6*b^10*c^2*d^6 - 8* a^7*b^9*c*d^7 + a^8*b^8*d^8)*(b*x + a)) + 8*b^8*d*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^ 3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3 *d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9) + 1/105*(1443*b^7 *d^8 + 9366*(b^9*c*d^7 - a*b^8*d^8)/((b*x + a)*b) + 25578*(b^11*c^2*d^6 - 2*a*b^10*c*d^7 + a^2*b^9*d^8)/((b*x + a)^2*b^2) + 37730*(b^13*c^3*d^5 - 3* a*b^12*c^2*d^6 + 3*a^2*b^11*c*d^7 - a^3*b^10*d^8)/((b*x + a)^3*b^3) + 3185 0*(b^15*c^4*d^4 - 4*a*b^14*c^3*d^5 + 6*a^2*b^13*c^2*d^6 - 4*a^3*b^12*c*d^7 + a^4*b^11*d^8)/((b*x + a)^4*b^4) + 14700*(b^17*c^5*d^3 - 5*a*b^16*c^4*d^ 4 + 10*a^2*b^15*c^3*d^5 - 10*a^3*b^14*c^2*d^6 + 5*a^4*b^13*c*d^7 - a^5*b^1 2*d^8)/((b*x + a)^5*b^5) + 2940*(b^19*c^6*d^2 - 6*a*b^18*c^5*d^3 + 15*a^2* b^17*c^4*d^4 - 20*a^3*b^16*c^3*d^5 + 15*a^4*b^15*c^2*d^6 - 6*a^5*b^14*c*d^ 7 + a^6*b^13*d^8)/((b*x + a)^6*b^6))/((b*c - a*d)^9*(b*c/(b*x + a) - a*d/( b*x + a) + d)^7)
Time = 1.64 (sec) , antiderivative size = 1738, normalized size of antiderivative = 7.52 \[ \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx=\text {Too large to display} \]
(16*b^7*d*atanh((a^9*d^9 + b^9*c^9 + 20*a^2*b^7*c^7*d^2 - 28*a^3*b^6*c^6*d ^3 + 14*a^4*b^5*c^5*d^4 + 14*a^5*b^4*c^4*d^5 - 28*a^6*b^3*c^3*d^6 + 20*a^7 *b^2*c^2*d^7 - 7*a*b^8*c^8*d - 7*a^8*b*c*d^8)/(a*d - b*c)^9 + (2*b*d*x*(a^ 8*d^8 + b^8*c^8 + 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*b^7*c^7*d - 8*a^7*b*c *d^7))/(a*d - b*c)^9))/(a*d - b*c)^9 - ((15*a^7*d^7 + 105*b^7*c^7 - 1497*a ^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^4*b^3*c^3*d^4 + 463*a^5*b^2 *c^2*d^5 + 1443*a*b^6*c^6*d - 125*a^6*b*c*d^6)/(105*(a^8*d^8 + b^8*c^8 + 2 8*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*b^7*c^7*d - 8*a^7*b*c*d^7)) + (4*b^5*x^5 *(107*b^2*c^2*d^5 - a^2*d^7 + 20*a*b*c*d^6))/(3*(a^8*d^8 + b^8*c^8 + 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*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*b^2*x^2*(2* a^5*d^7 + 669*b^5*c^5*d^2 + 786*a*b^4*c^4*d^3 - 264*a^2*b^3*c^3*d^4 + 86*a ^3*b^2*c^2*d^5 - 19*a^4*b*c*d^6))/(15*(a^8*d^8 + b^8*c^8 + 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*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*b^4*x^4*(a^3*d^7 + 31 9*b^3*c^3*d^4 + 113*a*b^2*c^2*d^5 - 13*a^2*b*c*d^6))/(3*(a^8*d^8 + b^8*c^8 + 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*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*...