Integrand size = 21, antiderivative size = 92 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=-\frac {F^{a+\frac {b}{c+d x}} \left (24 (c+d x)^4-24 b (c+d x)^3 \log (F)+12 b^2 (c+d x)^2 \log ^2(F)-4 b^3 (c+d x) \log ^3(F)+b^4 \log ^4(F)\right )}{b^5 d (c+d x)^4 \log ^5(F)} \] Output:
-F^(a+b/(d*x+c))*(24*(d*x+c)^4-24*b*(d*x+c)^3*ln(F)+12*b^2*(d*x+c)^2*ln(F) ^2-4*b^3*(d*x+c)*ln(F)^3+b^4*ln(F)^4)/b^5/d/(d*x+c)^4/ln(F)^5
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.32 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{c+d x}\right )}{b^5 d \log ^5(F)} \] Input:
Integrate[F^(a + b/(c + d*x))/(c + d*x)^6,x]
Output:
-((F^a*Gamma[5, -((b*Log[F])/(c + d*x))])/(b^5*d*Log[F]^5))
Time = 0.40 (sec) , antiderivative size = 92, 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.
| \(\displaystyle \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx\) |
| \(\Big \downarrow \) 2647 |
| \(\displaystyle -\frac {F^{a+\frac {b}{c+d x}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)+12 b^2 \log ^2(F) (c+d x)^2-24 b \log (F) (c+d x)^3+24 (c+d x)^4\right )}{b^5 d \log ^5(F) (c+d x)^4}\) |
Input:
Int[F^(a + b/(c + d*x))/(c + d*x)^6,x]
Output:
-((F^(a + b/(c + d*x))*(24*(c + d*x)^4 - 24*b*(c + d*x)^3*Log[F] + 12*b^2* (c + d*x)^2*Log[F]^2 - 4*b^3*(c + d*x)*Log[F]^3 + b^4*Log[F]^4))/(b^5*d*(c + d*x)^4*Log[F]^5))
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. \(188\) vs. \(2(92)=184\).
Time = 0.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.05
| method | result | size |
| risch | \(-\frac {\left (b^{4} \ln \left (F \right )^{4}-4 \ln \left (F \right )^{3} b^{3} d x +12 \ln \left (F \right )^{2} b^{2} d^{2} x^{2}-24 \ln \left (F \right ) b \,d^{3} x^{3}+24 d^{4} x^{4}-4 \ln \left (F \right )^{3} b^{3} c +24 \ln \left (F \right )^{2} b^{2} c d x -72 \ln \left (F \right ) b c \,d^{2} x^{2}+96 c \,d^{3} x^{3}+12 \ln \left (F \right )^{2} b^{2} c^{2}-72 \ln \left (F \right ) b \,c^{2} d x +144 c^{2} d^{2} x^{2}-24 \ln \left (F \right ) b \,c^{3}+96 c^{3} d x +24 c^{4}\right ) F^{\frac {a d x +a c +b}{d x +c}}}{b^{5} \ln \left (F \right )^{5} d \left (d x +c \right )^{4}}\) | \(189\) |
| norman | \(\frac {-\frac {24 d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {\left (b^{4} \ln \left (F \right )^{4}-8 \ln \left (F \right )^{3} b^{3} c +36 \ln \left (F \right )^{2} b^{2} c^{2}-96 \ln \left (F \right ) b \,c^{3}+120 c^{4}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {4 d \left (\ln \left (F \right )^{3} b^{3}-9 \ln \left (F \right )^{2} b^{2} c +36 \ln \left (F \right ) b \,c^{2}-60 c^{3}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {12 d^{2} \left (\ln \left (F \right )^{2} b^{2}-8 b c \ln \left (F \right )+20 c^{2}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {24 d^{3} \left (b \ln \left (F \right )-5 c \right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {\left (b^{4} \ln \left (F \right )^{4}-4 \ln \left (F \right )^{3} b^{3} c +12 \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b \,c^{3}+24 c^{4}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{b^{5} \ln \left (F \right )^{5} d}}{\left (d x +c \right )^{5}}\) | \(329\) |
