Integrand size = 21, antiderivative size = 123 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)} \]
2*F^(a+b/(d*x+c)^3)/b^4/d/ln(F)^4-2*F^(a+b/(d*x+c)^3)/b^3/d/(d*x+c)^3/ln(F )^3+F^(a+b/(d*x+c)^3)/b^2/d/(d*x+c)^6/ln(F)^2-1/3*F^(a+b/(d*x+c)^3)/b/d/(d *x+c)^9/ln(F)
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\frac {F^{a+\frac {b}{(c+d x)^3}} \left (6-\frac {6 b \log (F)}{(c+d x)^3}+\frac {3 b^2 \log ^2(F)}{(c+d x)^6}-\frac {b^3 \log ^3(F)}{(c+d x)^9}\right )}{3 b^4 d \log ^4(F)} \]
(F^(a + b/(c + d*x)^3)*(6 - (6*b*Log[F])/(c + d*x)^3 + (3*b^2*Log[F]^2)/(c + d*x)^6 - (b^3*Log[F]^3)/(c + d*x)^9))/(3*b^4*d*Log[F]^4)
Time = 0.46 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2641, 2641, 2641, 2638}
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.
| \(\displaystyle \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{10}}dx}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {3 \left (-\frac {2 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^7}dx}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6}\right )}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^4}dx}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3}\right )}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6}\right )}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9}\) |
| \(\Big \downarrow \) 2638 |
| \(\displaystyle -\frac {3 \left (-\frac {2 \left (\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3}\right )}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6}\right )}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9}\) |
-1/3*F^(a + b/(c + d*x)^3)/(b*d*(c + d*x)^9*Log[F]) - (3*(-1/3*F^(a + b/(c + d*x)^3)/(b*d*(c + d*x)^6*Log[F]) - (2*(F^(a + b/(c + d*x)^3)/(3*b^2*d*L og[F]^2) - F^(a + b/(c + d*x)^3)/(3*b*d*(c + d*x)^3*Log[F])))/(b*Log[F]))) /(b*Log[F])
3.4.50.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e + f*x)^n*(F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n *Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ [d*e - c*f, 0]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ .), x_Symbol] :> Simp[(c + d*x)^(m - n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*L og[F])), x] - Simp[(m - n + 1)/(b*n*Log[F]) Int[(c + d*x)^(m - n)*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/ n)] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n , 0])
Leaf count of result is larger than twice the leaf count of optimal. \(306\) vs. \(2(121)=242\).
Time = 6.03 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.50
| method | result | size |
| risch | \(-\frac {\left (-6 d^{9} x^{9}-54 c \,d^{8} x^{8}-216 c^{2} d^{7} x^{7}-504 c^{3} d^{6} x^{6}+6 \ln \left (F \right ) b \,d^{6} x^{6}-756 c^{4} d^{5} x^{5}+36 \ln \left (F \right ) b c \,d^{5} x^{5}-756 c^{5} d^{4} x^{4}+90 \ln \left (F \right ) b \,c^{2} d^{4} x^{4}-504 c^{6} d^{3} x^{3}+120 \ln \left (F \right ) b \,c^{3} d^{3} x^{3}-216 c^{7} d^{2} x^{2}-3 \ln \left (F \right )^{2} b^{2} d^{3} x^{3}+90 \ln \left (F \right ) b \,c^{4} d^{2} x^{2}-54 c^{8} d x -9 \ln \left (F \right )^{2} b^{2} c \,d^{2} x^{2}+36 \ln \left (F \right ) b \,c^{5} d x -6 c^{9}-9 \ln \left (F \right )^{2} b^{2} c^{2} d x +6 \ln \left (F \right ) b \,c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 b^{4} \ln \left (F \right )^{4} d \left (d x +c \right )^{9}}\) | \(307\) |
