3.40 \(\int (a+b (F^{g (e+f x)})^n)^3 (c+d x)^2 \, dx\)

Optimal. Leaf size=366 \[ \frac {a^3 (c+d x)^3}{3 d}-\frac {6 a^2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 a b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {3 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {2 b^3 d (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)} \]

[Out]

1/3*a^3*(d*x+c)^3/d+6*a^2*b*d^2*(F^(f*g*x+e*g))^n/f^3/g^3/n^3/ln(F)^3+3/4*a*b^2*d^2*(F^(f*g*x+e*g))^(2*n)/f^3/
g^3/n^3/ln(F)^3+2/27*b^3*d^2*(F^(f*g*x+e*g))^(3*n)/f^3/g^3/n^3/ln(F)^3-6*a^2*b*d*(F^(f*g*x+e*g))^n*(d*x+c)/f^2
/g^2/n^2/ln(F)^2-3/2*a*b^2*d*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f^2/g^2/n^2/ln(F)^2-2/9*b^3*d*(F^(f*g*x+e*g))^(3*n)
*(d*x+c)/f^2/g^2/n^2/ln(F)^2+3*a^2*b*(F^(f*g*x+e*g))^n*(d*x+c)^2/f/g/n/ln(F)+3/2*a*b^2*(F^(f*g*x+e*g))^(2*n)*(
d*x+c)^2/f/g/n/ln(F)+1/3*b^3*(F^(f*g*x+e*g))^(3*n)*(d*x+c)^2/f/g/n/ln(F)

________________________________________________________________________________________

Rubi [A]  time = 0.46, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2183, 2176, 2194} \[ -\frac {6 a^2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {a^3 (c+d x)^3}{3 d}-\frac {3 a b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {3 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {2 b^3 d (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2,x]

[Out]

(a^3*(c + d*x)^3)/(3*d) + (6*a^2*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log[F]^3) + (3*a*b^2*d^2*(F^(e*g + f*
g*x))^(2*n))/(4*f^3*g^3*n^3*Log[F]^3) + (2*b^3*d^2*(F^(e*g + f*g*x))^(3*n))/(27*f^3*g^3*n^3*Log[F]^3) - (6*a^2
*b*d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) - (3*a*b^2*d*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(2*
f^2*g^2*n^2*Log[F]^2) - (2*b^3*d*(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(9*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*
g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/(2*f*g*n*Log[F]) + (
b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^2)/(3*f*g*n*Log[F])

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2183

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx &=\int \left (a^3 (c+d x)^2+3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2+3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2+b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2\right ) \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx+\left (3 a b^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2 \, dx+b^3 \int \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2 \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)}-\frac {\left (6 a^2 b d\right ) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f g n \log (F)}-\frac {\left (3 a b^2 d\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x) \, dx}{f g n \log (F)}-\frac {\left (2 b^3 d\right ) \int \left (F^{e g+f g x}\right )^{3 n} (c+d x) \, dx}{3 f g n \log (F)}\\ &=\frac {a^3 (c+d x)^3}{3 d}-\frac {6 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f^2 g^2 n^2 \log ^2(F)}-\frac {2 b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)}+\frac {\left (6 a^2 b d^2\right ) \int \left (F^{e g+f g x}\right )^n \, dx}{f^2 g^2 n^2 \log ^2(F)}+\frac {\left (3 a b^2 d^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} \, dx}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {\left (2 b^3 d^2\right ) \int \left (F^{e g+f g x}\right )^{3 n} \, dx}{9 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {a^3 (c+d x)^3}{3 d}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f^2 g^2 n^2 \log ^2(F)}-\frac {2 b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.53, size = 248, normalized size = 0.68 \[ a^3 c^2 x+a^3 c d x^2+\frac {1}{3} a^3 d^2 x^3+\frac {3 a^2 b \left (F^{g (e+f x)}\right )^n \left (f^2 g^2 n^2 \log ^2(F) (c+d x)^2-2 d f g n \log (F) (c+d x)+2 d^2\right )}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (2 f^2 g^2 n^2 \log ^2(F) (c+d x)^2-2 d f g n \log (F) (c+d x)+d^2\right )}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n} \left (9 f^2 g^2 n^2 \log ^2(F) (c+d x)^2-6 d f g n \log (F) (c+d x)+2 d^2\right )}{27 f^3 g^3 n^3 \log ^3(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2,x]

[Out]

a^3*c^2*x + a^3*c*d*x^2 + (a^3*d^2*x^3)/3 + (3*a^2*b*(F^(g*(e + f*x)))^n*(2*d^2 - 2*d*f*g*n*(c + d*x)*Log[F] +
f^2*g^2*n^2*(c + d*x)^2*Log[F]^2))/(f^3*g^3*n^3*Log[F]^3) + (3*a*b^2*(F^(g*(e + f*x)))^(2*n)*(d^2 - 2*d*f*g*n
*(c + d*x)*Log[F] + 2*f^2*g^2*n^2*(c + d*x)^2*Log[F]^2))/(4*f^3*g^3*n^3*Log[F]^3) + (b^3*(F^(g*(e + f*x)))^(3*
n)*(2*d^2 - 6*d*f*g*n*(c + d*x)*Log[F] + 9*f^2*g^2*n^2*(c + d*x)^2*Log[F]^2))/(27*f^3*g^3*n^3*Log[F]^3)

