\(\int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx\) [189]

Optimal result

Integrand size = 21, antiderivative size = 105 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx=-\frac {F^{a+b (c+d x)^2} \left (120-120 b (c+d x)^2 \log (F)+60 b^2 (c+d x)^4 \log ^2(F)-20 b^3 (c+d x)^6 \log ^3(F)+5 b^4 (c+d x)^8 \log ^4(F)-b^5 (c+d x)^{10} \log ^5(F)\right )}{2 b^6 d \log ^6(F)} \] Output:

-1/2*F^(a+b*(d*x+c)^2)*(120-120*b*(d*x+c)^2*ln(F)+60*b^2*(d*x+c)^4*ln(F)^2 
-20*b^3*(d*x+c)^6*ln(F)^3+5*b^4*(d*x+c)^8*ln(F)^4-b^5*(d*x+c)^10*ln(F)^5)/ 
b^6/d/ln(F)^6
 

Mathematica [C] (verified)

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

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.30 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx=-\frac {F^a \Gamma \left (6,-b (c+d x)^2 \log (F)\right )}{2 b^6 d \log ^6(F)} \] Input:

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^11,x]
 

Output:

-1/2*(F^a*Gamma[6, -(b*(c + d*x)^2*Log[F])])/(b^6*d*Log[F]^6)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 105, 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.

\(\displaystyle \int (c+d x)^{11} F^{a+b (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2647

\(\displaystyle -\frac {F^{a+b (c+d x)^2} \left (-b^5 \log ^5(F) (c+d x)^{10}+5 b^4 \log ^4(F) (c+d x)^8-20 b^3 \log ^3(F) (c+d x)^6+60 b^2 \log ^2(F) (c+d x)^4-120 b \log (F) (c+d x)^2+120\right )}{2 b^6 d \log ^6(F)}\)

Input:

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^11,x]
 

Output:

-1/2*(F^(a + b*(c + d*x)^2)*(120 - 120*b*(c + d*x)^2*Log[F] + 60*b^2*(c + 
d*x)^4*Log[F]^2 - 20*b^3*(c + d*x)^6*Log[F]^3 + 5*b^4*(c + d*x)^8*Log[F]^4 
 - b^5*(c + d*x)^10*Log[F]^5))/(b^6*d*Log[F]^6)
 

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. \(568\) vs. \(2(103)=206\).

Time = 1.03 (sec) , antiderivative size = 569, normalized size of antiderivative = 5.42