| parallelrisch | \(\frac {-36 \ln \left (F \right ) x^{5} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b c \,d^{21}-90 \ln \left (F \right ) x^{4} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{2} d^{20}-120 \ln \left (F \right ) x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{3} d^{19}-90 \ln \left (F \right ) x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{4} d^{18}+9 \ln \left (F \right )^{2} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c \,d^{18}-36 \ln \left (F \right ) x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{5} d^{17}+9 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{2} d^{17}+6 F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{9} d^{16}+6 x^{9} F^{a +\frac {b}{\left (d x +c \right )^{3}}} d^{25}+54 x^{8} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c \,d^{24}+216 x^{7} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{2} d^{23}+504 x^{6} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{3} d^{22}+756 x^{5} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{4} d^{21}+756 x^{4} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{5} d^{20}+504 x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{6} d^{19}+216 x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{7} d^{18}+54 x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{8} d^{17}-\ln \left (F \right )^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{3} d^{16}-6 \ln \left (F \right ) x^{6} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,d^{22}+3 \ln \left (F \right )^{2} x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} d^{19}-6 \ln \left (F \right ) F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{6} d^{16}+3 \ln \left (F \right )^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{3} d^{16}}{3 \left (d x +c \right )^{9} \ln \left (F \right )^{4} b^{4} d^{17}}\) | \(569\) |
| norman | \(\frac {\frac {2 d^{11} x^{12} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {24 d^{10} c \,x^{11} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {\left (-6 c^{9}+6 \ln \left (F \right ) b \,c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) c^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 b^{4} \ln \left (F \right )^{4} d}-\frac {c^{2} \left (-24 c^{9}+18 \ln \left (F \right ) b \,c^{6}-6 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{b^{4} \ln \left (F \right )^{4}}-\frac {d^{2} \left (-1320 c^{9}+504 \ln \left (F \right ) b \,c^{6}-60 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{4} b^{4}}+\frac {d^{5} \left (1848 c^{6}-168 \ln \left (F \right ) b \,c^{3}+\ln \left (F \right )^{2} b^{2}\right ) x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {2 d^{8} \left (-220 c^{3}+b \ln \left (F \right )\right ) x^{9} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {132 d^{9} c^{2} x^{10} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {c d \left (-132 c^{9}+72 \ln \left (F \right ) b \,c^{6}-15 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {3 c^{2} d^{3} \left (330 c^{6}-84 \ln \left (F \right ) b \,c^{3}+5 \ln \left (F \right )^{2} b^{2}\right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {6 c \,d^{4} \left (264 c^{6}-42 \ln \left (F \right ) b \,c^{3}+\ln \left (F \right )^{2} b^{2}\right ) x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {72 c^{2} d^{6} \left (-22 c^{3}+b \ln \left (F \right )\right ) x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {18 c \,d^{7} \left (-55 c^{3}+b \ln \left (F \right )\right ) x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}}{\left (d x +c \right )^{12}}\) | \(641\) |
-1/3*(-6*d^9*x^9-54*c*d^8*x^8-216*c^2*d^7*x^7-504*c^3*d^6*x^6+6*ln(F)*b*d^ 6*x^6-756*c^4*d^5*x^5+36*ln(F)*b*c*d^5*x^5-756*c^5*d^4*x^4+90*ln(F)*b*c^2* d^4*x^4-504*c^6*d^3*x^3+120*ln(F)*b*c^3*d^3*x^3-216*c^7*d^2*x^2-3*ln(F)^2* b^2*d^3*x^3+90*ln(F)*b*c^4*d^2*x^2-54*c^8*d*x-9*ln(F)^2*b^2*c*d^2*x^2+36*l n(F)*b*c^5*d*x-6*c^9-9*ln(F)^2*b^2*c^2*d*x+6*ln(F)*b*c^6-3*ln(F)^2*b^2*c^3 +ln(F)^3*b^3)/b^4/ln(F)^4/d/(d*x+c)^9*F^((a*d^3*x^3+3*a*c*d^2*x^2+3*a*c^2* d*x+a*c^3+b)/(d*x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (121) = 242\).