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 414, normalized size = 1.13 \[ \frac {36 \, {\left (a^{3} d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{3} c d f^{3} g^{3} n^{3} x^{2} + 3 \, a^{3} c^{2} f^{3} g^{3} n^{3} x\right )} \log \relax (F)^{3} + 4 \, {\left (2 \, b^{3} d^{2} + 9 \, {\left (b^{3} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d f^{2} g^{2} n^{2} x + b^{3} c^{2} f^{2} g^{2} n^{2}\right )} \log \relax (F)^{2} - 6 \, {\left (b^{3} d^{2} f g n x + b^{3} c d f g n\right )} \log \relax (F)\right )} F^{3 \, f g n x + 3 \, e g n} + 81 \, {\left (a b^{2} d^{2} + 2 \, {\left (a b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d f^{2} g^{2} n^{2} x + a b^{2} c^{2} f^{2} g^{2} n^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (a b^{2} d^{2} f g n x + a b^{2} c d f g n\right )} \log \relax (F)\right )} F^{2 \, f g n x + 2 \, e g n} + 324 \, {\left (2 \, a^{2} b d^{2} + {\left (a^{2} b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d f^{2} g^{2} n^{2} x + a^{2} b c^{2} f^{2} g^{2} n^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (a^{2} b d^{2} f g n x + a^{2} b c d f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n}}{108 \, f^{3} g^{3} n^{3} \log \relax (F)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^3*(d*x+c)^2,x, algorithm="fricas")

[Out]

1/108*(36*(a^3*d^2*f^3*g^3*n^3*x^3 + 3*a^3*c*d*f^3*g^3*n^3*x^2 + 3*a^3*c^2*f^3*g^3*n^3*x)*log(F)^3 + 4*(2*b^3*
d^2 + 9*(b^3*d^2*f^2*g^2*n^2*x^2 + 2*b^3*c*d*f^2*g^2*n^2*x + b^3*c^2*f^2*g^2*n^2)*log(F)^2 - 6*(b^3*d^2*f*g*n*
x + b^3*c*d*f*g*n)*log(F))*F^(3*f*g*n*x + 3*e*g*n) + 81*(a*b^2*d^2 + 2*(a*b^2*d^2*f^2*g^2*n^2*x^2 + 2*a*b^2*c*
d*f^2*g^2*n^2*x + a*b^2*c^2*f^2*g^2*n^2)*log(F)^2 - 2*(a*b^2*d^2*f*g*n*x + a*b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n
*x + 2*e*g*n) + 324*(2*a^2*b*d^2 + (a^2*b*d^2*f^2*g^2*n^2*x^2 + 2*a^2*b*c*d*f^2*g^2*n^2*x + a^2*b*c^2*f^2*g^2*
n^2)*log(F)^2 - 2*(a^2*b*d^2*f*g*n*x + a^2*b*c*d*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(f^3*g^3*n^3*log(F)^3)

________________________________________________________________________________________

giac [C]  time = 1.66, size = 8854, normalized size = 24.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^3*(d*x+c)^2,x, algorithm="giac")

[Out]