method	result	size
orering	\(\frac {\left (-120+120 \ln \left (F \right ) b \,c^{2}+120 \ln \left (F \right ) b \,d^{2} x^{2}+d^{10} x^{10} b^{5} \ln \left (F \right )^{5}-5 d^{8} x^{8} b^{4} \ln \left (F \right )^{4}+20 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}-60 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-140 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}-40 \ln \left (F \right )^{4} b^{4} c^{7} d x +120 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+300 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+400 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+300 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+120 \ln \left (F \right )^{3} b^{3} c^{5} d x -240 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-360 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-240 \ln \left (F \right )^{2} b^{2} c^{3} d x +10 d^{9} c \,x^{9} b^{5} \ln \left (F \right )^{5}+45 \ln \left (F \right )^{5} b^{5} c^{2} d^{8} x^{8}+120 \ln \left (F \right )^{5} b^{5} c^{3} d^{7} x^{7}+210 \ln \left (F \right )^{5} b^{5} c^{4} d^{6} x^{6}+252 \ln \left (F \right )^{5} b^{5} c^{5} d^{5} x^{5}+210 \ln \left (F \right )^{5} b^{5} c^{6} d^{4} x^{4}+120 \ln \left (F \right )^{5} b^{5} c^{7} d^{3} x^{3}-40 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}-5 \ln \left (F \right )^{4} b^{4} c^{8}+240 \ln \left (F \right ) b c d x +20 \ln \left (F \right )^{3} b^{3} c^{6}-60 \ln \left (F \right )^{2} b^{2} c^{4}+\ln \left (F \right )^{5} b^{5} c^{10}+45 \ln \left (F \right )^{5} b^{5} c^{8} d^{2} x^{2}-140 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+10 \ln \left (F \right )^{5} b^{5} c^{9} d x -280 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}-350 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}-280 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{a +b \left (d x +c \right )^{2}}}{2 d \ln \left (F \right )^{6} b^{6}}\)	\(569\)
gosper	\(\frac {\left (-120+120 \ln \left (F \right ) b \,c^{2}+120 \ln \left (F \right ) b \,d^{2} x^{2}+d^{10} x^{10} b^{5} \ln \left (F \right )^{5}-5 d^{8} x^{8} b^{4} \ln \left (F \right )^{4}+20 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}-60 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-140 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}-40 \ln \left (F \right )^{4} b^{4} c^{7} d x +120 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+300 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+400 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+300 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+120 \ln \left (F \right )^{3} b^{3} c^{5} d x -240 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-360 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-240 \ln \left (F \right )^{2} b^{2} c^{3} d x +10 d^{9} c \,x^{9} b^{5} \ln \left (F \right )^{5}+45 \ln \left (F \right )^{5} b^{5} c^{2} d^{8} x^{8}+120 \ln \left (F \right )^{5} b^{5} c^{3} d^{7} x^{7}+210 \ln \left (F \right )^{5} b^{5} c^{4} d^{6} x^{6}+252 \ln \left (F \right )^{5} b^{5} c^{5} d^{5} x^{5}+210 \ln \left (F \right )^{5} b^{5} c^{6} d^{4} x^{4}+120 \ln \left (F \right )^{5} b^{5} c^{7} d^{3} x^{3}-40 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}-5 \ln \left (F \right )^{4} b^{4} c^{8}+240 \ln \left (F \right ) b c d x +20 \ln \left (F \right )^{3} b^{3} c^{6}-60 \ln \left (F \right )^{2} b^{2} c^{4}+\ln \left (F \right )^{5} b^{5} c^{10}+45 \ln \left (F \right )^{5} b^{5} c^{8} d^{2} x^{2}-140 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+10 \ln \left (F \right )^{5} b^{5} c^{9} d x -280 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}-350 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}-280 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{6} \ln \left (F \right )^{6} d}\)	\(579\)
risch	\(\frac {\left (-120+120 \ln \left (F \right ) b \,c^{2}+120 \ln \left (F \right ) b \,d^{2} x^{2}+d^{10} x^{10} b^{5} \ln \left (F \right )^{5}-5 d^{8} x^{8} b^{4} \ln \left (F \right )^{4}+20 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}-60 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-140 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}-40 \ln \left (F \right )^{4} b^{4} c^{7} d x +120 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+300 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+400 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+300 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+120 \ln \left (F \right )^{3} b^{3} c^{5} d x -240 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-360 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-240 \ln \left (F \right )^{2} b^{2} c^{3} d x +10 d^{9} c \,x^{9} b^{5} \ln \left (F \right )^{5}+45 \ln \left (F \right )^{5} b^{5} c^{2} d^{8} x^{8}+120 \ln \left (F \right )^{5} b^{5} c^{3} d^{7} x^{7}+210 \ln \left (F \right )^{5} b^{5} c^{4} d^{6} x^{6}+252 \ln \left (F \right )^{5} b^{5} c^{5} d^{5} x^{5}+210 \ln \left (F \right )^{5} b^{5} c^{6} d^{4} x^{4}+120 \ln \left (F \right )^{5} b^{5} c^{7} d^{3} x^{3}-40 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}-5 \ln \left (F \right )^{4} b^{4} c^{8}+240 \ln \left (F \right ) b c d x +20 \ln \left (F \right )^{3} b^{3} c^{6}-60 \ln \left (F \right )^{2} b^{2} c^{4}+\ln \left (F \right )^{5} b^{5} c^{10}+45 \ln \left (F \right )^{5} b^{5} c^{8} d^{2} x^{2}-140 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+10 \ln \left (F \right )^{5} b^{5} c^{9} d x -280 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}-350 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}-280 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{6} \ln \left (F \right )^{6} d}\)	\(579\)
norman	\(\frac {\left (\ln \left (F \right )^{5} b^{5} c^{10}-5 \ln \left (F \right )^{4} b^{4} c^{8}+20 \ln \left (F \right )^{3} b^{3} c^{6}-60 \ln \left (F \right )^{2} b^{2} c^{4}+120 \ln \left (F \right ) b \,c^{2}-120\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 b^{6} \ln \left (F \right )^{6} d}+\frac {d^{9} x^{10} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}+\frac {5 c \left (\ln \left (F \right )^{4} b^{4} c^{8}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}-24 \ln \left (F \right ) b \,c^{2}+24\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {5 d^{3} \left (21 \ln \left (F \right )^{3} b^{3} c^{6}-35 \ln \left (F \right )^{2} b^{2} c^{4}+30 \ln \left (F \right ) b \,c^{2}-6\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {5 d^{5} \left (21 \ln \left (F \right )^{2} b^{2} c^{4}-14 \ln \left (F \right ) b \,c^{2}+2\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {5 d^{7} \left (9 \ln \left (F \right ) b \,c^{2}-1\right ) x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} b^{2}}+\frac {5 d \left (9 \ln \left (F \right )^{4} b^{4} c^{8}-28 \ln \left (F \right )^{3} b^{3} c^{6}+60 \ln \left (F \right )^{2} b^{2} c^{4}-72 \ln \left (F \right ) b \,c^{2}+24\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{5} b^{5}}+\frac {5 d^{8} c \,x^{9} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 c \,d^{4} \left (63 \ln \left (F \right )^{2} b^{2} c^{4}-70 \ln \left (F \right ) b \,c^{2}+30\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {20 c \,d^{6} \left (3 \ln \left (F \right ) b \,c^{2}-1\right ) x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {20 d^{2} c \left (3 \ln \left (F \right )^{3} b^{3} c^{6}-7 \ln \left (F \right )^{2} b^{2} c^{4}+10 \ln \left (F \right ) b \,c^{2}-6\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}\)	\(607\)
parallelrisch	\(\text {Expression too large to display}\)	\(1025\)