Time = 0.33 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.44 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\frac {{\left (6 \, d^{9} x^{9} + 54 \, c d^{8} x^{8} + 216 \, c^{2} d^{7} x^{7} + 504 \, c^{3} d^{6} x^{6} + 756 \, c^{4} d^{5} x^{5} + 756 \, c^{5} d^{4} x^{4} + 504 \, c^{6} d^{3} x^{3} + 216 \, c^{7} d^{2} x^{2} + 54 \, c^{8} d x + 6 \, c^{9} - b^{3} \log \left (F\right )^{3} + 3 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} - 6 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{4} d^{10} x^{9} + 9 \, b^{4} c d^{9} x^{8} + 36 \, b^{4} c^{2} d^{8} x^{7} + 84 \, b^{4} c^{3} d^{7} x^{6} + 126 \, b^{4} c^{4} d^{6} x^{5} + 126 \, b^{4} c^{5} d^{5} x^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} + 36 \, b^{4} c^{7} d^{3} x^{2} + 9 \, b^{4} c^{8} d^{2} x + b^{4} c^{9} d\right )} \log \left (F\right )^{4}} \]
1/3*(6*d^9*x^9 + 54*c*d^8*x^8 + 216*c^2*d^7*x^7 + 504*c^3*d^6*x^6 + 756*c^ 4*d^5*x^5 + 756*c^5*d^4*x^4 + 504*c^6*d^3*x^3 + 216*c^7*d^2*x^2 + 54*c^8*d *x + 6*c^9 - b^3*log(F)^3 + 3*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d *x + b^2*c^3)*log(F)^2 - 6*(b*d^6*x^6 + 6*b*c*d^5*x^5 + 15*b*c^2*d^4*x^4 + 20*b*c^3*d^3*x^3 + 15*b*c^4*d^2*x^2 + 6*b*c^5*d*x + b*c^6)*log(F))*F^((a* d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/((b^4*d^10*x^9 + 9*b^4*c*d^9*x^8 + 36*b^4*c^2*d^8*x^7 + 84*b^4*c^3*d^7*x^6 + 126*b^4*c^4*d^6*x^5 + 126*b^4*c^5*d^5*x^4 + 84*b^4* c^6*d^4*x^3 + 36*b^4*c^7*d^3*x^2 + 9*b^4*c^8*d^2*x + b^4*c^9*d)*log(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (109) = 218\).
Time = 0.39 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.93 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b^{3} \log {\left (F \right )}^{3} + 3 b^{2} c^{3} \log {\left (F \right )}^{2} + 9 b^{2} c^{2} d x \log {\left (F \right )}^{2} + 9 b^{2} c d^{2} x^{2} \log {\left (F \right )}^{2} + 3 b^{2} d^{3} x^{3} \log {\left (F \right )}^{2} - 6 b c^{6} \log {\left (F \right )} - 36 b c^{5} d x \log {\left (F \right )} - 90 b c^{4} d^{2} x^{2} \log {\left (F \right )} - 120 b c^{3} d^{3} x^{3} \log {\left (F \right )} - 90 b c^{2} d^{4} x^{4} \log {\left (F \right )} - 36 b c d^{5} x^{5} \log {\left (F \right )} - 6 b d^{6} x^{6} \log {\left (F \right )} + 6 c^{9} + 54 c^{8} d x + 216 c^{7} d^{2} x^{2} + 504 c^{6} d^{3} x^{3} + 756 c^{5} d^{4} x^{4} + 756 c^{4} d^{5} x^{5} + 504 c^{3} d^{6} x^{6} + 216 c^{2} d^{7} x^{7} + 54 c d^{8} x^{8} + 6 d^{9} x^{9}\right )}{3 b^{4} c^{9} d \log {\left (F \right )}^{4} + 27 b^{4} c^{8} d^{2} x \log {\left (F \right )}^{4} + 108 b^{4} c^{7} d^{3} x^{2} \log {\left (F \right )}^{4} + 252 b^{4} c^{6} d^{4} x^{3} \log {\left (F \right )}^{4} + 378 b^{4} c^{5} d^{5} x^{4} \log {\left (F \right )}^{4} + 378 b^{4} c^{4} d^{6} x^{5} \log {\left (F \right )}^{4} + 252 b^{4} c^{3} d^{7} x^{6} \log {\left (F \right )}^{4} + 108 b^{4} c^{2} d^{8} x^{7} \log {\left (F \right )}^{4} + 27 b^{4} c d^{9} x^{8} \log {\left (F \right )}^{4} + 3 b^{4} d^{10} x^{9} \log {\left (F \right )}^{4}} \]
F**(a + b/(c + d*x)**3)*(-b**3*log(F)**3 + 3*b**2*c**3*log(F)**2 + 9*b**2* c**2*d*x*log(F)**2 + 9*b**2*c*d**2*x**2*log(F)**2 + 3*b**2*d**3*x**3*log(F )**2 - 6*b*c**6*log(F) - 36*b*c**5*d*x*log(F) - 90*b*c**4*d**2*x**2*log(F) - 120*b*c**3*d**3*x**3*log(F) - 90*b*c**2*d**4*x**4*log(F) - 36*b*c*d**5* x**5*log(F) - 6*b*d**6*x**6*log(F) + 6*c**9 + 54*c**8*d*x + 216*c**7*d**2* x**2 + 504*c**6*d**3*x**3 + 756*c**5*d**4*x**4 + 756*c**4*d**5*x**5 + 504* c**3*d**6*x**6 + 216*c**2*d**7*x**7 + 54*c*d**8*x**8 + 6*d**9*x**9)/(3*b** 4*c**9*d*log(F)**4 + 27*b**4*c**8*d**2*x*log(F)**4 + 108*b**4*c**7*d**3*x* *2*log(F)**4 + 252*b**4*c**6*d**4*x**3*log(F)**4 + 378*b**4*c**5*d**5*x**4 *log(F)**4 + 378*b**4*c**4*d**6*x**5*log(F)**4 + 252*b**4*c**3*d**7*x**6*l og(F)**4 + 108*b**4*c**2*d**8*x**7*log(F)**4 + 27*b**4*c*d**9*x**8*log(F)* *4 + 3*b**4*d**10*x**9*log(F)**4)
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (121) = 242\).