1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + a^3*c^2*x - 1/27*((6*(3*pi*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 3*pi*b
^3*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 6*pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 6*pi*b^3*c*d*f^2*g^2*n^2*
x*log(abs(F)) + 3*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 3*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b^3*d^
2*f*g*n*x*sgn(F) + pi*b^3*d^2*f*g*n*x - pi*b^3*c*d*f*g*n*sgn(F) + pi*b^3*c*d*f*g*n)*(pi^3*f^3*g^3*n^3*sgn(F) -
3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3
*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi
^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2) - (9*pi^2
*b^3*d^2*f^2*g^2*n^2*x^2*sgn(F) - 9*pi^2*b^3*d^2*f^2*g^2*n^2*x^2 + 18*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 +
18*pi^2*b^3*c*d*f^2*g^2*n^2*x*sgn(F) - 18*pi^2*b^3*c*d*f^2*g^2*n^2*x + 36*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))^2
+ 9*pi^2*b^3*c^2*f^2*g^2*n^2*sgn(F) - 9*pi^2*b^3*c^2*f^2*g^2*n^2 + 18*b^3*c^2*f^2*g^2*n^2*log(abs(F))^2 - 12*b
^3*d^2*f*g*n*x*log(abs(F)) - 12*b^3*c*d*f*g*n*log(abs(F)) + 4*b^3*d^2)*(3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F)
- 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*l
og(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*s
gn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2))*cos(-3/2*pi*f*g*n*x*sgn(F) + 3/2*pi*
f*g*n*x - 3/2*pi*g*n*e*sgn(F) + 3/2*pi*g*n*e) - ((9*pi^2*b^3*d^2*f^2*g^2*n^2*x^2*sgn(F) - 9*pi^2*b^3*d^2*f^2*g
^2*n^2*x^2 + 18*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 18*pi^2*b^3*c*d*f^2*g^2*n^2*x*sgn(F) - 18*pi^2*b^3*c*d
*f^2*g^2*n^2*x + 36*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 9*pi^2*b^3*c^2*f^2*g^2*n^2*sgn(F) - 9*pi^2*b^3*c^2*f
^2*g^2*n^2 + 18*b^3*c^2*f^2*g^2*n^2*log(abs(F))^2 - 12*b^3*d^2*f*g*n*x*log(abs(F)) - 12*b^3*c*d*f*g*n*log(abs(
F)) + 4*b^3*d^2)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^
3*g^3*n^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3
+ 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F))
+ 2*f^3*g^3*n^3*log(abs(F))^3)^2) + 6*(3*pi*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 3*pi*b^3*d^2*f^2*g^2*
n^2*x^2*log(abs(F)) + 6*pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 6*pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F)) +
3*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 3*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b^3*d^2*f*g*n*x*sgn(F
) + pi*b^3*d^2*f*g*n*x - pi*b^3*c*d*f*g*n*sgn(F) + pi*b^3*c*d*f*g*n)*(3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) -
3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log
(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn
(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2))*sin(-3/2*pi*f*g*n*x*sgn(F) + 3/2*pi*f*
g*n*x - 3/2*pi*g*n*e*sgn(F) + 3/2*pi*g*n*e))*e^(3*f*g*n*x*log(abs(F)) + 3*g*n*e*log(abs(F))) + 1/2*I*((36*I*pi
^2*b^3*d^2*f^2*g^2*n^2*x^2*sgn(F) - 72*pi*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 36*I*pi^2*b^3*d^2*f^2*g
^2*n^2*x^2 + 72*pi*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 72*I*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 72*I*pi^
2*b^3*c*d*f^2*g^2*n^2*x*sgn(F) - 144*pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 72*I*pi^2*b^3*c*d*f^2*g^2*n
^2*x + 144*pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F)) + 144*I*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 36*I*pi^2*b^3*c^
2*f^2*g^2*n^2*sgn(F) - 72*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 36*I*pi^2*b^3*c^2*f^2*g^2*n^2 + 72*pi*b^
3*c^2*f^2*g^2*n^2*log(abs(F)) + 72*I*b^3*c^2*f^2*g^2*n^2*log(abs(F))^2 + 24*pi*b^3*d^2*f*g*n*x*sgn(F) - 24*pi*
b^3*d^2*f*g*n*x - 48*I*b^3*d^2*f*g*n*x*log(abs(F)) + 24*pi*b^3*c*d*f*g*n*sgn(F) - 24*pi*b^3*c*d*f*g*n - 48*I*b
^3*c*d*f*g*n*log(abs(F)) + 16*I*b^3*d^2)*e^(3/2*I*pi*f*g*n*x*sgn(F) - 3/2*I*pi*f*g*n*x + 3/2*I*pi*g*n*e*sgn(F)
- 3/2*I*pi*g*n*e)/(-108*I*pi^3*f^3*g^3*n^3*sgn(F) + 324*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) + 324*I*pi*f^3*g^
3*n^3*log(abs(F))^2*sgn(F) + 108*I*pi^3*f^3*g^3*n^3 - 324*pi^2*f^3*g^3*n^3*log(abs(F)) - 324*I*pi*f^3*g^3*n^3*
log(abs(F))^2 + 216*f^3*g^3*n^3*log(abs(F))^3) - (36*I*pi^2*b^3*d^2*f^2*g^2*n^2*x^2*sgn(F) + 72*pi*b^3*d^2*f^2
*g^2*n^2*x^2*log(abs(F))*sgn(F) - 36*I*pi^2*b^3*d^2*f^2*g^2*n^2*x^2 - 72*pi*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F)
) + 72*I*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 72*I*pi^2*b^3*c*d*f^2*g^2*n^2*x*sgn(F) + 144*pi*b^3*c*d*f^2*g
^2*n^2*x*log(abs(F))*sgn(F) - 72*I*pi^2*b^3*c*d*f^2*g^2*n^2*x - 144*pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F)) + 144
*I*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 36*I*pi^2*b^3*c^2*f^2*g^2*n^2*sgn(F) + 72*pi*b^3*c^2*f^2*g^2*n^2*log(
abs(F))*sgn(F) - 36*I*pi^2*b^3*c^2*f^2*g^2*n^2 - 72*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F)) + 72*I*b^3*c^2*f^2*g^2*
n^2*log(abs(F))^2 - 24*pi*b^3*d^2*f*g*n*x*sgn(F) + 24*pi*b^3*d^2*f*g*n*x - 48*I*b^3*d^2*f*g*n*x*log(abs(F)) -
24*pi*b^3*c*d*f*g*n*sgn(F) + 24*pi*b^3*c*d*f*g*n - 48*I*b^3*c*d*f*g*n*log(abs(F)) + 16*I*b^3*d^2)*e^(-3/2*I*pi
*f*g*n*x*sgn(F) + 3/2*I*pi*f*g*n*x - 3/2*I*pi*g*n*e*sgn(F) + 3/2*I*pi*g*n*e)/(108*I*pi^3*f^3*g^3*n^3*sgn(F) +
324*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 324*I*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - 108*I*pi^3*f^3*g^3*n^3 -
324*pi^2*f^3*g^3*n^3*log(abs(F)) + 324*I*pi*f^3*g^3*n^3*log(abs(F))^2 + 216*f^3*g^3*n^3*log(abs(F))^3))*e^(3*