Input:

int(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/2/d*(-120+120*ln(F)*b*c^2+120*ln(F)*b*d^2*x^2+d^10*x^10*b^5*ln(F)^5-5*d^ 
8*x^8*b^4*ln(F)^4+20*d^6*x^6*b^3*ln(F)^3-60*d^4*x^4*b^2*ln(F)^2-140*ln(F)^ 
4*b^4*c^6*d^2*x^2-40*ln(F)^4*b^4*c^7*d*x+120*c*d^5*x^5*b^3*ln(F)^3+300*ln( 
F)^3*b^3*c^2*d^4*x^4+400*ln(F)^3*b^3*c^3*d^3*x^3+300*ln(F)^3*b^3*c^4*d^2*x 
^2+120*ln(F)^3*b^3*c^5*d*x-240*d^3*c*x^3*b^2*ln(F)^2-360*ln(F)^2*b^2*c^2*d 
^2*x^2-240*ln(F)^2*b^2*c^3*d*x+10*d^9*c*x^9*b^5*ln(F)^5+45*ln(F)^5*b^5*c^2 
*d^8*x^8+120*ln(F)^5*b^5*c^3*d^7*x^7+210*ln(F)^5*b^5*c^4*d^6*x^6+252*ln(F) 
^5*b^5*c^5*d^5*x^5+210*ln(F)^5*b^5*c^6*d^4*x^4+120*ln(F)^5*b^5*c^7*d^3*x^3 
-40*c*d^7*x^7*b^4*ln(F)^4-5*ln(F)^4*b^4*c^8+240*ln(F)*b*c*d*x+20*ln(F)^3*b 
^3*c^6-60*ln(F)^2*b^2*c^4+ln(F)^5*b^5*c^10+45*ln(F)^5*b^5*c^8*d^2*x^2-140* 
ln(F)^4*b^4*c^2*d^6*x^6+10*ln(F)^5*b^5*c^9*d*x-280*ln(F)^4*b^4*c^3*d^5*x^5 
-350*ln(F)^4*b^4*c^4*d^4*x^4-280*ln(F)^4*b^4*c^5*d^3*x^3)/ln(F)^6/b^6*F^(a 
+b*(d*x+c)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (102) = 204\).

Time = 0.09 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.46 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx=\frac {{\left ({\left (b^{5} d^{10} x^{10} + 10 \, b^{5} c d^{9} x^{9} + 45 \, b^{5} c^{2} d^{8} x^{8} + 120 \, b^{5} c^{3} d^{7} x^{7} + 210 \, b^{5} c^{4} d^{6} x^{6} + 252 \, b^{5} c^{5} d^{5} x^{5} + 210 \, b^{5} c^{6} d^{4} x^{4} + 120 \, b^{5} c^{7} d^{3} x^{3} + 45 \, b^{5} c^{8} d^{2} x^{2} + 10 \, b^{5} c^{9} d x + b^{5} c^{10}\right )} \log \left (F\right )^{5} - 5 \, {\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} + 20 \, {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} - 60 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 120 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 120\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{6} d \log \left (F\right )^{6}} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x, algorithm="fricas")
 

