\(\int F^{a+b (c+d x)^2} (c+d x)^9 \, dx\) [190]

Optimal result

Integrand size = 21, antiderivative size = 88 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {F^{a+b (c+d x)^2} \left (24-24 b (c+d x)^2 \log (F)+12 b^2 (c+d x)^4 \log ^2(F)-4 b^3 (c+d x)^6 \log ^3(F)+b^4 (c+d x)^8 \log ^4(F)\right )}{2 b^5 d \log ^5(F)} \] Output:

1/2*F^(a+b*(d*x+c)^2)*(24-24*b*(d*x+c)^2*ln(F)+12*b^2*(d*x+c)^4*ln(F)^2-4* 
b^3*(d*x+c)^6*ln(F)^3+b^4*(d*x+c)^8*ln(F)^4)/b^5/d/ln(F)^5
 

Mathematica [C] (verified)

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

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {F^a \Gamma \left (5,-b (c+d x)^2 \log (F)\right )}{2 b^5 d \log ^5(F)} \] Input:

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]
 

Output:

(F^a*Gamma[5, -(b*(c + d*x)^2*Log[F])])/(2*b^5*d*Log[F]^5)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 88, 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.

\(\displaystyle \int (c+d x)^9 F^{a+b (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2647

\(\displaystyle \frac {F^{a+b (c+d x)^2} \left (b^4 \log ^4(F) (c+d x)^8-4 b^3 \log ^3(F) (c+d x)^6+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^2+24\right )}{2 b^5 d \log ^5(F)}\)

Input:

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]
 

Output:

(F^(a + b*(c + d*x)^2)*(24 - 24*b*(c + d*x)^2*Log[F] + 12*b^2*(c + d*x)^4* 
Log[F]^2 - 4*b^3*(c + d*x)^6*Log[F]^3 + b^4*(c + d*x)^8*Log[F]^4))/(2*b^5* 
d*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. \(385\) vs. \(2(86)=172\).