f*g*n*x*log(abs(F)) + 3*g*n*e*log(abs(F))) - 3/2*(((2*pi*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 2*pi*a
*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*pi*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 4*pi*a*b^2*c*d*f^2*g^
2*n^2*x*log(abs(F)) + 2*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F)) -
pi*a*b^2*d^2*f*g*n*x*sgn(F) + pi*a*b^2*d^2*f*g*n*x - pi*a*b^2*c*d*f*g*n*sgn(F) + pi*a*b^2*c*d*f*g*n)*(pi^3*f^
3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)/
((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(ab
s(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F
))^3)^2) - (pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2 + 2*a*b^2*d^2*f^2*g^2*n^2*x
^2*log(abs(F))^2 + 2*pi^2*a*b^2*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*a*b^2*c*d*f^2*g^2*n^2*x + 4*a*b^2*c*d*f^2*g^
2*n^2*x*log(abs(F))^2 + pi^2*a*b^2*c^2*f^2*g^2*n^2*sgn(F) - pi^2*a*b^2*c^2*f^2*g^2*n^2 + 2*a*b^2*c^2*f^2*g^2*n
^2*log(abs(F))^2 - 2*a*b^2*d^2*f*g*n*x*log(abs(F)) - 2*a*b^2*c*d*f*g*n*log(abs(F)) + a*b^2*d^2)*(3*pi^2*f^3*g^
3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3*n^3*sg
n(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*
f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2))*cos(-pi*f*g
*n*x*sgn(F) + pi*f*g*n*x - pi*g*n*e*sgn(F) + pi*g*n*e) - ((pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*a*b^2*
d^2*f^2*g^2*n^2*x^2 + 2*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*a*b^2*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi
^2*a*b^2*c*d*f^2*g^2*n^2*x + 4*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*a*b^2*c^2*f^2*g^2*n^2*sgn(F) - pi^
2*a*b^2*c^2*f^2*g^2*n^2 + 2*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2 - 2*a*b^2*d^2*f*g*n*x*log(abs(F)) - 2*a*b^2*c*
d*f*g*n*log(abs(F)) + a*b^2*d^2)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g
^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - p
i^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*
n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2) + (2*pi*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 2*pi*
a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*pi*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 4*pi*a*b^2*c*d*f^2*g
^2*n^2*x*log(abs(F)) + 2*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))
- pi*a*b^2*d^2*f*g*n*x*sgn(F) + pi*a*b^2*d^2*f*g*n*x - pi*a*b^2*c*d*f*g*n*sgn(F) + pi*a*b^2*c*d*f*g*n)*(3*pi^2
*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3
*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (
3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2))*sin(
-pi*f*g*n*x*sgn(F) + pi*f*g*n*x - pi*g*n*e*sgn(F) + pi*g*n*e))*e^(2*f*g*n*x*log(abs(F)) + 2*g*n*e*log(abs(F)))
+ 1/2*I*((6*I*pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) - 12*pi*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 6*
I*pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2 + 12*pi*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 12*I*a*b^2*d^2*f^2*g^2*n^2*x^
2*log(abs(F))^2 + 12*I*pi^2*a*b^2*c*d*f^2*g^2*n^2*x*sgn(F) - 24*pi*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F)
- 12*I*pi^2*a*b^2*c*d*f^2*g^2*n^2*x + 24*pi*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F)) + 24*I*a*b^2*c*d*f^2*g^2*n^2*x
*log(abs(F))^2 + 6*I*pi^2*a*b^2*c^2*f^2*g^2*n^2*sgn(F) - 12*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*I*
pi^2*a*b^2*c^2*f^2*g^2*n^2 + 12*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F)) + 12*I*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))^
2 + 6*pi*a*b^2*d^2*f*g*n*x*sgn(F) - 6*pi*a*b^2*d^2*f*g*n*x - 12*I*a*b^2*d^2*f*g*n*x*log(abs(F)) + 6*pi*a*b^2*c
*d*f*g*n*sgn(F) - 6*pi*a*b^2*c*d*f*g*n - 12*I*a*b^2*c*d*f*g*n*log(abs(F)) + 6*I*a*b^2*d^2)*e^(I*pi*f*g*n*x*sgn
(F) - I*pi*f*g*n*x + I*pi*g*n*e*sgn(F) - I*pi*g*n*e)/(-4*I*pi^3*f^3*g^3*n^3*sgn(F) + 12*pi^2*f^3*g^3*n^3*log(a
bs(F))*sgn(F) + 12*I*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) + 4*I*pi^3*f^3*g^3*n^3 - 12*pi^2*f^3*g^3*n^3*log(abs(
F)) - 12*I*pi*f^3*g^3*n^3*log(abs(F))^2 + 8*f^3*g^3*n^3*log(abs(F))^3) - (6*I*pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2*s
gn(F) + 12*pi*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 6*I*pi^2*a*b^2*d^2*f^2*g^2*n^2*x^2 - 12*pi*a*b^2*
d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 12*I*a*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 12*I*pi^2*a*b^2*c*d*f^2*g^2*n
^2*x*sgn(F) + 24*pi*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 12*I*pi^2*a*b^2*c*d*f^2*g^2*n^2*x - 24*pi*a*b
^2*c*d*f^2*g^2*n^2*x*log(abs(F)) + 24*I*a*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 6*I*pi^2*a*b^2*c^2*f^2*g^2*n^2
*sgn(F) + 12*pi*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*I*pi^2*a*b^2*c^2*f^2*g^2*n^2 - 12*pi*a*b^2*c^2*f^
2*g^2*n^2*log(abs(F)) + 12*I*a*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2 - 6*pi*a*b^2*d^2*f*g*n*x*sgn(F) + 6*pi*a*b^2*
d^2*f*g*n*x - 12*I*a*b^2*d^2*f*g*n*x*log(abs(F)) - 6*pi*a*b^2*c*d*f*g*n*sgn(F) + 6*pi*a*b^2*c*d*f*g*n - 12*I*a
*b^2*c*d*f*g*n*log(abs(F)) + 6*I*a*b^2*d^2)*e^(-I*pi*f*g*n*x*sgn(F) + I*pi*f*g*n*x - I*pi*g*n*e*sgn(F) + I*pi*
g*n*e)/(4*I*pi^3*f^3*g^3*n^3*sgn(F) + 12*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 12*I*pi*f^3*g^3*n^3*log(abs(F))
^2*sgn(F) - 4*I*pi^3*f^3*g^3*n^3 - 12*pi^2*f^3*g^3*n^3*log(abs(F)) + 12*I*pi*f^3*g^3*n^3*log(abs(F))^2 + 8*f^3
*g^3*n^3*log(abs(F))^3))*e^(2*f*g*n*x*log(abs(F)) + 2*g*n*e*log(abs(F))) - 3*((2*(pi*a^2*b*d^2*f^2*g^2*n^2*x^2
*log(abs(F))*sgn(F) - pi*a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 2*pi*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(