Output:

1/2*((b^5*d^10*x^10 + 10*b^5*c*d^9*x^9 + 45*b^5*c^2*d^8*x^8 + 120*b^5*c^3* 
d^7*x^7 + 210*b^5*c^4*d^6*x^6 + 252*b^5*c^5*d^5*x^5 + 210*b^5*c^6*d^4*x^4 
+ 120*b^5*c^7*d^3*x^3 + 45*b^5*c^8*d^2*x^2 + 10*b^5*c^9*d*x + b^5*c^10)*lo 
g(F)^5 - 5*(b^4*d^8*x^8 + 8*b^4*c*d^7*x^7 + 28*b^4*c^2*d^6*x^6 + 56*b^4*c^ 
3*d^5*x^5 + 70*b^4*c^4*d^4*x^4 + 56*b^4*c^5*d^3*x^3 + 28*b^4*c^6*d^2*x^2 + 
 8*b^4*c^7*d*x + b^4*c^8)*log(F)^4 + 20*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + 1 
5*b^3*c^2*d^4*x^4 + 20*b^3*c^3*d^3*x^3 + 15*b^3*c^4*d^2*x^2 + 6*b^3*c^5*d* 
x + b^3*c^6)*log(F)^3 - 60*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2* 
x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(F)^2 + 120*(b*d^2*x^2 + 2*b*c*d*x + b*c 
^2)*log(F) - 120)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)/(b^6*d*log(F)^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (105) = 210\).

Time = 0.24 (sec) , antiderivative size = 794, normalized size of antiderivative = 7.56 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx =\text {Too large to display} \] Input:

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**11,x)
 

Output:

Piecewise((F**(a + b*(c + d*x)**2)*(b**5*c**10*log(F)**5 + 10*b**5*c**9*d* 
x*log(F)**5 + 45*b**5*c**8*d**2*x**2*log(F)**5 + 120*b**5*c**7*d**3*x**3*l 
og(F)**5 + 210*b**5*c**6*d**4*x**4*log(F)**5 + 252*b**5*c**5*d**5*x**5*log 
(F)**5 + 210*b**5*c**4*d**6*x**6*log(F)**5 + 120*b**5*c**3*d**7*x**7*log(F 
)**5 + 45*b**5*c**2*d**8*x**8*log(F)**5 + 10*b**5*c*d**9*x**9*log(F)**5 + 
b**5*d**10*x**10*log(F)**5 - 5*b**4*c**8*log(F)**4 - 40*b**4*c**7*d*x*log( 
F)**4 - 140*b**4*c**6*d**2*x**2*log(F)**4 - 280*b**4*c**5*d**3*x**3*log(F) 
**4 - 350*b**4*c**4*d**4*x**4*log(F)**4 - 280*b**4*c**3*d**5*x**5*log(F)** 
4 - 140*b**4*c**2*d**6*x**6*log(F)**4 - 40*b**4*c*d**7*x**7*log(F)**4 - 5* 
b**4*d**8*x**8*log(F)**4 + 20*b**3*c**6*log(F)**3 + 120*b**3*c**5*d*x*log( 
F)**3 + 300*b**3*c**4*d**2*x**2*log(F)**3 + 400*b**3*c**3*d**3*x**3*log(F) 
**3 + 300*b**3*c**2*d**4*x**4*log(F)**3 + 120*b**3*c*d**5*x**5*log(F)**3 + 
 20*b**3*d**6*x**6*log(F)**3 - 60*b**2*c**4*log(F)**2 - 240*b**2*c**3*d*x* 
log(F)**2 - 360*b**2*c**2*d**2*x**2*log(F)**2 - 240*b**2*c*d**3*x**3*log(F 
)**2 - 60*b**2*d**4*x**4*log(F)**2 + 120*b*c**2*log(F) + 240*b*c*d*x*log(F 
) + 120*b*d**2*x**2*log(F) - 120)/(2*b**6*d*log(F)**6), Ne(b**6*d*log(F)** 
6, 0)), (c**11*x + 11*c**10*d*x**2/2 + 55*c**9*d**2*x**3/3 + 165*c**8*d**3 
*x**4/4 + 66*c**7*d**4*x**5 + 77*c**6*d**5*x**6 + 66*c**5*d**6*x**7 + 165* 
c**4*d**7*x**8/4 + 55*c**3*d**8*x**9/3 + 11*c**2*d**9*x**10/2 + c*d**10*x* 
*11 + d**11*x**12/12, True))
 