Time = 0.55 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.39

method	result	size
orering	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}-24 \ln \left (F \right ) b \,d^{2} x^{2}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}+28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x +8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+\ln \left (F \right )^{4} b^{4} c^{8}-48 \ln \left (F \right ) b c d x -4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{a +b \left (d x +c \right )^{2}}}{2 d \ln \left (F \right )^{5} b^{5}}\)	\(386\)
gosper	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}-24 \ln \left (F \right ) b \,d^{2} x^{2}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}+28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x +8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+\ln \left (F \right )^{4} b^{4} c^{8}-48 \ln \left (F \right ) b c d x -4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{5} \ln \left (F \right )^{5} d}\)	\(396\)
risch	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}-24 \ln \left (F \right ) b \,d^{2} x^{2}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}+28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x +8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+\ln \left (F \right )^{4} b^{4} c^{8}-48 \ln \left (F \right ) b c d x -4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{5} \ln \left (F \right )^{5} d}\)	\(396\)
norman	\(\frac {d^{3} \left (35 \ln \left (F \right )^{2} b^{2} c^{4}-30 \ln \left (F \right ) b \,c^{2}+6\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {\left (\ln \left (F \right )^{4} b^{4} c^{8}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}-24 \ln \left (F \right ) b \,c^{2}+24\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 b^{5} \ln \left (F \right )^{5} d}+\frac {d^{7} x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}+\frac {4 c \left (\ln \left (F \right )^{3} b^{3} c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,c^{2}-6\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {2 d^{5} \left (7 \ln \left (F \right ) b \,c^{2}-1\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {2 d \left (7 \ln \left (F \right )^{3} b^{3} c^{6}-15 \ln \left (F \right )^{2} b^{2} c^{4}+18 \ln \left (F \right ) b \,c^{2}-6\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {4 d^{6} c \,x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {4 c \,d^{4} \left (7 \ln \left (F \right ) b \,c^{2}-3\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {4 d^{2} c \left (7 \ln \left (F \right )^{2} b^{2} c^{4}-10 \ln \left (F \right ) b \,c^{2}+6\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}\)	\(441\)
parallelrisch	\(\frac {24 F^{a +b \left (d x +c \right )^{2}}+d^{8} F^{a +b \left (d x +c \right )^{2}} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} F^{a +b \left (d x +c \right )^{2}} x^{6} b^{3} \ln \left (F \right )^{3}+12 d^{4} F^{a +b \left (d x +c \right )^{2}} x^{4} b^{2} \ln \left (F \right )^{2}-24 d^{2} F^{a +b \left (d x +c \right )^{2}} x^{2} b \ln \left (F \right )+\ln \left (F \right )^{4} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{8}-4 \ln \left (F \right )^{3} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{6}+12 \ln \left (F \right )^{2} F^{a +b \left (d x +c \right )^{2}} b^{2} c^{4}-24 \ln \left (F \right ) F^{a +b \left (d x +c \right )^{2}} b \,c^{2}+8 c \,d^{7} F^{a +b \left (d x +c \right )^{2}} x^{7} b^{4} \ln \left (F \right )^{4}+28 \ln \left (F \right )^{4} x^{6} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{2} d^{6}+56 \ln \left (F \right )^{4} x^{5} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{3} d^{5}+70 \ln \left (F \right )^{4} x^{4} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{4} d^{4}+56 \ln \left (F \right )^{4} x^{3} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{5} d^{3}+28 \ln \left (F \right )^{4} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{4} c^{6} d^{2}+8 \ln \left (F \right )^{4} x \,F^{a +b \left (d x +c \right )^{2}} b^{4} c^{7} d -24 c \,d^{5} F^{a +b \left (d x +c \right )^{2}} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} x^{4} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{2} d^{4}-80 \ln \left (F \right )^{3} x^{3} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{3} d^{3}-60 \ln \left (F \right )^{3} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{4} d^{2}-24 \ln \left (F \right )^{3} x \,F^{a +b \left (d x +c \right )^{2}} b^{3} c^{5} d +48 d^{3} c \,F^{a +b \left (d x +c \right )^{2}} x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{2} c^{2} d^{2}+48 \ln \left (F \right )^{2} x \,F^{a +b \left (d x +c \right )^{2}} b^{2} c^{3} d -48 c \,F^{a +b \left (d x +c \right )^{2}} x b \ln \left (F \right ) d}{2 b^{5} \ln \left (F \right )^{5} d}\)	\(699\)

Input:

int(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/2/d*(24-24*ln(F)*b*c^2-24*ln(F)*b*d^2*x^2+d^8*x^8*b^4*ln(F)^4-4*d^6*x^6* 
b^3*ln(F)^3+12*d^4*x^4*b^2*ln(F)^2+28*ln(F)^4*b^4*c^6*d^2*x^2+8*ln(F)^4*b^ 
4*c^7*d*x-24*c*d^5*x^5*b^3*ln(F)^3-60*ln(F)^3*b^3*c^2*d^4*x^4-80*ln(F)^3*b 
^3*c^3*d^3*x^3-60*ln(F)^3*b^3*c^4*d^2*x^2-24*ln(F)^3*b^3*c^5*d*x+48*d^3*c* 
x^3*b^2*ln(F)^2+72*ln(F)^2*b^2*c^2*d^2*x^2+48*ln(F)^2*b^2*c^3*d*x+8*c*d^7* 
x^7*b^4*ln(F)^4+ln(F)^4*b^4*c^8-48*ln(F)*b*c*d*x-4*ln(F)^3*b^3*c^6+12*ln(F 
)^2*b^2*c^4+28*ln(F)^4*b^4*c^2*d^6*x^6+56*ln(F)^4*b^4*c^3*d^5*x^5+70*ln(F) 
^4*b^4*c^4*d^4*x^4+56*ln(F)^4*b^4*c^5*d^3*x^3)/ln(F)^5/b^5*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. 324 vs. \(2 (86) = 172\).

Time = 0.08 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.68 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {{\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{5} d \log \left (F\right )^{5}} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (87) = 174\).

