\(\int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx\) [285]

Optimal result

Integrand size = 21, antiderivative size = 96 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx=-\frac {F^{a+\frac {b}{(c+d x)^3}} \left (24 (c+d x)^{12}-24 b (c+d x)^9 \log (F)+12 b^2 (c+d x)^6 \log ^2(F)-4 b^3 (c+d x)^3 \log ^3(F)+b^4 \log ^4(F)\right )}{3 b^5 d (c+d x)^{12} \log ^5(F)} \] Output:

-1/3*F^(a+b/(d*x+c)^3)*(24*(d*x+c)^12-24*b*(d*x+c)^9*ln(F)+12*b^2*(d*x+c)^ 
6*ln(F)^2-4*b^3*(d*x+c)^3*ln(F)^3+b^4*ln(F)^4)/b^5/d/(d*x+c)^12/ln(F)^5
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)} \] Input:

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^16,x]
 

Output:

-1/3*(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^3)])/(b^5*d*Log[F]^5)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 96, 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.

\(\displaystyle \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx\)

\(\Big \downarrow \) 2647

\(\displaystyle -\frac {F^{a+\frac {b}{(c+d x)^3}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)^3+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^9+24 (c+d x)^{12}\right )}{3 b^5 d \log ^5(F) (c+d x)^{12}}\)

Input:

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^16,x]
 

Output:

-1/3*(F^(a + b/(c + d*x)^3)*(24*(c + d*x)^12 - 24*b*(c + d*x)^9*Log[F] + 1 
2*b^2*(c + d*x)^6*Log[F]^2 - 4*b^3*(c + d*x)^3*Log[F]^3 + b^4*Log[F]^4))/( 
b^5*d*(c + d*x)^12*Log[F]^5)
 

Defintions of rubi rules used

rule 2647

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(492\) vs. \(2(94)=188\).

Time = 10.55 (sec) , antiderivative size = 493, normalized size of antiderivative = 5.14

method	result	size
risch	\(-\frac {\left (12 \ln \left (F \right )^{2} b^{2} c^{6}+24 d^{12} x^{12}+24 c^{12}+b^{4} \ln \left (F \right )^{4}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-3024 \ln \left (F \right ) b \,c^{5} d^{4} x^{4}-2016 \ln \left (F \right ) b \,c^{6} d^{3} x^{3}-864 \ln \left (F \right ) b \,c^{7} d^{2} x^{2}-216 \ln \left (F \right ) b \,c^{8} d x -12 \ln \left (F \right )^{3} b^{3} c \,d^{2} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{2} d x +288 c \,d^{11} x^{11}+1584 c^{2} d^{10} x^{10}+5280 c^{3} d^{9} x^{9}+11880 c^{4} d^{8} x^{8}+19008 c^{5} d^{7} x^{7}+22176 c^{6} d^{6} x^{6}+19008 c^{7} d^{5} x^{5}+11880 c^{8} d^{4} x^{4}+5280 c^{9} d^{3} x^{3}+1584 c^{10} d^{2} x^{2}+288 c^{11} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x -24 \ln \left (F \right ) b \,c^{9}-4 \ln \left (F \right )^{3} b^{3} c^{3}-216 \ln \left (F \right ) b c \,d^{8} x^{8}-864 \ln \left (F \right ) b \,c^{2} d^{7} x^{7}-2016 \ln \left (F \right ) b \,c^{3} d^{6} x^{6}-3024 \ln \left (F \right ) b \,c^{4} d^{5} x^{5}-24 \ln \left (F \right ) b \,d^{9} x^{9}-4 \ln \left (F \right )^{3} b^{3} d^{3} x^{3}\right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 b^{5} \ln \left (F \right )^{5} d \left (d x +c \right )^{12}}\)	\(493\)
norman	\(\text {Expression too large to display}\)	\(889\)
parallelrisch	\(\frac {-288 x^{11} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c \,d^{30}-1584 x^{10} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{2} d^{29}-5280 x^{9} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{3} d^{28}-11880 x^{8} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{4} d^{27}-19008 x^{7} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{5} d^{26}-22176 x^{6} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{6} d^{25}-19008 x^{5} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{7} d^{24}-11880 x^{4} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{8} d^{23}-5280 x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{9} d^{22}-1584 x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{10} d^{21}-288 x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{11} d^{20}-\ln \left (F \right )^{4} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{4} d^{19}-24 F^{a +\frac {b}{\left (d x +c \right )^{3}}} c^{12} d^{19}-24 x^{12} F^{a +\frac {b}{\left (d x +c \right )^{3}}} d^{31}+24 \ln \left (F \right ) x^{9} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,d^{28}-12 \ln \left (F \right )^{2} x^{6} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} d^{25}+4 \ln \left (F \right )^{3} x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{3} d^{22}+24 \ln \left (F \right ) F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{9} d^{19}-12 \ln \left (F \right )^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{6} d^{19}+4 \ln \left (F \right )^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{3} c^{3} d^{19}+216 \ln \left (F \right ) x^{8} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b c \,d^{27}+864 \ln \left (F \right ) x^{7} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{2} d^{26}+2016 \ln \left (F \right ) x^{6} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{3} d^{25}+3024 \ln \left (F \right ) x^{5} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{4} d^{24}-72 \ln \left (F \right )^{2} x^{5} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c \,d^{24}+3024 \ln \left (F \right ) x^{4} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{5} d^{23}-180 \ln \left (F \right )^{2} x^{4} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{2} d^{23}+2016 \ln \left (F \right ) x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{6} d^{22}-240 \ln \left (F \right )^{2} x^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{3} d^{22}+864 \ln \left (F \right ) x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{7} d^{21}-180 \ln \left (F \right )^{2} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{4} d^{21}+216 \ln \left (F \right ) x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} b \,c^{8} d^{20}+12 \ln \left (F \right )^{3} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{3} c \,d^{21}-72 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{2} c^{5} d^{20}+12 \ln \left (F \right )^{3} x \,F^{a +\frac {b}{\left (d x +c \right )^{3}}} b^{3} c^{2} d^{20}}{3 \left (d x +c \right )^{12} \ln \left (F \right )^{5} b^{5} d^{20}}\)	\(929\)