Maxima [C] (verification not implemented)

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

Time = 1.87 (sec) , antiderivative size = 5261, normalized size of antiderivative = 50.10 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx=\text {Too large to display} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x, algorithm="maxima")
 

Output:

-11/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F 
)/(b*d^2))) - 1)*log(F)^2/((b*log(F))^(3/2)*d^2*sqrt(-(b*d^2*x + b*c*d)^2* 
log(F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*log(F)/((b*log(F))^(3 
/2)*d))*F^a*c^10/sqrt(b*log(F)) + 55/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2 
*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*log(F)) 
^(5/2)*d^3*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^((b*d^2*x + b* 
c*d)^2/(b*d^2))*b^2*c*log(F)^2/((b*log(F))^(5/2)*d^2) - (b*d^2*x + b*c*d)^ 
3*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F))^(5/ 
2)*d^5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*c^9*d/sqrt(b*log( 
F)) - 165/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c* 
d)^2*log(F)/(b*d^2))) - 1)*log(F)^4/((b*log(F))^(7/2)*d^4*sqrt(-(b*d^2*x + 
 b*c*d)^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^3*c^2*log 
(F)^3/((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, -(b*d^ 
2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^2*x + 
 b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x + b*c*d)^2*log(F 
)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^8*d^2/sqrt(b*log(F)) + 1 
65*(sqrt(pi)*(b*d^2*x + b*c*d)*b^4*c^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log( 
F)/(b*d^2))) - 1)*log(F)^5/((b*log(F))^(9/2)*d^5*sqrt(-(b*d^2*x + b*c*d)^2 
*log(F)/(b*d^2))) - 4*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^4*c^3*log(F)^4/((b 
*log(F))^(9/2)*d^4) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*gamma(3/2, -(b*d^2*...
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.38 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx=\frac {{\left (b^{5} d^{10} {\left (x + \frac {c}{d}\right )}^{10} \log \left (F\right )^{5} - 5 \, b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{8} \log \left (F\right )^{4} + 20 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \left (F\right )^{3} - 60 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} + 120 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) - 120\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{6} d \log \left (F\right )^{6}} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/2*(b^5*d^10*(x + c/d)^10*log(F)^5 - 5*b^4*d^8*(x + c/d)^8*log(F)^4 + 20* 
b^3*d^6*(x + c/d)^6*log(F)^3 - 60*b^2*d^4*(x + c/d)^4*log(F)^2 + 120*b*d^2 
*(x + c/d)^2*log(F) - 120)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2* 
log(F) + a*log(F))/(b^6*d*log(F)^6)
 

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 553, normalized size of antiderivative = 5.27 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx=\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (c^{10}+10\,c^9\,d\,x+45\,c^8\,d^2\,x^2+120\,c^7\,d^3\,x^3+210\,c^6\,d^4\,x^4+252\,c^5\,d^5\,x^5+210\,c^4\,d^6\,x^6+120\,c^3\,d^7\,x^7+45\,c^2\,d^8\,x^8+10\,c\,d^9\,x^9+d^{10}\,x^{10}\right )}{2\,b\,d\,\ln \left (F\right )}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (5\,c^8+40\,c^7\,d\,x+140\,c^6\,d^2\,x^2+280\,c^5\,d^3\,x^3+350\,c^4\,d^4\,x^4+280\,c^3\,d^5\,x^5+140\,c^2\,d^6\,x^6+40\,c\,d^7\,x^7+5\,d^8\,x^8\right )}{2\,b^2\,d\,{\ln \left (F\right )}^2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (60\,c^4+240\,c^3\,d\,x+360\,c^2\,d^2\,x^2+240\,c\,d^3\,x^3+60\,d^4\,x^4\right )}{2\,b^4\,d\,{\ln \left (F\right )}^4}-\frac {60\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}}{b^6\,d\,{\ln \left (F\right )}^6}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (20\,c^6+120\,c^5\,d\,x+300\,c^4\,d^2\,x^2+400\,c^3\,d^3\,x^3+300\,c^2\,d^4\,x^4+120\,c\,d^5\,x^5+20\,d^6\,x^6\right )}{2\,b^3\,d\,{\ln \left (F\right )}^3}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (120\,c^2+240\,c\,d\,x+120\,d^2\,x^2\right )}{2\,b^5\,d\,{\ln \left (F\right )}^5} \] Input:

int(F^(a + b*(c + d*x)^2)*(c + d*x)^11,x)
 