Time = 0.34 (sec) , antiderivative size = 507, normalized size of antiderivative = 4.12 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\frac {{\left (6 \, F^{a} d^{9} x^{9} + 54 \, F^{a} c d^{8} x^{8} + 216 \, F^{a} c^{2} d^{7} x^{7} + 6 \, F^{a} c^{9} - 6 \, F^{a} b c^{6} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{3} \log \left (F\right )^{2} + 6 \, {\left (84 \, F^{a} c^{3} d^{6} - F^{a} b d^{6} \log \left (F\right )\right )} x^{6} - F^{a} b^{3} \log \left (F\right )^{3} + 36 \, {\left (21 \, F^{a} c^{4} d^{5} - F^{a} b c d^{5} \log \left (F\right )\right )} x^{5} + 18 \, {\left (42 \, F^{a} c^{5} d^{4} - 5 \, F^{a} b c^{2} d^{4} \log \left (F\right )\right )} x^{4} + 3 \, {\left (168 \, F^{a} c^{6} d^{3} - 40 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + F^{a} b^{2} d^{3} \log \left (F\right )^{2}\right )} x^{3} + 9 \, {\left (24 \, F^{a} c^{7} d^{2} - 10 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + F^{a} b^{2} c d^{2} \log \left (F\right )^{2}\right )} x^{2} + 9 \, {\left (6 \, F^{a} c^{8} d - 4 \, F^{a} b c^{5} d \log \left (F\right ) + F^{a} b^{2} c^{2} d \log \left (F\right )^{2}\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{4} d^{10} x^{9} \log \left (F\right )^{4} + 9 \, b^{4} c d^{9} x^{8} \log \left (F\right )^{4} + 36 \, b^{4} c^{2} d^{8} x^{7} \log \left (F\right )^{4} + 84 \, b^{4} c^{3} d^{7} x^{6} \log \left (F\right )^{4} + 126 \, b^{4} c^{4} d^{6} x^{5} \log \left (F\right )^{4} + 126 \, b^{4} c^{5} d^{5} x^{4} \log \left (F\right )^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} \log \left (F\right )^{4} + 36 \, b^{4} c^{7} d^{3} x^{2} \log \left (F\right )^{4} + 9 \, b^{4} c^{8} d^{2} x \log \left (F\right )^{4} + b^{4} c^{9} d \log \left (F\right )^{4}\right )}} \]
1/3*(6*F^a*d^9*x^9 + 54*F^a*c*d^8*x^8 + 216*F^a*c^2*d^7*x^7 + 6*F^a*c^9 - 6*F^a*b*c^6*log(F) + 3*F^a*b^2*c^3*log(F)^2 + 6*(84*F^a*c^3*d^6 - F^a*b*d^ 6*log(F))*x^6 - F^a*b^3*log(F)^3 + 36*(21*F^a*c^4*d^5 - F^a*b*c*d^5*log(F) )*x^5 + 18*(42*F^a*c^5*d^4 - 5*F^a*b*c^2*d^4*log(F))*x^4 + 3*(168*F^a*c^6* d^3 - 40*F^a*b*c^3*d^3*log(F) + F^a*b^2*d^3*log(F)^2)*x^3 + 9*(24*F^a*c^7* d^2 - 10*F^a*b*c^4*d^2*log(F) + F^a*b^2*c*d^2*log(F)^2)*x^2 + 9*(6*F^a*c^8 *d - 4*F^a*b*c^5*d*log(F) + F^a*b^2*c^2*d*log(F)^2)*x)*F^(b/(d^3*x^3 + 3*c *d^2*x^2 + 3*c^2*d*x + c^3))/(b^4*d^10*x^9*log(F)^4 + 9*b^4*c*d^9*x^8*log( F)^4 + 36*b^4*c^2*d^8*x^7*log(F)^4 + 84*b^4*c^3*d^7*x^6*log(F)^4 + 126*b^4 *c^4*d^6*x^5*log(F)^4 + 126*b^4*c^5*d^5*x^4*log(F)^4 + 84*b^4*c^6*d^4*x^3* log(F)^4 + 36*b^4*c^7*d^3*x^2*log(F)^4 + 9*b^4*c^8*d^2*x*log(F)^4 + b^4*c^ 9*d*log(F)^4)