F) - 2*pi*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*a^2*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*a^2*b*c^2*f^2
*g^2*n^2*log(abs(F)) - pi*a^2*b*d^2*f*g*n*x*sgn(F) + pi*a^2*b*d^2*f*g*n*x - pi*a^2*b*c*d*f*g*n*sgn(F) + pi*a^2
*b*c*d*f*g*n)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g
^3*n^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3
*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2
*f^3*g^3*n^3*log(abs(F))^3)^2) - (pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2 + 2*a
^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*a^2*b*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*a^2*b*c*d*f^2*g^2*n^2*
x + 4*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*a^2*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*a^2*b*c^2*f^2*g^2*n^2 +
2*a^2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*a^2*b*d^2*f*g*n*x*log(abs(F)) - 4*a^2*b*c*d*f*g*n*log(abs(F)) + 4*a
^2*b*d^2)*(3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^
3)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log
(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(ab
s(F))^3)^2))*cos(-1/2*pi*f*g*n*x*sgn(F) + 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*sgn(F) + 1/2*pi*g*n*e) - ((pi^2*a^2*b*
d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2 + 2*a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi
^2*a^2*b*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*a^2*b*c*d*f^2*g^2*n^2*x + 4*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 +
pi^2*a^2*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*a^2*b*c^2*f^2*g^2*n^2 + 2*a^2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*a^
2*b*d^2*f*g*n*x*log(abs(F)) - 4*a^2*b*c*d*f*g*n*log(abs(F)) + 4*a^2*b*d^2)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3
*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) -
3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^
3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2) + 2*(pi*a^2*b*d^2*
f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 2*pi*a^2*b*c*d*f^2*g^2*n^2*x*l
og(abs(F))*sgn(F) - 2*pi*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*a^2*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - p
i*a^2*b*c^2*f^2*g^2*n^2*log(abs(F)) - pi*a^2*b*d^2*f*g*n*x*sgn(F) + pi*a^2*b*d^2*f*g*n*x - pi*a^2*b*c*d*f*g*n*
sgn(F) + pi*a^2*b*c*d*f*g*n)*(3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g
^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3
*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2
*f^3*g^3*n^3*log(abs(F))^3)^2))*sin(-1/2*pi*f*g*n*x*sgn(F) + 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*sgn(F) + 1/2*pi*g*n
*e))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(abs(F))) + 1/2*I*((12*I*pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - 24*pi*
a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 12*I*pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2 + 24*pi*a^2*b*d^2*f^2*g^2*n
^2*x^2*log(abs(F)) + 24*I*a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 24*I*pi^2*a^2*b*c*d*f^2*g^2*n^2*x*sgn(F) -
48*pi*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 24*I*pi^2*a^2*b*c*d*f^2*g^2*n^2*x + 48*pi*a^2*b*c*d*f^2*g^
2*n^2*x*log(abs(F)) + 48*I*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 12*I*pi^2*a^2*b*c^2*f^2*g^2*n^2*sgn(F) - 24
*pi*a^2*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 12*I*pi^2*a^2*b*c^2*f^2*g^2*n^2 + 24*pi*a^2*b*c^2*f^2*g^2*n^2*l
og(abs(F)) + 24*I*a^2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 + 24*pi*a^2*b*d^2*f*g*n*x*sgn(F) - 24*pi*a^2*b*d^2*f*g*n
*x - 48*I*a^2*b*d^2*f*g*n*x*log(abs(F)) + 24*pi*a^2*b*c*d*f*g*n*sgn(F) - 24*pi*a^2*b*c*d*f*g*n - 48*I*a^2*b*c*
d*f*g*n*log(abs(F)) + 48*I*a^2*b*d^2)*e^(1/2*I*pi*f*g*n*x*sgn(F) - 1/2*I*pi*f*g*n*x + 1/2*I*pi*g*n*e*sgn(F) -
1/2*I*pi*g*n*e)/(-4*I*pi^3*f^3*g^3*n^3*sgn(F) + 12*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) + 12*I*pi*f^3*g^3*n^3*l
og(abs(F))^2*sgn(F) + 4*I*pi^3*f^3*g^3*n^3 - 12*pi^2*f^3*g^3*n^3*log(abs(F)) - 12*I*pi*f^3*g^3*n^3*log(abs(F))
^2 + 8*f^3*g^3*n^3*log(abs(F))^3) - (12*I*pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) + 24*pi*a^2*b*d^2*f^2*g^2*n^2*
x^2*log(abs(F))*sgn(F) - 12*I*pi^2*a^2*b*d^2*f^2*g^2*n^2*x^2 - 24*pi*a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 2
4*I*a^2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 24*I*pi^2*a^2*b*c*d*f^2*g^2*n^2*x*sgn(F) + 48*pi*a^2*b*c*d*f^2*g
^2*n^2*x*log(abs(F))*sgn(F) - 24*I*pi^2*a^2*b*c*d*f^2*g^2*n^2*x - 48*pi*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F)) +
48*I*a^2*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 12*I*pi^2*a^2*b*c^2*f^2*g^2*n^2*sgn(F) + 24*pi*a^2*b*c^2*f^2*g^2*
n^2*log(abs(F))*sgn(F) - 12*I*pi^2*a^2*b*c^2*f^2*g^2*n^2 - 24*pi*a^2*b*c^2*f^2*g^2*n^2*log(abs(F)) + 24*I*a^2*
b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 24*pi*a^2*b*d^2*f*g*n*x*sgn(F) + 24*pi*a^2*b*d^2*f*g*n*x - 48*I*a^2*b*d^2*f*
g*n*x*log(abs(F)) - 24*pi*a^2*b*c*d*f*g*n*sgn(F) + 24*pi*a^2*b*c*d*f*g*n - 48*I*a^2*b*c*d*f*g*n*log(abs(F)) +
48*I*a^2*b*d^2)*e^(-1/2*I*pi*f*g*n*x*sgn(F) + 1/2*I*pi*f*g*n*x - 1/2*I*pi*g*n*e*sgn(F) + 1/2*I*pi*g*n*e)/(4*I*
pi^3*f^3*g^3*n^3*sgn(F) + 12*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 12*I*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) -
4*I*pi^3*f^3*g^3*n^3 - 12*pi^2*f^3*g^3*n^3*log(abs(F)) + 12*I*pi*f^3*g^3*n^3*log(abs(F))^2 + 8*f^3*g^3*n^3*log
(abs(F))^3))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(abs(F)))

