Integrand size = 21, antiderivative size = 113 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\frac {F^{a+\frac {b}{(c+d x)^2}} \left (120 (c+d x)^{10}-120 b (c+d x)^8 \log (F)+60 b^2 (c+d x)^6 \log ^2(F)-20 b^3 (c+d x)^4 \log ^3(F)+5 b^4 (c+d x)^2 \log ^4(F)-b^5 \log ^5(F)\right )}{2 b^6 d (c+d x)^{10} \log ^6(F)} \]
1/2*F^(a+b/(d*x+c)^2)*(120*(d*x+c)^10-120*b*(d*x+c)^8*ln(F)+60*b^2*(d*x+c) ^6*ln(F)^2-20*b^3*(d*x+c)^4*ln(F)^3+5*b^4*(d*x+c)^2*ln(F)^4-b^5*ln(F)^5)/b ^6/d/(d*x+c)^10/ln(F)^6
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.27 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\frac {F^a \Gamma \left (6,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)} \]
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2647}
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.
| \(\displaystyle \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx\) |
| \(\Big \downarrow \) 2647 |
| \(\displaystyle \frac {F^{a+\frac {b}{(c+d x)^2}} \left (-b^5 \log ^5(F)+5 b^4 \log ^4(F) (c+d x)^2-20 b^3 \log ^3(F) (c+d x)^4+60 b^2 \log ^2(F) (c+d x)^6-120 b \log (F) (c+d x)^8+120 (c+d x)^{10}\right )}{2 b^6 d \log ^6(F) (c+d x)^{10}}\) |
(F^(a + b/(c + d*x)^2)*(120*(c + d*x)^10 - 120*b*(c + d*x)^8*Log[F] + 60*b ^2*(c + d*x)^6*Log[F]^2 - 20*b^3*(c + d*x)^4*Log[F]^3 + 5*b^4*(c + d*x)^2* Log[F]^4 - b^5*Log[F]^5))/(2*b^6*d*(c + d*x)^10*Log[F]^6)
3.4.26.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> With[{p = Simplify[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n* ((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Log[F]] ]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] && !TrueQ[$UseGamma]
Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(111)=222\).
Time = 4.50 (sec) , antiderivative size = 502, normalized size of antiderivative = 4.44
| method | result | size |
| risch | \(-\frac {\left (120 \ln \left (F \right ) b \,c^{8}+960 \ln \left (F \right ) b c \,d^{7} x^{7}+3360 \ln \left (F \right ) b \,c^{2} d^{6} x^{6}+6720 \ln \left (F \right ) b \,c^{3} d^{5} x^{5}+8400 \ln \left (F \right ) b \,c^{4} d^{4} x^{4}+80 \ln \left (F \right )^{3} b^{3} c \,d^{3} x^{3}+6720 \ln \left (F \right ) b \,c^{5} d^{3} x^{3}+120 \ln \left (F \right )^{3} b^{3} c^{2} d^{2} x^{2}+3360 \ln \left (F \right ) b \,c^{6} d^{2} x^{2}-10 \ln \left (F \right )^{4} b^{4} c d x +80 \ln \left (F \right )^{3} b^{3} c^{3} d x +960 \ln \left (F \right ) b \,c^{7} d x -120 d^{10} x^{10}-120 c^{10}+b^{5} \ln \left (F \right )^{5}+120 \ln \left (F \right ) b \,d^{8} x^{8}+20 \ln \left (F \right )^{3} b^{3} d^{4} x^{4}-5 \ln \left (F \right )^{4} b^{4} d^{2} x^{2}-5 \ln \left (F \right )^{4} b^{4} c^{2}-60 \ln \left (F \right )^{2} b^{2} c^{6}-1200 c \,d^{9} x^{9}-5400 c^{2} d^{8} x^{8}-14400 c^{3} d^{7} x^{7}-25200 c^{4} d^{6} x^{6}-30240 c^{5} d^{5} x^{5}-25200 c^{6} d^{4} x^{4}-14400 c^{7} d^{3} x^{3}-5400 c^{8} d^{2} x^{2}-1200 c^{9} d x -60 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-360 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-900 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-1200 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-900 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-360 \ln \left (F \right )^{2} b^{2} c^{5} d x +20 \ln \left (F \right )^{3} b^{3} c^{4}\right ) F^{\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}}}{2 b^{6} \ln \left (F \right )^{6} d \left (d x +c \right )^{10}}\) | \(502\) |
| norman | \(\text {Expression too large to display}\) | \(797\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(968\) |
-1/2*(120*ln(F)*b*c^8+960*ln(F)*b*c*d^7*x^7+3360*ln(F)*b*c^2*d^6*x^6+6720* ln(F)*b*c^3*d^5*x^5+8400*ln(F)*b*c^4*d^4*x^4+80*ln(F)^3*b^3*c*d^3*x^3+6720 *ln(F)*b*c^5*d^3*x^3+120*ln(F)^3*b^3*c^2*d^2*x^2+3360*ln(F)*b*c^6*d^2*x^2- 10*ln(F)^4*b^4*c*d*x+80*ln(F)^3*b^3*c^3*d*x+960*ln(F)*b*c^7*d*x-120*d^10*x ^10-120*c^10+b^5*ln(F)^5+120*ln(F)*b*d^8*x^8+20*ln(F)^3*b^3*d^4*x^4-5*ln(F )^4*b^4*d^2*x^2-5*ln(F)^4*b^4*c^2-60*ln(F)^2*b^2*c^6-1200*c*d^9*x^9-5400*c ^2*d^8*x^8-14400*c^3*d^7*x^7-25200*c^4*d^6*x^6-30240*c^5*d^5*x^5-25200*c^6 *d^4*x^4-14400*c^7*d^3*x^3-5400*c^8*d^2*x^2-1200*c^9*d*x-60*d^6*x^6*ln(F)^ 2*b^2-360*c*d^5*x^5*ln(F)^2*b^2-900*c^2*d^4*x^4*ln(F)^2*b^2-1200*ln(F)^2*b ^2*c^3*d^3*x^3-900*ln(F)^2*b^2*c^4*d^2*x^2-360*ln(F)^2*b^2*c^5*d*x+20*ln(F )^3*b^3*c^4)/b^6/ln(F)^6/d/(d*x+c)^10*F^((a*d^2*x^2+2*a*c*d*x+a*c^2+b)/(d* x+c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (111) = 222\).
Time = 0.30 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.16 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\frac {{\left (120 \, d^{10} x^{10} + 1200 \, c d^{9} x^{9} + 5400 \, c^{2} d^{8} x^{8} + 14400 \, c^{3} d^{7} x^{7} + 25200 \, c^{4} d^{6} x^{6} + 30240 \, c^{5} d^{5} x^{5} + 25200 \, c^{6} d^{4} x^{4} + 14400 \, c^{7} d^{3} x^{3} + 5400 \, c^{8} d^{2} x^{2} + 1200 \, c^{9} d x + 120 \, c^{10} - b^{5} \log \left (F\right )^{5} + 5 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \left (F\right )^{4} - 20 \, {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \left (F\right )^{3} + 60 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 120 \, {\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{6} d^{11} x^{10} + 10 \, b^{6} c d^{10} x^{9} + 45 \, b^{6} c^{2} d^{9} x^{8} + 120 \, b^{6} c^{3} d^{8} x^{7} + 210 \, b^{6} c^{4} d^{7} x^{6} + 252 \, b^{6} c^{5} d^{6} x^{5} + 210 \, b^{6} c^{6} d^{5} x^{4} + 120 \, b^{6} c^{7} d^{4} x^{3} + 45 \, b^{6} c^{8} d^{3} x^{2} + 10 \, b^{6} c^{9} d^{2} x + b^{6} c^{10} d\right )} \log \left (F\right )^{6}} \]
1/2*(120*d^10*x^10 + 1200*c*d^9*x^9 + 5400*c^2*d^8*x^8 + 14400*c^3*d^7*x^7 + 25200*c^4*d^6*x^6 + 30240*c^5*d^5*x^5 + 25200*c^6*d^4*x^4 + 14400*c^7*d ^3*x^3 + 5400*c^8*d^2*x^2 + 1200*c^9*d*x + 120*c^10 - b^5*log(F)^5 + 5*(b^ 4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)*log(F)^4 - 20*(b^3*d^4*x^4 + 4*b^3*c*d^ 3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*log(F)^3 + 60*(b^2*d^ 6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2 *c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 - 120*(b*d^8*x^8 + 8*b*c* d^7*x^7 + 28*b*c^2*d^6*x^6 + 56*b*c^3*d^5*x^5 + 70*b*c^4*d^4*x^4 + 56*b*c^ 5*d^3*x^3 + 28*b*c^6*d^2*x^2 + 8*b*c^7*d*x + b*c^8)*log(F))*F^((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))/((b^6*d^11*x^10 + 10*b ^6*c*d^10*x^9 + 45*b^6*c^2*d^9*x^8 + 120*b^6*c^3*d^8*x^7 + 210*b^6*c^4*d^7 *x^6 + 252*b^6*c^5*d^6*x^5 + 210*b^6*c^6*d^5*x^4 + 120*b^6*c^7*d^4*x^3 + 4 5*b^6*c^8*d^3*x^2 + 10*b^6*c^9*d^2*x + b^6*c^10*d)*log(F)^6)
Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (110) = 220\).
Time = 1.55 (sec) , antiderivative size = 745, normalized size of antiderivative = 6.59 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{5} \log {\left (F \right )}^{5} + 5 b^{4} c^{2} \log {\left (F \right )}^{4} + 10 b^{4} c d x \log {\left (F \right )}^{4} + 5 b^{4} d^{2} x^{2} \log {\left (F \right )}^{4} - 20 b^{3} c^{4} \log {\left (F \right )}^{3} - 80 b^{3} c^{3} d x \log {\left (F \right )}^{3} - 120 b^{3} c^{2} d^{2} x^{2} \log {\left (F \right )}^{3} - 80 b^{3} c d^{3} x^{3} \log {\left (F \right )}^{3} - 20 b^{3} d^{4} x^{4} \log {\left (F \right )}^{3} + 60 b^{2} c^{6} \log {\left (F \right )}^{2} + 360 b^{2} c^{5} d x \log {\left (F \right )}^{2} + 900 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} + 1200 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} + 900 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 360 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} + 60 b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} - 120 b c^{8} \log {\left (F \right )} - 960 b c^{7} d x \log {\left (F \right )} - 3360 b c^{6} d^{2} x^{2} \log {\left (F \right )} - 6720 b c^{5} d^{3} x^{3} \log {\left (F \right )} - 8400 b c^{4} d^{4} x^{4} \log {\left (F \right )} - 6720 b c^{3} d^{5} x^{5} \log {\left (F \right )} - 3360 b c^{2} d^{6} x^{6} \log {\left (F \right )} - 960 b c d^{7} x^{7} \log {\left (F \right )} - 120 b d^{8} x^{8} \log {\left (F \right )} + 120 c^{10} + 1200 c^{9} d x + 5400 c^{8} d^{2} x^{2} + 14400 c^{7} d^{3} x^{3} + 25200 c^{6} d^{4} x^{4} + 30240 c^{5} d^{5} x^{5} + 25200 c^{4} d^{6} x^{6} + 14400 c^{3} d^{7} x^{7} + 5400 c^{2} d^{8} x^{8} + 1200 c d^{9} x^{9} + 120 d^{10} x^{10}\right )}{2 b^{6} c^{10} d \log {\left (F \right )}^{6} + 20 b^{6} c^{9} d^{2} x \log {\left (F \right )}^{6} + 90 b^{6} c^{8} d^{3} x^{2} \log {\left (F \right )}^{6} + 240 b^{6} c^{7} d^{4} x^{3} \log {\left (F \right )}^{6} + 420 b^{6} c^{6} d^{5} x^{4} \log {\left (F \right )}^{6} + 504 b^{6} c^{5} d^{6} x^{5} \log {\left (F \right )}^{6} + 420 b^{6} c^{4} d^{7} x^{6} \log {\left (F \right )}^{6} + 240 b^{6} c^{3} d^{8} x^{7} \log {\left (F \right )}^{6} + 90 b^{6} c^{2} d^{9} x^{8} \log {\left (F \right )}^{6} + 20 b^{6} c d^{10} x^{9} \log {\left (F \right )}^{6} + 2 b^{6} d^{11} x^{10} \log {\left (F \right )}^{6}} \]
F**(a + b/(c + d*x)**2)*(-b**5*log(F)**5 + 5*b**4*c**2*log(F)**4 + 10*b**4 *c*d*x*log(F)**4 + 5*b**4*d**2*x**2*log(F)**4 - 20*b**3*c**4*log(F)**3 - 8 0*b**3*c**3*d*x*log(F)**3 - 120*b**3*c**2*d**2*x**2*log(F)**3 - 80*b**3*c* d**3*x**3*log(F)**3 - 20*b**3*d**4*x**4*log(F)**3 + 60*b**2*c**6*log(F)**2 + 360*b**2*c**5*d*x*log(F)**2 + 900*b**2*c**4*d**2*x**2*log(F)**2 + 1200* b**2*c**3*d**3*x**3*log(F)**2 + 900*b**2*c**2*d**4*x**4*log(F)**2 + 360*b* *2*c*d**5*x**5*log(F)**2 + 60*b**2*d**6*x**6*log(F)**2 - 120*b*c**8*log(F) - 960*b*c**7*d*x*log(F) - 3360*b*c**6*d**2*x**2*log(F) - 6720*b*c**5*d**3 *x**3*log(F) - 8400*b*c**4*d**4*x**4*log(F) - 6720*b*c**3*d**5*x**5*log(F) - 3360*b*c**2*d**6*x**6*log(F) - 960*b*c*d**7*x**7*log(F) - 120*b*d**8*x* *8*log(F) + 120*c**10 + 1200*c**9*d*x + 5400*c**8*d**2*x**2 + 14400*c**7*d **3*x**3 + 25200*c**6*d**4*x**4 + 30240*c**5*d**5*x**5 + 25200*c**4*d**6*x **6 + 14400*c**3*d**7*x**7 + 5400*c**2*d**8*x**8 + 1200*c*d**9*x**9 + 120* d**10*x**10)/(2*b**6*c**10*d*log(F)**6 + 20*b**6*c**9*d**2*x*log(F)**6 + 9 0*b**6*c**8*d**3*x**2*log(F)**6 + 240*b**6*c**7*d**4*x**3*log(F)**6 + 420* b**6*c**6*d**5*x**4*log(F)**6 + 504*b**6*c**5*d**6*x**5*log(F)**6 + 420*b* *6*c**4*d**7*x**6*log(F)**6 + 240*b**6*c**3*d**8*x**7*log(F)**6 + 90*b**6* c**2*d**9*x**8*log(F)**6 + 20*b**6*c*d**10*x**9*log(F)**6 + 2*b**6*d**11*x **10*log(F)**6)
Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (111) = 222\).