\[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{13}} \,d x } \]
Time = 1.28 (sec) , antiderivative size = 422, normalized size of antiderivative = 3.43 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx=\frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {2\,x^9}{b^4\,d\,{\ln \left (F\right )}^4}-\frac {b^3\,{\ln \left (F\right )}^3-3\,b^2\,c^3\,{\ln \left (F\right )}^2+6\,b\,c^6\,\ln \left (F\right )-6\,c^9}{3\,b^4\,d^{10}\,{\ln \left (F\right )}^4}+\frac {18\,c\,x^8}{b^4\,d^2\,{\ln \left (F\right )}^4}+\frac {72\,c^2\,x^7}{b^4\,d^3\,{\ln \left (F\right )}^4}+\frac {x^3\,\left (b^2\,{\ln \left (F\right )}^2-40\,b\,c^3\,\ln \left (F\right )+168\,c^6\right )}{b^4\,d^7\,{\ln \left (F\right )}^4}-\frac {2\,x^6\,\left (b\,\ln \left (F\right )-84\,c^3\right )}{b^4\,d^4\,{\ln \left (F\right )}^4}+\frac {3\,c^2\,x\,\left (b^2\,{\ln \left (F\right )}^2-4\,b\,c^3\,\ln \left (F\right )+6\,c^6\right )}{b^4\,d^9\,{\ln \left (F\right )}^4}+\frac {3\,c\,x^2\,\left (b^2\,{\ln \left (F\right )}^2-10\,b\,c^3\,\ln \left (F\right )+24\,c^6\right )}{b^4\,d^8\,{\ln \left (F\right )}^4}-\frac {12\,c\,x^5\,\left (b\,\ln \left (F\right )-21\,c^3\right )}{b^4\,d^5\,{\ln \left (F\right )}^4}-\frac {6\,c^2\,x^4\,\left (5\,b\,\ln \left (F\right )-42\,c^3\right )}{b^4\,d^6\,{\ln \left (F\right )}^4}\right )}{x^9+\frac {c^9}{d^9}+\frac {9\,c\,x^8}{d}+\frac {9\,c^8\,x}{d^8}+\frac {36\,c^2\,x^7}{d^2}+\frac {84\,c^3\,x^6}{d^3}+\frac {126\,c^4\,x^5}{d^4}+\frac {126\,c^5\,x^4}{d^5}+\frac {84\,c^6\,x^3}{d^6}+\frac {36\,c^7\,x^2}{d^7}} \]
(F^a*F^(b/(c^3 + d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x))*((2*x^9)/(b^4*d*log(F )^4) - (b^3*log(F)^3 - 6*c^9 + 6*b*c^6*log(F) - 3*b^2*c^3*log(F)^2)/(3*b^4 *d^10*log(F)^4) + (18*c*x^8)/(b^4*d^2*log(F)^4) + (72*c^2*x^7)/(b^4*d^3*lo g(F)^4) + (x^3*(b^2*log(F)^2 + 168*c^6 - 40*b*c^3*log(F)))/(b^4*d^7*log(F) ^4) - (2*x^6*(b*log(F) - 84*c^3))/(b^4*d^4*log(F)^4) + (3*c^2*x*(b^2*log(F )^2 + 6*c^6 - 4*b*c^3*log(F)))/(b^4*d^9*log(F)^4) + (3*c*x^2*(b^2*log(F)^2 + 24*c^6 - 10*b*c^3*log(F)))/(b^4*d^8*log(F)^4) - (12*c*x^5*(b*log(F) - 2 1*c^3))/(b^4*d^5*log(F)^4) - (6*c^2*x^4*(5*b*log(F) - 42*c^3))/(b^4*d^6*lo g(F)^4)))/(x^9 + c^9/d^9 + (9*c*x^8)/d + (9*c^8*x)/d^8 + (36*c^2*x^7)/d^2 + (84*c^3*x^6)/d^3 + (126*c^4*x^5)/d^4 + (126*c^5*x^4)/d^5 + (84*c^6*x^3)/ d^6 + (36*c^7*x^2)/d^7)