________________________________________________________________________________________

maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{3} \left (d x +c \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*(F^((f*x+e)*g))^n+a)^3*(d*x+c)^2,x)

[Out]

int((b*(F^((f*x+e)*g))^n+a)^3*(d*x+c)^2,x)

________________________________________________________________________________________

maxima [A]  time = 1.81, size = 558, normalized size = 1.52 \[ \frac {1}{3} \, a^{3} d^{2} x^{3} + a^{3} c d x^{2} + a^{3} c^{2} x + \frac {3 \, {\left (F^{f g x + e g}\right )}^{n} a^{2} b c^{2}}{f g n \log \relax (F)} + \frac {3 \, {\left (F^{f g x + e g}\right )}^{2 \, n} a b^{2} c^{2}}{2 \, f g n \log \relax (F)} + \frac {{\left (F^{f g x + e g}\right )}^{3 \, n} b^{3} c^{2}}{3 \, f g n \log \relax (F)} + \frac {6 \, {\left ({\left (F^{e g}\right )}^{n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} a^{2} b c d}{f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {3 \, {\left (2 \, {\left (F^{e g}\right )}^{2 \, n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{2 \, n}\right )} {\left (F^{f g x}\right )}^{2 \, n} a b^{2} c d}{2 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {2 \, {\left (3 \, {\left (F^{e g}\right )}^{3 \, n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{3 \, n}\right )} {\left (F^{f g x}\right )}^{3 \, n} b^{3} c d}{9 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {3 \, {\left ({\left (F^{e g}\right )}^{n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} - 2 \, {\left (F^{e g}\right )}^{n} f g n x \log \relax (F) + 2 \, {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} a^{2} b d^{2}}{f^{3} g^{3} n^{3} \log \relax (F)^{3}} + \frac {3 \, {\left (2 \, {\left (F^{e g}\right )}^{2 \, n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} - 2 \, {\left (F^{e g}\right )}^{2 \, n} f g n x \log \relax (F) + {\left (F^{e g}\right )}^{2 \, n}\right )} {\left (F^{f g x}\right )}^{2 \, n} a b^{2} d^{2}}{4 \, f^{3} g^{3} n^{3} \log \relax (F)^{3}} + \frac {{\left (9 \, {\left (F^{e g}\right )}^{3 \, n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} - 6 \, {\left (F^{e g}\right )}^{3 \, n} f g n x \log \relax (F) + 2 \, {\left (F^{e g}\right )}^{3 \, n}\right )} {\left (F^{f g x}\right )}^{3 \, n} b^{3} d^{2}}{27 \, f^{3} g^{3} n^{3} \log \relax (F)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^3*(d*x+c)^2,x, algorithm="maxima")