Time = 0.19 (sec) , antiderivative size = 556, normalized size of antiderivative = 6.32 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b^{4} c^{8} \log {\left (F \right )}^{4} + 8 b^{4} c^{7} d x \log {\left (F \right )}^{4} + 28 b^{4} c^{6} d^{2} x^{2} \log {\left (F \right )}^{4} + 56 b^{4} c^{5} d^{3} x^{3} \log {\left (F \right )}^{4} + 70 b^{4} c^{4} d^{4} x^{4} \log {\left (F \right )}^{4} + 56 b^{4} c^{3} d^{5} x^{5} \log {\left (F \right )}^{4} + 28 b^{4} c^{2} d^{6} x^{6} \log {\left (F \right )}^{4} + 8 b^{4} c d^{7} x^{7} \log {\left (F \right )}^{4} + b^{4} d^{8} x^{8} \log {\left (F \right )}^{4} - 4 b^{3} c^{6} \log {\left (F \right )}^{3} - 24 b^{3} c^{5} d x \log {\left (F \right )}^{3} - 60 b^{3} c^{4} d^{2} x^{2} \log {\left (F \right )}^{3} - 80 b^{3} c^{3} d^{3} x^{3} \log {\left (F \right )}^{3} - 60 b^{3} c^{2} d^{4} x^{4} \log {\left (F \right )}^{3} - 24 b^{3} c d^{5} x^{5} \log {\left (F \right )}^{3} - 4 b^{3} d^{6} x^{6} \log {\left (F \right )}^{3} + 12 b^{2} c^{4} \log {\left (F \right )}^{2} + 48 b^{2} c^{3} d x \log {\left (F \right )}^{2} + 72 b^{2} c^{2} d^{2} x^{2} \log {\left (F \right )}^{2} + 48 b^{2} c d^{3} x^{3} \log {\left (F \right )}^{2} + 12 b^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 24 b c^{2} \log {\left (F \right )} - 48 b c d x \log {\left (F \right )} - 24 b d^{2} x^{2} \log {\left (F \right )} + 24\right )}{2 b^{5} d \log {\left (F \right )}^{5}} & \text {for}\: b^{5} d \log {\left (F \right )}^{5} \neq 0 \\c^{9} x + \frac {9 c^{8} d x^{2}}{2} + 12 c^{7} d^{2} x^{3} + 21 c^{6} d^{3} x^{4} + \frac {126 c^{5} d^{4} x^{5}}{5} + 21 c^{4} d^{5} x^{6} + 12 c^{3} d^{6} x^{7} + \frac {9 c^{2} d^{7} x^{8}}{2} + c d^{8} x^{9} + \frac {d^{9} x^{10}}{10} & \text {otherwise} \end {cases} \] Input:

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**9,x)
 

Output:

Piecewise((F**(a + b*(c + d*x)**2)*(b**4*c**8*log(F)**4 + 8*b**4*c**7*d*x* 
log(F)**4 + 28*b**4*c**6*d**2*x**2*log(F)**4 + 56*b**4*c**5*d**3*x**3*log( 
F)**4 + 70*b**4*c**4*d**4*x**4*log(F)**4 + 56*b**4*c**3*d**5*x**5*log(F)** 
4 + 28*b**4*c**2*d**6*x**6*log(F)**4 + 8*b**4*c*d**7*x**7*log(F)**4 + b**4 
*d**8*x**8*log(F)**4 - 4*b**3*c**6*log(F)**3 - 24*b**3*c**5*d*x*log(F)**3 
- 60*b**3*c**4*d**2*x**2*log(F)**3 - 80*b**3*c**3*d**3*x**3*log(F)**3 - 60 
*b**3*c**2*d**4*x**4*log(F)**3 - 24*b**3*c*d**5*x**5*log(F)**3 - 4*b**3*d* 
*6*x**6*log(F)**3 + 12*b**2*c**4*log(F)**2 + 48*b**2*c**3*d*x*log(F)**2 + 
72*b**2*c**2*d**2*x**2*log(F)**2 + 48*b**2*c*d**3*x**3*log(F)**2 + 12*b**2 
*d**4*x**4*log(F)**2 - 24*b*c**2*log(F) - 48*b*c*d*x*log(F) - 24*b*d**2*x* 
*2*log(F) + 24)/(2*b**5*d*log(F)**5), Ne(b**5*d*log(F)**5, 0)), (c**9*x + 
9*c**8*d*x**2/2 + 12*c**7*d**2*x**3 + 21*c**6*d**3*x**4 + 126*c**5*d**4*x* 
*5/5 + 21*c**4*d**5*x**6 + 12*c**3*d**6*x**7 + 9*c**2*d**7*x**8/2 + c*d**8 
*x**9 + d**9*x**10/10, True))
 