Input:

int(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x,method=_RETURNVERBOSE)
 

Output:

-1/3*(12*ln(F)^2*b^2*c^6+24*d^12*x^12+24*c^12+b^4*ln(F)^4+12*d^6*x^6*ln(F) 
^2*b^2-3024*ln(F)*b*c^5*d^4*x^4-2016*ln(F)*b*c^6*d^3*x^3-864*ln(F)*b*c^7*d 
^2*x^2-216*ln(F)*b*c^8*d*x-12*ln(F)^3*b^3*c*d^2*x^2-12*ln(F)^3*b^3*c^2*d*x 
+288*c*d^11*x^11+1584*c^2*d^10*x^10+5280*c^3*d^9*x^9+11880*c^4*d^8*x^8+190 
08*c^5*d^7*x^7+22176*c^6*d^6*x^6+19008*c^7*d^5*x^5+11880*c^8*d^4*x^4+5280* 
c^9*d^3*x^3+1584*c^10*d^2*x^2+288*c^11*d*x+72*c*d^5*x^5*ln(F)^2*b^2+180*c^ 
2*d^4*x^4*ln(F)^2*b^2+240*ln(F)^2*b^2*c^3*d^3*x^3+180*ln(F)^2*b^2*c^4*d^2* 
x^2+72*ln(F)^2*b^2*c^5*d*x-24*ln(F)*b*c^9-4*ln(F)^3*b^3*c^3-216*ln(F)*b*c* 
d^8*x^8-864*ln(F)*b*c^2*d^7*x^7-2016*ln(F)*b*c^3*d^6*x^6-3024*ln(F)*b*c^4* 
d^5*x^5-24*ln(F)*b*d^9*x^9-4*ln(F)^3*b^3*d^3*x^3)/b^5/ln(F)^5/d/(d*x+c)^12 
*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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (94) = 188\).