Output:

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(c^10 + d^10*x^10 + 10*c*d^9*x^ 
9 + 45*c^8*d^2*x^2 + 120*c^7*d^3*x^3 + 210*c^6*d^4*x^4 + 252*c^5*d^5*x^5 + 
 210*c^4*d^6*x^6 + 120*c^3*d^7*x^7 + 45*c^2*d^8*x^8 + 10*c^9*d*x))/(2*b*d* 
log(F)) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(5*c^8 + 5*d^8*x^8 + 
40*c*d^7*x^7 + 140*c^6*d^2*x^2 + 280*c^5*d^3*x^3 + 350*c^4*d^4*x^4 + 280*c 
^3*d^5*x^5 + 140*c^2*d^6*x^6 + 40*c^7*d*x))/(2*b^2*d*log(F)^2) - (F^(b*d^2 
*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(60*c^4 + 60*d^4*x^4 + 240*c*d^3*x^3 + 3 
60*c^2*d^2*x^2 + 240*c^3*d*x))/(2*b^4*d*log(F)^4) - (60*F^(b*d^2*x^2)*F^a* 
F^(b*c^2)*F^(2*b*c*d*x))/(b^6*d*log(F)^6) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F 
^(2*b*c*d*x)*(20*c^6 + 20*d^6*x^6 + 120*c*d^5*x^5 + 300*c^4*d^2*x^2 + 400* 
c^3*d^3*x^3 + 300*c^2*d^4*x^4 + 120*c^5*d*x))/(2*b^3*d*log(F)^3) + (F^(b*d 
^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(120*c^2 + 120*d^2*x^2 + 240*c*d*x))/( 
2*b^5*d*log(F)^5)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 578, normalized size of antiderivative = 5.50 \[ \int F^{a+b (c+d x)^2} (c+d x)^{11} \, dx =\text {Too large to display} \] Input:

int(F^(a+b*(d*x+c)^2)*(d*x+c)^11,x)
 

Output:

(f**(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*(log(f)**5*b**5*c**10 + 10*log( 
f)**5*b**5*c**9*d*x + 45*log(f)**5*b**5*c**8*d**2*x**2 + 120*log(f)**5*b** 
5*c**7*d**3*x**3 + 210*log(f)**5*b**5*c**6*d**4*x**4 + 252*log(f)**5*b**5* 
c**5*d**5*x**5 + 210*log(f)**5*b**5*c**4*d**6*x**6 + 120*log(f)**5*b**5*c* 
*3*d**7*x**7 + 45*log(f)**5*b**5*c**2*d**8*x**8 + 10*log(f)**5*b**5*c*d**9 
*x**9 + log(f)**5*b**5*d**10*x**10 - 5*log(f)**4*b**4*c**8 - 40*log(f)**4* 
b**4*c**7*d*x - 140*log(f)**4*b**4*c**6*d**2*x**2 - 280*log(f)**4*b**4*c** 
5*d**3*x**3 - 350*log(f)**4*b**4*c**4*d**4*x**4 - 280*log(f)**4*b**4*c**3* 
d**5*x**5 - 140*log(f)**4*b**4*c**2*d**6*x**6 - 40*log(f)**4*b**4*c*d**7*x 
**7 - 5*log(f)**4*b**4*d**8*x**8 + 20*log(f)**3*b**3*c**6 + 120*log(f)**3* 
b**3*c**5*d*x + 300*log(f)**3*b**3*c**4*d**2*x**2 + 400*log(f)**3*b**3*c** 
3*d**3*x**3 + 300*log(f)**3*b**3*c**2*d**4*x**4 + 120*log(f)**3*b**3*c*d** 
5*x**5 + 20*log(f)**3*b**3*d**6*x**6 - 60*log(f)**2*b**2*c**4 - 240*log(f) 
**2*b**2*c**3*d*x - 360*log(f)**2*b**2*c**2*d**2*x**2 - 240*log(f)**2*b**2 
*c*d**3*x**3 - 60*log(f)**2*b**2*d**4*x**4 + 120*log(f)*b*c**2 + 240*log(f 
)*b*c*d*x + 120*log(f)*b*d**2*x**2 - 120))/(2*log(f)**6*b**6*d)