| parallelrisch | \(\frac {-\ln \left (F \right )^{4} F^{a +\frac {b}{d x +c}} b^{4} d^{7}+4 \ln \left (F \right )^{3} x \,F^{a +\frac {b}{d x +c}} b^{3} d^{8}-12 \ln \left (F \right )^{2} x^{2} F^{a +\frac {b}{d x +c}} b^{2} d^{9}+24 \ln \left (F \right ) x^{3} F^{a +\frac {b}{d x +c}} b \,d^{10}-24 d^{11} F^{a +\frac {b}{d x +c}} x^{4}+4 \ln \left (F \right )^{3} F^{a +\frac {b}{d x +c}} b^{3} c \,d^{7}-24 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{d x +c}} b^{2} c \,d^{8}+72 \ln \left (F \right ) x^{2} F^{a +\frac {b}{d x +c}} b c \,d^{9}-96 d^{10} c \,F^{a +\frac {b}{d x +c}} x^{3}-12 \ln \left (F \right )^{2} F^{a +\frac {b}{d x +c}} b^{2} c^{2} d^{7}+72 \ln \left (F \right ) x \,F^{a +\frac {b}{d x +c}} b \,c^{2} d^{8}-144 d^{9} c^{2} F^{a +\frac {b}{d x +c}} x^{2}+24 \ln \left (F \right ) F^{a +\frac {b}{d x +c}} b \,c^{3} d^{7}-96 c^{3} F^{a +\frac {b}{d x +c}} x \,d^{8}-24 c^{4} F^{a +\frac {b}{d x +c}} d^{7}}{\left (d x +c \right )^{4} b^{5} \ln \left (F \right )^{5} d^{8}}\) | \(388\) |
Input:
int(F^(a+b/(d*x+c))/(d*x+c)^6,x,method=_RETURNVERBOSE)
Output:
-(b^4*ln(F)^4-4*ln(F)^3*b^3*d*x+12*ln(F)^2*b^2*d^2*x^2-24*ln(F)*b*d^3*x^3+ 24*d^4*x^4-4*ln(F)^3*b^3*c+24*ln(F)^2*b^2*c*d*x-72*ln(F)*b*c*d^2*x^2+96*c* d^3*x^3+12*ln(F)^2*b^2*c^2-72*ln(F)*b*c^2*d*x+144*c^2*d^2*x^2-24*ln(F)*b*c ^3+96*c^3*d*x+24*c^4)/b^5/ln(F)^5/d/(d*x+c)^4*F^((a*d*x+a*c+b)/(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (92) = 184\).
Time = 0.08 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.38 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=-\frac {{\left (24 \, d^{4} x^{4} + b^{4} \log \left (F\right )^{4} + 96 \, c d^{3} x^{3} + 144 \, c^{2} d^{2} x^{2} + 96 \, c^{3} d x + 24 \, c^{4} - 4 \, {\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{5} d^{5} x^{4} + 4 \, b^{5} c d^{4} x^{3} + 6 \, b^{5} c^{2} d^{3} x^{2} + 4 \, b^{5} c^{3} d^{2} x + b^{5} c^{4} d\right )} \log \left (F\right )^{5}} \] Input:
integrate(F^(a+b/(d*x+c))/(d*x+c)^6,x, algorithm="fricas")
Output:
-(24*d^4*x^4 + b^4*log(F)^4 + 96*c*d^3*x^3 + 144*c^2*d^2*x^2 + 96*c^3*d*x + 24*c^4 - 4*(b^3*d*x + b^3*c)*log(F)^3 + 12*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(F)^2 - 24*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*l og(F))*F^((a*d*x + a*c + b)/(d*x + c))/((b^5*d^5*x^4 + 4*b^5*c*d^4*x^3 + 6 *b^5*c^2*d^3*x^2 + 4*b^5*c^3*d^2*x + b^5*c^4*d)*log(F)^5)
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (90) = 180\).
Time = 0.14 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.96 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=\frac {F^{a + \frac {b}{c + d x}} \left (- b^{4} \log {\left (F \right )}^{4} + 4 b^{3} c \log {\left (F \right )}^{3} + 4 b^{3} d x \log {\left (F \right )}^{3} - 12 b^{2} c^{2} \log {\left (F \right )}^{2} - 24 b^{2} c d x \log {\left (F \right )}^{2} - 12 b^{2} d^{2} x^{2} \log {\left (F \right )}^{2} + 24 b c^{3} \log {\left (F \right )} + 72 b c^{2} d x \log {\left (F \right )} + 72 b c d^{2} x^{2} \log {\left (F \right )} + 24 b d^{3} x^{3} \log {\left (F \right )} - 24 c^{4} - 96 c^{3} d x - 144 c^{2} d^{2} x^{2} - 96 c d^{3} x^{3} - 24 d^{4} x^{4}\right )}{b^{5} c^{4} d \log {\left (F \right )}^{5} + 4 b^{5} c^{3} d^{2} x \log {\left (F \right )}^{5} + 6 b^{5} c^{2} d^{3} x^{2} \log {\left (F \right )}^{5} + 4 b^{5} c d^{4} x^{3} \log {\left (F \right )}^{5} + b^{5} d^{5} x^{4} \log {\left (F \right )}^{5}} \] Input:
integrate(F**(a+b/(d*x+c))/(d*x+c)**6,x)
Output:
F**(a + b/(c + d*x))*(-b**4*log(F)**4 + 4*b**3*c*log(F)**3 + 4*b**3*d*x*lo g(F)**3 - 12*b**2*c**2*log(F)**2 - 24*b**2*c*d*x*log(F)**2 - 12*b**2*d**2* x**2*log(F)**2 + 24*b*c**3*log(F) + 72*b*c**2*d*x*log(F) + 72*b*c*d**2*x** 2*log(F) + 24*b*d**3*x**3*log(F) - 24*c**4 - 96*c**3*d*x - 144*c**2*d**2*x **2 - 96*c*d**3*x**3 - 24*d**4*x**4)/(b**5*c**4*d*log(F)**5 + 4*b**5*c**3* d**2*x*log(F)**5 + 6*b**5*c**2*d**3*x**2*log(F)**5 + 4*b**5*c*d**4*x**3*lo g(F)**5 + b**5*d**5*x**4*log(F)**5)
\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{6}} \,d x } \] Input:
integrate(F^(a+b/(d*x+c))/(d*x+c)^6,x, algorithm="maxima")
Output:
integrate(F^(a + b/(d*x + c))/(d*x + c)^6, x)
\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{6}} \,d x } \] Input:
integrate(F^(a+b/(d*x+c))/(d*x+c)^6,x, algorithm="giac")
Output:
integrate(F^(a + b/(d*x + c))/(d*x + c)^6, x)
Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.51 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=-\frac {F^{a+\frac {b}{c+d\,x}}\,\left (\frac {b^4\,{\ln \left (F\right )}^4-4\,b^3\,c\,{\ln \left (F\right )}^3+12\,b^2\,c^2\,{\ln \left (F\right )}^2-24\,b\,c^3\,\ln \left (F\right )+24\,c^4}{b^5\,d^5\,{\ln \left (F\right )}^5}+\frac {24\,x^4}{b^5\,d\,{\ln \left (F\right )}^5}+\frac {x^2\,\left (12\,b^2\,{\ln \left (F\right )}^2-72\,b\,c\,\ln \left (F\right )+144\,c^2\right )}{b^5\,d^3\,{\ln \left (F\right )}^5}+\frac {x^3\,\left (96\,c-24\,b\,\ln \left (F\right )\right )}{b^5\,d^2\,{\ln \left (F\right )}^5}-\frac {4\,x\,\left (b^3\,{\ln \left (F\right )}^3-6\,b^2\,c\,{\ln \left (F\right )}^2+18\,b\,c^2\,\ln \left (F\right )-24\,c^3\right )}{b^5\,d^4\,{\ln \left (F\right )}^5}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \] Input:
int(F^(a + b/(c + d*x))/(c + d*x)^6,x)
Output:
-(F^(a + b/(c + d*x))*((b^4*log(F)^4 + 24*c^4 - 24*b*c^3*log(F) - 4*b^3*c* log(F)^3 + 12*b^2*c^2*log(F)^2)/(b^5*d^5*log(F)^5) + (24*x^4)/(b^5*d*log(F )^5) + (x^2*(12*b^2*log(F)^2 + 144*c^2 - 72*b*c*log(F)))/(b^5*d^3*log(F)^5 ) + (x^3*(96*c - 24*b*log(F)))/(b^5*d^2*log(F)^5) - (4*x*(b^3*log(F)^3 - 2 4*c^3 + 18*b*c^2*log(F) - 6*b^2*c*log(F)^2))/(b^5*d^4*log(F)^5)))/(x^4 + c ^4/d^4 + (4*c*x^3)/d + (4*c^3*x)/d^3 + (6*c^2*x^2)/d^2)
Time = 0.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.40 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx=\frac {f^{\frac {a d x +a c +b}{d x +c}} \left (-\mathrm {log}\left (f \right )^{4} b^{4}+4 \mathrm {log}\left (f \right )^{3} b^{3} c +4 \mathrm {log}\left (f \right )^{3} b^{3} d x -12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2}-24 \mathrm {log}\left (f \right )^{2} b^{2} c d x -12 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} x^{2}+24 \,\mathrm {log}\left (f \right ) b \,c^{3}+72 \,\mathrm {log}\left (f \right ) b \,c^{2} d x +72 \,\mathrm {log}\left (f \right ) b c \,d^{2} x^{2}+24 \,\mathrm {log}\left (f \right ) b \,d^{3} x^{3}-24 c^{4}-96 c^{3} d x -144 c^{2} d^{2} x^{2}-96 c \,d^{3} x^{3}-24 d^{4} x^{4}\right )}{\mathrm {log}\left (f \right )^{5} b^{5} d \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] Input:
int(F^(a+b/(d*x+c))/(d*x+c)^6,x)
Output:
(f**((a*c + a*d*x + b)/(c + d*x))*( - log(f)**4*b**4 + 4*log(f)**3*b**3*c + 4*log(f)**3*b**3*d*x - 12*log(f)**2*b**2*c**2 - 24*log(f)**2*b**2*c*d*x - 12*log(f)**2*b**2*d**2*x**2 + 24*log(f)*b*c**3 + 72*log(f)*b*c**2*d*x + 72*log(f)*b*c*d**2*x**2 + 24*log(f)*b*d**3*x**3 - 24*c**4 - 96*c**3*d*x - 144*c**2*d**2*x**2 - 96*c*d**3*x**3 - 24*d**4*x**4))/(log(f)**5*b**5*d*(c* *4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4))