Time = 0.12 (sec) , antiderivative size = 621, normalized size of antiderivative = 6.47 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx=-\frac {{\left (24 \, d^{12} x^{12} + 288 \, c d^{11} x^{11} + 1584 \, c^{2} d^{10} x^{10} + 5280 \, c^{3} d^{9} x^{9} + 11880 \, c^{4} d^{8} x^{8} + 19008 \, c^{5} d^{7} x^{7} + 22176 \, c^{6} d^{6} x^{6} + 19008 \, c^{7} d^{5} x^{5} + 11880 \, c^{8} d^{4} x^{4} + 5280 \, c^{9} d^{3} x^{3} + 1584 \, c^{10} d^{2} x^{2} + 288 \, c^{11} d x + 24 \, c^{12} + b^{4} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{5} d^{13} x^{12} + 12 \, b^{5} c d^{12} x^{11} + 66 \, b^{5} c^{2} d^{11} x^{10} + 220 \, b^{5} c^{3} d^{10} x^{9} + 495 \, b^{5} c^{4} d^{9} x^{8} + 792 \, b^{5} c^{5} d^{8} x^{7} + 924 \, b^{5} c^{6} d^{7} x^{6} + 792 \, b^{5} c^{7} d^{6} x^{5} + 495 \, b^{5} c^{8} d^{5} x^{4} + 220 \, b^{5} c^{9} d^{4} x^{3} + 66 \, b^{5} c^{10} d^{3} x^{2} + 12 \, b^{5} c^{11} d^{2} x + b^{5} c^{12} d\right )} \log \left (F\right )^{5}} \] Input:

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="fricas")
 

Output:

-1/3*(24*d^12*x^12 + 288*c*d^11*x^11 + 1584*c^2*d^10*x^10 + 5280*c^3*d^9*x 
^9 + 11880*c^4*d^8*x^8 + 19008*c^5*d^7*x^7 + 22176*c^6*d^6*x^6 + 19008*c^7 
*d^5*x^5 + 11880*c^8*d^4*x^4 + 5280*c^9*d^3*x^3 + 1584*c^10*d^2*x^2 + 288* 
c^11*d*x + 24*c^12 + b^4*log(F)^4 - 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b 
^3*c^2*d*x + b^3*c^3)*log(F)^3 + 12*(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 - 24*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d^7*x^7 + 84* 
b*c^3*d^6*x^6 + 126*b*c^4*d^5*x^5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 
 36*b*c^7*d^2*x^2 + 9*b*c^8*d*x + b*c^9)*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^5*d^13*x^12 + 12*b^5*c*d^12*x^11 + 66*b^5*c^2*d^11*x^10 + 220*b^5*c^3 
*d^10*x^9 + 495*b^5*c^4*d^9*x^8 + 792*b^5*c^5*d^8*x^7 + 924*b^5*c^6*d^7*x^ 
6 + 792*b^5*c^7*d^6*x^5 + 495*b^5*c^8*d^5*x^4 + 220*b^5*c^9*d^4*x^3 + 66*b 
^5*c^10*d^3*x^2 + 12*b^5*c^11*d^2*x + b^5*c^12*d)*log(F)^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (95) = 190\).

Time = 0.57 (sec) , antiderivative size = 760, normalized size of antiderivative = 7.92 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx =\text {Too large to display} \] Input:

integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**16,x)
 

Output:

F**(a + b/(c + d*x)**3)*(-b**4*log(F)**4 + 4*b**3*c**3*log(F)**3 + 12*b**3 
*c**2*d*x*log(F)**3 + 12*b**3*c*d**2*x**2*log(F)**3 + 4*b**3*d**3*x**3*log 
(F)**3 - 12*b**2*c**6*log(F)**2 - 72*b**2*c**5*d*x*log(F)**2 - 180*b**2*c* 
*4*d**2*x**2*log(F)**2 - 240*b**2*c**3*d**3*x**3*log(F)**2 - 180*b**2*c**2 
*d**4*x**4*log(F)**2 - 72*b**2*c*d**5*x**5*log(F)**2 - 12*b**2*d**6*x**6*l 
og(F)**2 + 24*b*c**9*log(F) + 216*b*c**8*d*x*log(F) + 864*b*c**7*d**2*x**2 
*log(F) + 2016*b*c**6*d**3*x**3*log(F) + 3024*b*c**5*d**4*x**4*log(F) + 30 
24*b*c**4*d**5*x**5*log(F) + 2016*b*c**3*d**6*x**6*log(F) + 864*b*c**2*d** 
7*x**7*log(F) + 216*b*c*d**8*x**8*log(F) + 24*b*d**9*x**9*log(F) - 24*c**1 
2 - 288*c**11*d*x - 1584*c**10*d**2*x**2 - 5280*c**9*d**3*x**3 - 11880*c** 
8*d**4*x**4 - 19008*c**7*d**5*x**5 - 22176*c**6*d**6*x**6 - 19008*c**5*d** 
7*x**7 - 11880*c**4*d**8*x**8 - 5280*c**3*d**9*x**9 - 1584*c**2*d**10*x**1 
0 - 288*c*d**11*x**11 - 24*d**12*x**12)/(3*b**5*c**12*d*log(F)**5 + 36*b** 
5*c**11*d**2*x*log(F)**5 + 198*b**5*c**10*d**3*x**2*log(F)**5 + 660*b**5*c 
**9*d**4*x**3*log(F)**5 + 1485*b**5*c**8*d**5*x**4*log(F)**5 + 2376*b**5*c 
**7*d**6*x**5*log(F)**5 + 2772*b**5*c**6*d**7*x**6*log(F)**5 + 2376*b**5*c 
**5*d**8*x**7*log(F)**5 + 1485*b**5*c**4*d**9*x**8*log(F)**5 + 660*b**5*c* 
*3*d**10*x**9*log(F)**5 + 198*b**5*c**2*d**11*x**10*log(F)**5 + 36*b**5*c* 
d**12*x**11*log(F)**5 + 3*b**5*d**13*x**12*log(F)**5)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (94) = 188\).

Time = 0.06 (sec) , antiderivative size = 770, normalized size of antiderivative = 8.02 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx =\text {Too large to display} \] Input:

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="maxima")
 

Output:

-1/3*(24*F^a*d^12*x^12 + 288*F^a*c*d^11*x^11 + 1584*F^a*c^2*d^10*x^10 + 24 
*F^a*c^12 - 24*F^a*b*c^9*log(F) + 12*F^a*b^2*c^6*log(F)^2 + 24*(220*F^a*c^ 
3*d^9 - F^a*b*d^9*log(F))*x^9 - 4*F^a*b^3*c^3*log(F)^3 + 216*(55*F^a*c^4*d 
^8 - F^a*b*c*d^8*log(F))*x^8 + F^a*b^4*log(F)^4 + 864*(22*F^a*c^5*d^7 - F^ 
a*b*c^2*d^7*log(F))*x^7 + 12*(1848*F^a*c^6*d^6 - 168*F^a*b*c^3*d^6*log(F) 
+ F^a*b^2*d^6*log(F)^2)*x^6 + 72*(264*F^a*c^7*d^5 - 42*F^a*b*c^4*d^5*log(F 
) + F^a*b^2*c*d^5*log(F)^2)*x^5 + 36*(330*F^a*c^8*d^4 - 84*F^a*b*c^5*d^4*l 
og(F) + 5*F^a*b^2*c^2*d^4*log(F)^2)*x^4 + 4*(1320*F^a*c^9*d^3 - 504*F^a*b* 
c^6*d^3*log(F) + 60*F^a*b^2*c^3*d^3*log(F)^2 - F^a*b^3*d^3*log(F)^3)*x^3 + 
 12*(132*F^a*c^10*d^2 - 72*F^a*b*c^7*d^2*log(F) + 15*F^a*b^2*c^4*d^2*log(F 
)^2 - F^a*b^3*c*d^2*log(F)^3)*x^2 + 12*(24*F^a*c^11*d - 18*F^a*b*c^8*d*log 
(F) + 6*F^a*b^2*c^5*d*log(F)^2 - F^a*b^3*c^2*d*log(F)^3)*x)*F^(b/(d^3*x^3 
+ 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(b^5*d^13*x^12*log(F)^5 + 12*b^5*c*d^12* 
x^11*log(F)^5 + 66*b^5*c^2*d^11*x^10*log(F)^5 + 220*b^5*c^3*d^10*x^9*log(F 
)^5 + 495*b^5*c^4*d^9*x^8*log(F)^5 + 792*b^5*c^5*d^8*x^7*log(F)^5 + 924*b^ 
5*c^6*d^7*x^6*log(F)^5 + 792*b^5*c^7*d^6*x^5*log(F)^5 + 495*b^5*c^8*d^5*x^ 
4*log(F)^5 + 220*b^5*c^9*d^4*x^3*log(F)^5 + 66*b^5*c^10*d^3*x^2*log(F)^5 + 
 12*b^5*c^11*d^2*x*log(F)^5 + b^5*c^12*d*log(F)^5)
 