[Out]

1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + a^3*c^2*x + 3*(F^(f*g*x + e*g))^n*a^2*b*c^2/(f*g*n*log(F)) + 3/2*(F^(f*g*x + e
*g))^(2*n)*a*b^2*c^2/(f*g*n*log(F)) + 1/3*(F^(f*g*x + e*g))^(3*n)*b^3*c^2/(f*g*n*log(F)) + 6*((F^(e*g))^n*f*g*
n*x*log(F) - (F^(e*g))^n)*(F^(f*g*x))^n*a^2*b*c*d/(f^2*g^2*n^2*log(F)^2) + 3/2*(2*(F^(e*g))^(2*n)*f*g*n*x*log(
F) - (F^(e*g))^(2*n))*(F^(f*g*x))^(2*n)*a*b^2*c*d/(f^2*g^2*n^2*log(F)^2) + 2/9*(3*(F^(e*g))^(3*n)*f*g*n*x*log(
F) - (F^(e*g))^(3*n))*(F^(f*g*x))^(3*n)*b^3*c*d/(f^2*g^2*n^2*log(F)^2) + 3*((F^(e*g))^n*f^2*g^2*n^2*x^2*log(F)
^2 - 2*(F^(e*g))^n*f*g*n*x*log(F) + 2*(F^(e*g))^n)*(F^(f*g*x))^n*a^2*b*d^2/(f^3*g^3*n^3*log(F)^3) + 3/4*(2*(F^
(e*g))^(2*n)*f^2*g^2*n^2*x^2*log(F)^2 - 2*(F^(e*g))^(2*n)*f*g*n*x*log(F) + (F^(e*g))^(2*n))*(F^(f*g*x))^(2*n)*
a*b^2*d^2/(f^3*g^3*n^3*log(F)^3) + 1/27*(9*(F^(e*g))^(3*n)*f^2*g^2*n^2*x^2*log(F)^2 - 6*(F^(e*g))^(3*n)*f*g*n*
x*log(F) + 2*(F^(e*g))^(3*n))*(F^(f*g*x))^(3*n)*b^3*d^2/(f^3*g^3*n^3*log(F)^3)

________________________________________________________________________________________

mupad [B]  time = 3.99, size = 399, normalized size = 1.09 \[ {\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}\,\left (\frac {b^3\,\left (9\,c^2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-6\,c\,d\,f\,g\,n\,\ln \relax (F)+2\,d^2\right )}{27\,f^3\,g^3\,n^3\,{\ln \relax (F)}^3}+\frac {b^3\,d^2\,x^2}{3\,f\,g\,n\,\ln \relax (F)}-\frac {2\,b^3\,d\,x\,\left (d-3\,c\,f\,g\,n\,\ln \relax (F)\right )}{9\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {3\,a^2\,b\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-2\,c\,d\,f\,g\,n\,\ln \relax (F)+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \relax (F)}^3}+\frac {3\,a^2\,b\,d^2\,x^2}{f\,g\,n\,\ln \relax (F)}-\frac {6\,a^2\,b\,d\,x\,\left (d-c\,f\,g\,n\,\ln \relax (F)\right )}{f^2\,g^2\,n^2\,{\ln \relax (F)}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {3\,a\,b^2\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-2\,c\,d\,f\,g\,n\,\ln \relax (F)+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \relax (F)}^3}+\frac {3\,a\,b^2\,d^2\,x^2}{2\,f\,g\,n\,\ln \relax (F)}-\frac {3\,a\,b^2\,d\,x\,\left (d-2\,c\,f\,g\,n\,\ln \relax (F)\right )}{2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2}\right )+a^3\,c^2\,x+\frac {a^3\,d^2\,x^3}{3}+a^3\,c\,d\,x^2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2,x)

[Out]

(F^(f*g*x)*F^(e*g))^(3*n)*((b^3*(2*d^2 + 9*c^2*f^2*g^2*n^2*log(F)^2 - 6*c*d*f*g*n*log(F)))/(27*f^3*g^3*n^3*log
(F)^3) + (b^3*d^2*x^2)/(3*f*g*n*log(F)) - (2*b^3*d*x*(d - 3*c*f*g*n*log(F)))/(9*f^2*g^2*n^2*log(F)^2)) + (F^(f
*g*x)*F^(e*g))^n*((3*a^2*b*(2*d^2 + c^2*f^2*g^2*n^2*log(F)^2 - 2*c*d*f*g*n*log(F)))/(f^3*g^3*n^3*log(F)^3) + (
3*a^2*b*d^2*x^2)/(f*g*n*log(F)) - (6*a^2*b*d*x*(d - c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F)^2)) + (F^(f*g*x)*F^(e
*g))^(2*n)*((3*a*b^2*(d^2 + 2*c^2*f^2*g^2*n^2*log(F)^2 - 2*c*d*f*g*n*log(F)))/(4*f^3*g^3*n^3*log(F)^3) + (3*a*
b^2*d^2*x^2)/(2*f*g*n*log(F)) - (3*a*b^2*d*x*(d - 2*c*f*g*n*log(F)))/(2*f^2*g^2*n^2*log(F)^2)) + a^3*c^2*x + (
a^3*d^2*x^3)/3 + a^3*c*d*x^2