Maxima [C] (verification not implemented)

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

Time = 1.38 (sec) , antiderivative size = 3727, normalized size of antiderivative = 42.35 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\text {Too large to display} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x, algorithm="maxima")
 

Output:

-9/2*(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*l 
og(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^8/sqrt(b*log(F)) + 18*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(er 
f(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*ga 
mma(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^7*d/sqrt(b*log(F)) 
- 42*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*lo 
g(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^6*d^2/sqrt(b*log(F)) + 63*(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*x + b*c*...
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.41 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {{\left (b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{8} \log \left (F\right )^{4} - 4 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \left (F\right )^{3} + 12 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} - 24 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) + 24\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{5} d \log \left (F\right )^{5}} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/2*(b^4*d^8*(x + c/d)^8*log(F)^4 - 4*b^3*d^6*(x + c/d)^6*log(F)^3 + 12*b^ 
2*d^4*(x + c/d)^4*log(F)^2 - 24*b*d^2*(x + c/d)^2*log(F) + 24)*e^(b*d^2*x^ 
2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b^5*d*log(F)^5)
 

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 391, normalized size of antiderivative = 4.44 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {12\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^3\,{\ln \left (F\right )}^3\,\left (4\,c^6+24\,c^5\,d\,x+60\,c^4\,d^2\,x^2+80\,c^3\,d^3\,x^3+60\,c^2\,d^4\,x^4+24\,c\,d^5\,x^5+4\,d^6\,x^6\right )}{2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^4\,{\ln \left (F\right )}^4\,\left (c^8+8\,c^7\,d\,x+28\,c^6\,d^2\,x^2+56\,c^5\,d^3\,x^3+70\,c^4\,d^4\,x^4+56\,c^3\,d^5\,x^5+28\,c^2\,d^6\,x^6+8\,c\,d^7\,x^7+d^8\,x^8\right )}{2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b\,\ln \left (F\right )\,\left (24\,c^2+48\,c\,d\,x+24\,d^2\,x^2\right )}{2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^2\,{\ln \left (F\right )}^2\,\left (12\,c^4+48\,c^3\,d\,x+72\,c^2\,d^2\,x^2+48\,c\,d^3\,x^3+12\,d^4\,x^4\right )}{2}}{b^5\,d\,{\ln \left (F\right )}^5} \] Input:

int(F^(a + b*(c + d*x)^2)*(c + d*x)^9,x)
 

Output:

(12*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x) - (F^(b*d^2*x^2)*F^a*F^(b*c^ 
2)*F^(2*b*c*d*x)*b^3*log(F)^3*(4*c^6 + 4*d^6*x^6 + 24*c*d^5*x^5 + 60*c^4*d 
^2*x^2 + 80*c^3*d^3*x^3 + 60*c^2*d^4*x^4 + 24*c^5*d*x))/2 + (F^(b*d^2*x^2) 
*F^a*F^(b*c^2)*F^(2*b*c*d*x)*b^4*log(F)^4*(c^8 + d^8*x^8 + 8*c*d^7*x^7 + 2 
8*c^6*d^2*x^2 + 56*c^5*d^3*x^3 + 70*c^4*d^4*x^4 + 56*c^3*d^5*x^5 + 28*c^2* 
d^6*x^6 + 8*c^7*d*x))/2 - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*b*log 
(F)*(24*c^2 + 24*d^2*x^2 + 48*c*d*x))/2 + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^( 
2*b*c*d*x)*b^2*log(F)^2*(12*c^4 + 12*d^4*x^4 + 48*c*d^3*x^3 + 72*c^2*d^2*x 
^2 + 48*c^3*d*x))/2)/(b^5*d*log(F)^5)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 395, normalized size of antiderivative = 4.49 \[ \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx=\frac {f^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a} \left (24-48 \,\mathrm {log}\left (f \right ) b c d x +8 \mathrm {log}\left (f \right )^{4} b^{4} c^{7} d x +28 \mathrm {log}\left (f \right )^{4} b^{4} c^{6} d^{2} x^{2}+56 \mathrm {log}\left (f \right )^{4} b^{4} c^{5} d^{3} x^{3}+70 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \mathrm {log}\left (f \right )^{4} b^{4} c^{3} d^{5} x^{5}+28 \mathrm {log}\left (f \right )^{4} b^{4} c^{2} d^{6} x^{6}+8 \mathrm {log}\left (f \right )^{4} b^{4} c \,d^{7} x^{7}-24 \mathrm {log}\left (f \right )^{3} b^{3} c^{5} d x -60 \mathrm {log}\left (f \right )^{3} b^{3} c^{4} d^{2} x^{2}-80 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \mathrm {log}\left (f \right )^{3} b^{3} c^{2} d^{4} x^{4}-24 \mathrm {log}\left (f \right )^{3} b^{3} c \,d^{5} x^{5}+48 \mathrm {log}\left (f \right )^{2} b^{2} c^{3} d x +72 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \mathrm {log}\left (f \right )^{2} b^{2} c \,d^{3} x^{3}+\mathrm {log}\left (f \right )^{4} b^{4} d^{8} x^{8}-4 \mathrm {log}\left (f \right )^{3} b^{3} d^{6} x^{6}+12 \mathrm {log}\left (f \right )^{2} b^{2} d^{4} x^{4}-24 \,\mathrm {log}\left (f \right ) b \,d^{2} x^{2}+\mathrm {log}\left (f \right )^{4} b^{4} c^{8}-4 \mathrm {log}\left (f \right )^{3} b^{3} c^{6}+12 \mathrm {log}\left (f \right )^{2} b^{2} c^{4}-24 \,\mathrm {log}\left (f \right ) b \,c^{2}\right )}{2 \mathrm {log}\left (f \right )^{5} b^{5} d} \] Input:

int(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x)
 

Output:

(f**(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*(log(f)**4*b**4*c**8 + 8*log(f) 
**4*b**4*c**7*d*x + 28*log(f)**4*b**4*c**6*d**2*x**2 + 56*log(f)**4*b**4*c 
**5*d**3*x**3 + 70*log(f)**4*b**4*c**4*d**4*x**4 + 56*log(f)**4*b**4*c**3* 
d**5*x**5 + 28*log(f)**4*b**4*c**2*d**6*x**6 + 8*log(f)**4*b**4*c*d**7*x** 
7 + log(f)**4*b**4*d**8*x**8 - 4*log(f)**3*b**3*c**6 - 24*log(f)**3*b**3*c 
**5*d*x - 60*log(f)**3*b**3*c**4*d**2*x**2 - 80*log(f)**3*b**3*c**3*d**3*x 
**3 - 60*log(f)**3*b**3*c**2*d**4*x**4 - 24*log(f)**3*b**3*c*d**5*x**5 - 4 
*log(f)**3*b**3*d**6*x**6 + 12*log(f)**2*b**2*c**4 + 48*log(f)**2*b**2*c** 
3*d*x + 72*log(f)**2*b**2*c**2*d**2*x**2 + 48*log(f)**2*b**2*c*d**3*x**3 + 
 12*log(f)**2*b**2*d**4*x**4 - 24*log(f)*b*c**2 - 48*log(f)*b*c*d*x - 24*l 
og(f)*b*d**2*x**2 + 24))/(2*log(f)**5*b**5*d)