Giac [F]

\[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{16}} \,d x } \] Input:

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="giac")
 

Output:

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^16, x)
 

Mupad [B] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 620, normalized size of antiderivative = 6.46 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, 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 {b^4\,{\ln \left (F\right )}^4-4\,b^3\,c^3\,{\ln \left (F\right )}^3+12\,b^2\,c^6\,{\ln \left (F\right )}^2-24\,b\,c^9\,\ln \left (F\right )+24\,c^{12}}{3\,b^5\,d^{13}\,{\ln \left (F\right )}^5}+\frac {8\,x^{12}}{b^5\,d\,{\ln \left (F\right )}^5}+\frac {96\,c\,x^{11}}{b^5\,d^2\,{\ln \left (F\right )}^5}+\frac {528\,c^2\,x^{10}}{b^5\,d^3\,{\ln \left (F\right )}^5}-\frac {4\,x^3\,\left (b^3\,{\ln \left (F\right )}^3-60\,b^2\,c^3\,{\ln \left (F\right )}^2+504\,b\,c^6\,\ln \left (F\right )-1320\,c^9\right )}{3\,b^5\,d^{10}\,{\ln \left (F\right )}^5}+\frac {4\,x^6\,\left (b^2\,{\ln \left (F\right )}^2-168\,b\,c^3\,\ln \left (F\right )+1848\,c^6\right )}{b^5\,d^7\,{\ln \left (F\right )}^5}-\frac {8\,x^9\,\left (b\,\ln \left (F\right )-220\,c^3\right )}{b^5\,d^4\,{\ln \left (F\right )}^5}-\frac {4\,c^2\,x\,\left (b^3\,{\ln \left (F\right )}^3-6\,b^2\,c^3\,{\ln \left (F\right )}^2+18\,b\,c^6\,\ln \left (F\right )-24\,c^9\right )}{b^5\,d^{12}\,{\ln \left (F\right )}^5}-\frac {4\,c\,x^2\,\left (b^3\,{\ln \left (F\right )}^3-15\,b^2\,c^3\,{\ln \left (F\right )}^2+72\,b\,c^6\,\ln \left (F\right )-132\,c^9\right )}{b^5\,d^{11}\,{\ln \left (F\right )}^5}+\frac {24\,c\,x^5\,\left (b^2\,{\ln \left (F\right )}^2-42\,b\,c^3\,\ln \left (F\right )+264\,c^6\right )}{b^5\,d^8\,{\ln \left (F\right )}^5}-\frac {72\,c\,x^8\,\left (b\,\ln \left (F\right )-55\,c^3\right )}{b^5\,d^5\,{\ln \left (F\right )}^5}+\frac {12\,c^2\,x^4\,\left (5\,b^2\,{\ln \left (F\right )}^2-84\,b\,c^3\,\ln \left (F\right )+330\,c^6\right )}{b^5\,d^9\,{\ln \left (F\right )}^5}-\frac {288\,c^2\,x^7\,\left (b\,\ln \left (F\right )-22\,c^3\right )}{b^5\,d^6\,{\ln \left (F\right )}^5}\right )}{x^{12}+\frac {c^{12}}{d^{12}}+\frac {12\,c\,x^{11}}{d}+\frac {12\,c^{11}\,x}{d^{11}}+\frac {66\,c^2\,x^{10}}{d^2}+\frac {220\,c^3\,x^9}{d^3}+\frac {495\,c^4\,x^8}{d^4}+\frac {792\,c^5\,x^7}{d^5}+\frac {924\,c^6\,x^6}{d^6}+\frac {792\,c^7\,x^5}{d^7}+\frac {495\,c^8\,x^4}{d^8}+\frac {220\,c^9\,x^3}{d^9}+\frac {66\,c^{10}\,x^2}{d^{10}}} \] Input:

int(F^(a + b/(c + d*x)^3)/(c + d*x)^16,x)
 