Time = 0.25 (sec) , antiderivative size = 740, normalized size of antiderivative = 6.55 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\frac {{\left (120 \, F^{a} d^{10} x^{10} + 1200 \, F^{a} c d^{9} x^{9} + 120 \, F^{a} c^{10} - 120 \, F^{a} b c^{8} \log \left (F\right ) + 60 \, F^{a} b^{2} c^{6} \log \left (F\right )^{2} - 20 \, F^{a} b^{3} c^{4} \log \left (F\right )^{3} + 5 \, F^{a} b^{4} c^{2} \log \left (F\right )^{4} - F^{a} b^{5} \log \left (F\right )^{5} + 120 \, {\left (45 \, F^{a} c^{2} d^{8} - F^{a} b d^{8} \log \left (F\right )\right )} x^{8} + 960 \, {\left (15 \, F^{a} c^{3} d^{7} - F^{a} b c d^{7} \log \left (F\right )\right )} x^{7} + 60 \, {\left (420 \, F^{a} c^{4} d^{6} - 56 \, F^{a} b c^{2} d^{6} \log \left (F\right ) + F^{a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 120 \, {\left (252 \, F^{a} c^{5} d^{5} - 56 \, F^{a} b c^{3} d^{5} \log \left (F\right ) + 3 \, F^{a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} + 20 \, {\left (1260 \, F^{a} c^{6} d^{4} - 420 \, F^{a} b c^{4} d^{4} \log \left (F\right ) + 45 \, F^{a} b^{2} c^{2} d^{4} \log \left (F\right )^{2} - F^{a} b^{3} d^{4} \log \left (F\right )^{3}\right )} x^{4} + 80 \, {\left (180 \, F^{a} c^{7} d^{3} - 84 \, F^{a} b c^{5} d^{3} \log \left (F\right ) + 15 \, F^{a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} - F^{a} b^{3} c d^{3} \log \left (F\right )^{3}\right )} x^{3} + 5 \, {\left (1080 \, F^{a} c^{8} d^{2} - 672 \, F^{a} b c^{6} d^{2} \log \left (F\right ) + 180 \, F^{a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} - 24 \, F^{a} b^{3} c^{2} d^{2} \log \left (F\right )^{3} + F^{a} b^{4} d^{2} \log \left (F\right )^{4}\right )} x^{2} + 10 \, {\left (120 \, F^{a} c^{9} d - 96 \, F^{a} b c^{7} d \log \left (F\right ) + 36 \, F^{a} b^{2} c^{5} d \log \left (F\right )^{2} - 8 \, F^{a} b^{3} c^{3} d \log \left (F\right )^{3} + F^{a} b^{4} c d \log \left (F\right )^{4}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{6} d^{11} x^{10} \log \left (F\right )^{6} + 10 \, b^{6} c d^{10} x^{9} \log \left (F\right )^{6} + 45 \, b^{6} c^{2} d^{9} x^{8} \log \left (F\right )^{6} + 120 \, b^{6} c^{3} d^{8} x^{7} \log \left (F\right )^{6} + 210 \, b^{6} c^{4} d^{7} x^{6} \log \left (F\right )^{6} + 252 \, b^{6} c^{5} d^{6} x^{5} \log \left (F\right )^{6} + 210 \, b^{6} c^{6} d^{5} x^{4} \log \left (F\right )^{6} + 120 \, b^{6} c^{7} d^{4} x^{3} \log \left (F\right )^{6} + 45 \, b^{6} c^{8} d^{3} x^{2} \log \left (F\right )^{6} + 10 \, b^{6} c^{9} d^{2} x \log \left (F\right )^{6} + b^{6} c^{10} d \log \left (F\right )^{6}\right )}} \]