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 653, normalized size = 1.78 \[ a^{3} c^{2} x + a^{3} c d x^{2} + \frac {a^{3} d^{2} x^{3}}{3} + \begin {cases} \frac {\left (36 b^{3} c^{2} f^{8} g^{8} n^{8} \log {\relax (F )}^{8} + 72 b^{3} c d f^{8} g^{8} n^{8} x \log {\relax (F )}^{8} - 24 b^{3} c d f^{7} g^{7} n^{7} \log {\relax (F )}^{7} + 36 b^{3} d^{2} f^{8} g^{8} n^{8} x^{2} \log {\relax (F )}^{8} - 24 b^{3} d^{2} f^{7} g^{7} n^{7} x \log {\relax (F )}^{7} + 8 b^{3} d^{2} f^{6} g^{6} n^{6} \log {\relax (F )}^{6}\right ) \left (F^{g \left (e + f x\right )}\right )^{3 n} + \left (162 a b^{2} c^{2} f^{8} g^{8} n^{8} \log {\relax (F )}^{8} + 324 a b^{2} c d f^{8} g^{8} n^{8} x \log {\relax (F )}^{8} - 162 a b^{2} c d f^{7} g^{7} n^{7} \log {\relax (F )}^{7} + 162 a b^{2} d^{2} f^{8} g^{8} n^{8} x^{2} \log {\relax (F )}^{8} - 162 a b^{2} d^{2} f^{7} g^{7} n^{7} x \log {\relax (F )}^{7} + 81 a b^{2} d^{2} f^{6} g^{6} n^{6} \log {\relax (F )}^{6}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (324 a^{2} b c^{2} f^{8} g^{8} n^{8} \log {\relax (F )}^{8} + 648 a^{2} b c d f^{8} g^{8} n^{8} x \log {\relax (F )}^{8} - 648 a^{2} b c d f^{7} g^{7} n^{7} \log {\relax (F )}^{7} + 324 a^{2} b d^{2} f^{8} g^{8} n^{8} x^{2} \log {\relax (F )}^{8} - 648 a^{2} b d^{2} f^{7} g^{7} n^{7} x \log {\relax (F )}^{7} + 648 a^{2} b d^{2} f^{6} g^{6} n^{6} \log {\relax (F )}^{6}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{108 f^{9} g^{9} n^{9} \log {\relax (F )}^{9}} & \text {for}\: 108 f^{9} g^{9} n^{9} \log {\relax (F )}^{9} \neq 0 \\x^{3} \left (a^{2} b d^{2} + a b^{2} d^{2} + \frac {b^{3} d^{2}}{3}\right ) + x^{2} \left (3 a^{2} b c d + 3 a b^{2} c d + b^{3} c d\right ) + x \left (3 a^{2} b c^{2} + 3 a b^{2} c^{2} + b^{3} c^{2}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F**(g*(f*x+e)))**n)**3*(d*x+c)**2,x)

[Out]

a**3*c**2*x + a**3*c*d*x**2 + a**3*d**2*x**3/3 + Piecewise((((36*b**3*c**2*f**8*g**8*n**8*log(F)**8 + 72*b**3*
c*d*f**8*g**8*n**8*x*log(F)**8 - 24*b**3*c*d*f**7*g**7*n**7*log(F)**7 + 36*b**3*d**2*f**8*g**8*n**8*x**2*log(F
)**8 - 24*b**3*d**2*f**7*g**7*n**7*x*log(F)**7 + 8*b**3*d**2*f**6*g**6*n**6*log(F)**6)*(F**(g*(e + f*x)))**(3*
n) + (162*a*b**2*c**2*f**8*g**8*n**8*log(F)**8 + 324*a*b**2*c*d*f**8*g**8*n**8*x*log(F)**8 - 162*a*b**2*c*d*f*
*7*g**7*n**7*log(F)**7 + 162*a*b**2*d**2*f**8*g**8*n**8*x**2*log(F)**8 - 162*a*b**2*d**2*f**7*g**7*n**7*x*log(
F)**7 + 81*a*b**2*d**2*f**6*g**6*n**6*log(F)**6)*(F**(g*(e + f*x)))**(2*n) + (324*a**2*b*c**2*f**8*g**8*n**8*l
og(F)**8 + 648*a**2*b*c*d*f**8*g**8*n**8*x*log(F)**8 - 648*a**2*b*c*d*f**7*g**7*n**7*log(F)**7 + 324*a**2*b*d*
*2*f**8*g**8*n**8*x**2*log(F)**8 - 648*a**2*b*d**2*f**7*g**7*n**7*x*log(F)**7 + 648*a**2*b*d**2*f**6*g**6*n**6
*log(F)**6)*(F**(g*(e + f*x)))**n)/(108*f**9*g**9*n**9*log(F)**9), Ne(108*f**9*g**9*n**9*log(F)**9, 0)), (x**3
*(a**2*b*d**2 + a*b**2*d**2 + b**3*d**2/3) + x**2*(3*a**2*b*c*d + 3*a*b**2*c*d + b**3*c*d) + x*(3*a**2*b*c**2
+ 3*a*b**2*c**2 + b**3*c**2), True))

________________________________________________________________________________________