Output:

-(F^a*F^(b/(c^3 + d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x))*((b^4*log(F)^4 + 24* 
c^12 - 24*b*c^9*log(F) - 4*b^3*c^3*log(F)^3 + 12*b^2*c^6*log(F)^2)/(3*b^5* 
d^13*log(F)^5) + (8*x^12)/(b^5*d*log(F)^5) + (96*c*x^11)/(b^5*d^2*log(F)^5 
) + (528*c^2*x^10)/(b^5*d^3*log(F)^5) - (4*x^3*(b^3*log(F)^3 - 1320*c^9 + 
504*b*c^6*log(F) - 60*b^2*c^3*log(F)^2))/(3*b^5*d^10*log(F)^5) + (4*x^6*(b 
^2*log(F)^2 + 1848*c^6 - 168*b*c^3*log(F)))/(b^5*d^7*log(F)^5) - (8*x^9*(b 
*log(F) - 220*c^3))/(b^5*d^4*log(F)^5) - (4*c^2*x*(b^3*log(F)^3 - 24*c^9 + 
 18*b*c^6*log(F) - 6*b^2*c^3*log(F)^2))/(b^5*d^12*log(F)^5) - (4*c*x^2*(b^ 
3*log(F)^3 - 132*c^9 + 72*b*c^6*log(F) - 15*b^2*c^3*log(F)^2))/(b^5*d^11*l 
og(F)^5) + (24*c*x^5*(b^2*log(F)^2 + 264*c^6 - 42*b*c^3*log(F)))/(b^5*d^8* 
log(F)^5) - (72*c*x^8*(b*log(F) - 55*c^3))/(b^5*d^5*log(F)^5) + (12*c^2*x^ 
4*(5*b^2*log(F)^2 + 330*c^6 - 84*b*c^3*log(F)))/(b^5*d^9*log(F)^5) - (288* 
c^2*x^7*(b*log(F) - 22*c^3))/(b^5*d^6*log(F)^5)))/(x^12 + c^12/d^12 + (12* 
c*x^11)/d + (12*c^11*x)/d^11 + (66*c^2*x^10)/d^2 + (220*c^3*x^9)/d^3 + (49 
5*c^4*x^8)/d^4 + (792*c^5*x^7)/d^5 + (924*c^6*x^6)/d^6 + (792*c^7*x^5)/d^7 
 + (495*c^8*x^4)/d^8 + (220*c^9*x^3)/d^9 + (66*c^10*x^2)/d^10)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 636, normalized size of antiderivative = 6.62 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx=\frac {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}}} \left (-24 d^{12} x^{12}+4 \mathrm {log}\left (f \right )^{3} b^{3} d^{3} x^{3}+24 \,\mathrm {log}\left (f \right ) b \,d^{9} x^{9}-72 \mathrm {log}\left (f \right )^{2} b^{2} c^{5} d x -180 \mathrm {log}\left (f \right )^{2} b^{2} c^{4} d^{2} x^{2}-240 \mathrm {log}\left (f \right )^{2} b^{2} c^{3} d^{3} x^{3}-180 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{4} x^{4}-72 \mathrm {log}\left (f \right )^{2} b^{2} c \,d^{5} x^{5}-12 \mathrm {log}\left (f \right )^{2} b^{2} c^{6}-24 c^{12}-12 \mathrm {log}\left (f \right )^{2} b^{2} d^{6} x^{6}-\mathrm {log}\left (f \right )^{4} b^{4}+12 \mathrm {log}\left (f \right )^{3} b^{3} c^{2} d x +12 \mathrm {log}\left (f \right )^{3} b^{3} c \,d^{2} x^{2}+216 \,\mathrm {log}\left (f \right ) b \,c^{8} d x +864 \,\mathrm {log}\left (f \right ) b \,c^{7} d^{2} x^{2}+2016 \,\mathrm {log}\left (f \right ) b \,c^{6} d^{3} x^{3}+3024 \,\mathrm {log}\left (f \right ) b \,c^{5} d^{4} x^{4}+3024 \,\mathrm {log}\left (f \right ) b \,c^{4} d^{5} x^{5}+2016 \,\mathrm {log}\left (f \right ) b \,c^{3} d^{6} x^{6}+864 \,\mathrm {log}\left (f \right ) b \,c^{2} d^{7} x^{7}+216 \,\mathrm {log}\left (f \right ) b c \,d^{8} x^{8}-1584 c^{2} d^{10} x^{10}-288 c \,d^{11} x^{11}+4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3}+24 \,\mathrm {log}\left (f \right ) b \,c^{9}-288 c^{11} d x -1584 c^{10} d^{2} x^{2}-5280 c^{9} d^{3} x^{3}-11880 c^{8} d^{4} x^{4}-19008 c^{7} d^{5} x^{5}-22176 c^{6} d^{6} x^{6}-19008 c^{5} d^{7} x^{7}-11880 c^{4} d^{8} x^{8}-5280 c^{3} d^{9} x^{9}\right )}{3 \mathrm {log}\left (f \right )^{5} b^{5} d \left (d^{12} x^{12}+12 c \,d^{11} x^{11}+66 c^{2} d^{10} x^{10}+220 c^{3} d^{9} x^{9}+495 c^{4} d^{8} x^{8}+792 c^{5} d^{7} x^{7}+924 c^{6} d^{6} x^{6}+792 c^{7} d^{5} x^{5}+495 c^{8} d^{4} x^{4}+220 c^{9} d^{3} x^{3}+66 c^{10} d^{2} x^{2}+12 c^{11} d x +c^{12}\right )} \] Input:

int(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x)
 

Output:

(f**((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3 
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*( - log(f)**4*b**4 + 4*log(f)**3*b 
**3*c**3 + 12*log(f)**3*b**3*c**2*d*x + 12*log(f)**3*b**3*c*d**2*x**2 + 4* 
log(f)**3*b**3*d**3*x**3 - 12*log(f)**2*b**2*c**6 - 72*log(f)**2*b**2*c**5 
*d*x - 180*log(f)**2*b**2*c**4*d**2*x**2 - 240*log(f)**2*b**2*c**3*d**3*x* 
*3 - 180*log(f)**2*b**2*c**2*d**4*x**4 - 72*log(f)**2*b**2*c*d**5*x**5 - 1 
2*log(f)**2*b**2*d**6*x**6 + 24*log(f)*b*c**9 + 216*log(f)*b*c**8*d*x + 86 
4*log(f)*b*c**7*d**2*x**2 + 2016*log(f)*b*c**6*d**3*x**3 + 3024*log(f)*b*c 
**5*d**4*x**4 + 3024*log(f)*b*c**4*d**5*x**5 + 2016*log(f)*b*c**3*d**6*x** 
6 + 864*log(f)*b*c**2*d**7*x**7 + 216*log(f)*b*c*d**8*x**8 + 24*log(f)*b*d 
**9*x**9 - 24*c**12 - 288*c**11*d*x - 1584*c**10*d**2*x**2 - 5280*c**9*d** 
3*x**3 - 11880*c**8*d**4*x**4 - 19008*c**7*d**5*x**5 - 22176*c**6*d**6*x** 
6 - 19008*c**5*d**7*x**7 - 11880*c**4*d**8*x**8 - 5280*c**3*d**9*x**9 - 15 
84*c**2*d**10*x**10 - 288*c*d**11*x**11 - 24*d**12*x**12))/(3*log(f)**5*b* 
*5*d*(c**12 + 12*c**11*d*x + 66*c**10*d**2*x**2 + 220*c**9*d**3*x**3 + 495 
*c**8*d**4*x**4 + 792*c**7*d**5*x**5 + 924*c**6*d**6*x**6 + 792*c**5*d**7* 
x**7 + 495*c**4*d**8*x**8 + 220*c**3*d**9*x**9 + 66*c**2*d**10*x**10 + 12* 
c*d**11*x**11 + d**12*x**12))