1/2*(120*F^a*d^10*x^10 + 1200*F^a*c*d^9*x^9 + 120*F^a*c^10 - 120*F^a*b*c^8 *log(F) + 60*F^a*b^2*c^6*log(F)^2 - 20*F^a*b^3*c^4*log(F)^3 + 5*F^a*b^4*c^ 2*log(F)^4 - F^a*b^5*log(F)^5 + 120*(45*F^a*c^2*d^8 - F^a*b*d^8*log(F))*x^ 8 + 960*(15*F^a*c^3*d^7 - F^a*b*c*d^7*log(F))*x^7 + 60*(420*F^a*c^4*d^6 - 56*F^a*b*c^2*d^6*log(F) + F^a*b^2*d^6*log(F)^2)*x^6 + 120*(252*F^a*c^5*d^5 - 56*F^a*b*c^3*d^5*log(F) + 3*F^a*b^2*c*d^5*log(F)^2)*x^5 + 20*(1260*F^a* c^6*d^4 - 420*F^a*b*c^4*d^4*log(F) + 45*F^a*b^2*c^2*d^4*log(F)^2 - F^a*b^3 *d^4*log(F)^3)*x^4 + 80*(180*F^a*c^7*d^3 - 84*F^a*b*c^5*d^3*log(F) + 15*F^ a*b^2*c^3*d^3*log(F)^2 - F^a*b^3*c*d^3*log(F)^3)*x^3 + 5*(1080*F^a*c^8*d^2 - 672*F^a*b*c^6*d^2*log(F) + 180*F^a*b^2*c^4*d^2*log(F)^2 - 24*F^a*b^3*c^ 2*d^2*log(F)^3 + F^a*b^4*d^2*log(F)^4)*x^2 + 10*(120*F^a*c^9*d - 96*F^a*b* c^7*d*log(F) + 36*F^a*b^2*c^5*d*log(F)^2 - 8*F^a*b^3*c^3*d*log(F)^3 + F^a* b^4*c*d*log(F)^4)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^6*d^11*x^10*log(F) ^6 + 10*b^6*c*d^10*x^9*log(F)^6 + 45*b^6*c^2*d^9*x^8*log(F)^6 + 120*b^6*c^ 3*d^8*x^7*log(F)^6 + 210*b^6*c^4*d^7*x^6*log(F)^6 + 252*b^6*c^5*d^6*x^5*lo g(F)^6 + 210*b^6*c^6*d^5*x^4*log(F)^6 + 120*b^6*c^7*d^4*x^3*log(F)^6 + 45* b^6*c^8*d^3*x^2*log(F)^6 + 10*b^6*c^9*d^2*x*log(F)^6 + b^6*c^10*d*log(F)^6 )
\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{13}} \,d x } \]
Time = 1.62 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.16 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx=\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {60\,x^{10}}{b^6\,d\,{\ln \left (F\right )}^6}-\frac {b^5\,{\ln \left (F\right )}^5-5\,b^4\,c^2\,{\ln \left (F\right )}^4+20\,b^3\,c^4\,{\ln \left (F\right )}^3-60\,b^2\,c^6\,{\ln \left (F\right )}^2+120\,b\,c^8\,\ln \left (F\right )-120\,c^{10}}{2\,b^6\,d^{11}\,{\ln \left (F\right )}^6}+\frac {600\,c\,x^9}{b^6\,d^2\,{\ln \left (F\right )}^6}+\frac {5\,x^2\,\left (b^4\,{\ln \left (F\right )}^4-24\,b^3\,c^2\,{\ln \left (F\right )}^3+180\,b^2\,c^4\,{\ln \left (F\right )}^2-672\,b\,c^6\,\ln \left (F\right )+1080\,c^8\right )}{2\,b^6\,d^9\,{\ln \left (F\right )}^6}-\frac {10\,x^4\,\left (b^3\,{\ln \left (F\right )}^3-45\,b^2\,c^2\,{\ln \left (F\right )}^2+420\,b\,c^4\,\ln \left (F\right )-1260\,c^6\right )}{b^6\,d^7\,{\ln \left (F\right )}^6}+\frac {30\,x^6\,\left (b^2\,{\ln \left (F\right )}^2-56\,b\,c^2\,\ln \left (F\right )+420\,c^4\right )}{b^6\,d^5\,{\ln \left (F\right )}^6}-\frac {60\,x^8\,\left (b\,\ln \left (F\right )-45\,c^2\right )}{b^6\,d^3\,{\ln \left (F\right )}^6}-\frac {40\,c\,x^3\,\left (b^3\,{\ln \left (F\right )}^3-15\,b^2\,c^2\,{\ln \left (F\right )}^2+84\,b\,c^4\,\ln \left (F\right )-180\,c^6\right )}{b^6\,d^8\,{\ln \left (F\right )}^6}+\frac {60\,c\,x^5\,\left (3\,b^2\,{\ln \left (F\right )}^2-56\,b\,c^2\,\ln \left (F\right )+252\,c^4\right )}{b^6\,d^6\,{\ln \left (F\right )}^6}-\frac {480\,c\,x^7\,\left (b\,\ln \left (F\right )-15\,c^2\right )}{b^6\,d^4\,{\ln \left (F\right )}^6}+\frac {5\,c\,x\,\left (b^4\,{\ln \left (F\right )}^4-8\,b^3\,c^2\,{\ln \left (F\right )}^3+36\,b^2\,c^4\,{\ln \left (F\right )}^2-96\,b\,c^6\,\ln \left (F\right )+120\,c^8\right )}{b^6\,d^{10}\,{\ln \left (F\right )}^6}\right )}{x^{10}+\frac {c^{10}}{d^{10}}+\frac {10\,c\,x^9}{d}+\frac {10\,c^9\,x}{d^9}+\frac {45\,c^2\,x^8}{d^2}+\frac {120\,c^3\,x^7}{d^3}+\frac {210\,c^4\,x^6}{d^4}+\frac {252\,c^5\,x^5}{d^5}+\frac {210\,c^6\,x^4}{d^6}+\frac {120\,c^7\,x^3}{d^7}+\frac {45\,c^8\,x^2}{d^8}} \]
(F^a*F^(b/(c^2 + d^2*x^2 + 2*c*d*x))*((60*x^10)/(b^6*d*log(F)^6) - (b^5*lo g(F)^5 - 120*c^10 + 120*b*c^8*log(F) - 60*b^2*c^6*log(F)^2 + 20*b^3*c^4*lo g(F)^3 - 5*b^4*c^2*log(F)^4)/(2*b^6*d^11*log(F)^6) + (600*c*x^9)/(b^6*d^2* log(F)^6) + (5*x^2*(b^4*log(F)^4 + 1080*c^8 - 672*b*c^6*log(F) + 180*b^2*c ^4*log(F)^2 - 24*b^3*c^2*log(F)^3))/(2*b^6*d^9*log(F)^6) - (10*x^4*(b^3*lo g(F)^3 - 1260*c^6 + 420*b*c^4*log(F) - 45*b^2*c^2*log(F)^2))/(b^6*d^7*log( F)^6) + (30*x^6*(b^2*log(F)^2 + 420*c^4 - 56*b*c^2*log(F)))/(b^6*d^5*log(F )^6) - (60*x^8*(b*log(F) - 45*c^2))/(b^6*d^3*log(F)^6) - (40*c*x^3*(b^3*lo g(F)^3 - 180*c^6 + 84*b*c^4*log(F) - 15*b^2*c^2*log(F)^2))/(b^6*d^8*log(F) ^6) + (60*c*x^5*(3*b^2*log(F)^2 + 252*c^4 - 56*b*c^2*log(F)))/(b^6*d^6*log (F)^6) - (480*c*x^7*(b*log(F) - 15*c^2))/(b^6*d^4*log(F)^6) + (5*c*x*(b^4* log(F)^4 + 120*c^8 - 96*b*c^6*log(F) + 36*b^2*c^4*log(F)^2 - 8*b^3*c^2*log (F)^3))/(b^6*d^10*log(F)^6)))/(x^10 + c^10/d^10 + (10*c*x^9)/d + (10*c^9*x )/d^9 + (45*c^2*x^8)/d^2 + (120*c^3*x^7)/d^3 + (210*c^4*x^6)/d^4 + (252*c^ 5*x^5)/d^5 + (210*c^6*x^4)/d^6 + (120*c^7*x^3)/d^7 + (45*c^8*x^2)/d^8)