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

Optimal. Leaf size=414 \[ -\frac {120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac {48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac {120 f^2 x F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {48 e f x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {60 f^2 x^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {6 e^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {24 e f x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {20 f^2 x^3 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {3 e^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {8 e f x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 x^4 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^4 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^5 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

-120*f^2*F^(b*d*x+b*c+a)/b^6/d^6/ln(F)^6+48*e*f*F^(b*d*x+b*c+a)/b^5/d^5/ln(F)^5+120*f^2*F^(b*d*x+b*c+a)*x/b^5/
d^5/ln(F)^5-6*e^2*F^(b*d*x+b*c+a)/b^4/d^4/ln(F)^4-48*e*f*F^(b*d*x+b*c+a)*x/b^4/d^4/ln(F)^4-60*f^2*F^(b*d*x+b*c
+a)*x^2/b^4/d^4/ln(F)^4+6*e^2*F^(b*d*x+b*c+a)*x/b^3/d^3/ln(F)^3+24*e*f*F^(b*d*x+b*c+a)*x^2/b^3/d^3/ln(F)^3+20*
f^2*F^(b*d*x+b*c+a)*x^3/b^3/d^3/ln(F)^3-3*e^2*F^(b*d*x+b*c+a)*x^2/b^2/d^2/ln(F)^2-8*e*f*F^(b*d*x+b*c+a)*x^3/b^
2/d^2/ln(F)^2-5*f^2*F^(b*d*x+b*c+a)*x^4/b^2/d^2/ln(F)^2+e^2*F^(b*d*x+b*c+a)*x^3/b/d/ln(F)+2*e*f*F^(b*d*x+b*c+a
)*x^4/b/d/ln(F)+f^2*F^(b*d*x+b*c+a)*x^5/b/d/ln(F)

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Rubi [A]  time = 0.67, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2196, 2176, 2194} \[ -\frac {3 e^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {6 e^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {8 e f x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {24 e f x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {48 e f x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {5 f^2 x^4 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {20 f^2 x^3 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {60 f^2 x^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {120 f^2 x F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac {e^2 x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^4 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^5 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-120*f^2*F^(a + b*c + b*d*x))/(b^6*d^6*Log[F]^6) + (48*e*f*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) + (120*f^2
*F^(a + b*c + b*d*x)*x)/(b^5*d^5*Log[F]^5) - (6*e^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (48*e*f*F^(a + b
*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) - (60*f^2*F^(a + b*c + b*d*x)*x^2)/(b^4*d^4*Log[F]^4) + (6*e^2*F^(a + b*c +
b*d*x)*x)/(b^3*d^3*Log[F]^3) + (24*e*f*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) + (20*f^2*F^(a + b*c + b*d*
x)*x^3)/(b^3*d^3*Log[F]^3) - (3*e^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (8*e*f*F^(a + b*c + b*d*x)*x
^3)/(b^2*d^2*Log[F]^2) - (5*f^2*F^(a + b*c + b*d*x)*x^4)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^3)/(b
*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^5)/(b*d*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 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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {align*} \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx &=\int \left (e^2 F^{a+b c+b d x} x^3+2 e f F^{a+b c+b d x} x^4+f^2 F^{a+b c+b d x} x^5\right ) \, dx\\ &=e^2 \int F^{a+b c+b d x} x^3 \, dx+(2 e f) \int F^{a+b c+b d x} x^4 \, dx+f^2 \int F^{a+b c+b d x} x^5 \, dx\\ &=\frac {e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^5}{b d \log (F)}-\frac {\left (3 e^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b d \log (F)}-\frac {(8 e f) \int F^{a+b c+b d x} x^3 \, dx}{b d \log (F)}-\frac {\left (5 f^2\right ) \int F^{a+b c+b d x} x^4 \, dx}{b d \log (F)}\\ &=-\frac {3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^5}{b d \log (F)}+\frac {\left (6 e^2\right ) \int F^{a+b c+b d x} x \, dx}{b^2 d^2 \log ^2(F)}+\frac {(24 e f) \int F^{a+b c+b d x} x^2 \, dx}{b^2 d^2 \log ^2(F)}+\frac {\left (20 f^2\right ) \int F^{a+b c+b d x} x^3 \, dx}{b^2 d^2 \log ^2(F)}\\ &=\frac {6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac {20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac {3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^5}{b d \log (F)}-\frac {\left (6 e^2\right ) \int F^{a+b c+b d x} \, dx}{b^3 d^3 \log ^3(F)}-\frac {(48 e f) \int F^{a+b c+b d x} x \, dx}{b^3 d^3 \log ^3(F)}-\frac {\left (60 f^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b^3 d^3 \log ^3(F)}\\ &=-\frac {6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {48 e f F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}-\frac {60 f^2 F^{a+b c+b d x} x^2}{b^4 d^4 \log ^4(F)}+\frac {6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac {20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac {3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^5}{b d \log (F)}+\frac {(48 e f) \int F^{a+b c+b d x} \, dx}{b^4 d^4 \log ^4(F)}+\frac {\left (120 f^2\right ) \int F^{a+b c+b d x} x \, dx}{b^4 d^4 \log ^4(F)}\\ &=\frac {48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac {120 f^2 F^{a+b c+b d x} x}{b^5 d^5 \log ^5(F)}-\frac {6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {48 e f F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}-\frac {60 f^2 F^{a+b c+b d x} x^2}{b^4 d^4 \log ^4(F)}+\frac {6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac {20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac {3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^5}{b d \log (F)}-\frac {\left (120 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^5 d^5 \log ^5(F)}\\ &=-\frac {120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac {48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac {120 f^2 F^{a+b c+b d x} x}{b^5 d^5 \log ^5(F)}-\frac {6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {48 e f F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}-\frac {60 f^2 F^{a+b c+b d x} x^2}{b^4 d^4 \log ^4(F)}+\frac {6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac {20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac {3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac {5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^5}{b d \log (F)}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 159, normalized size = 0.38 \[ \frac {F^{a+b (c+d x)} \left (b^5 d^5 x^3 \log ^5(F) (e+f x)^2-b^4 d^4 x^2 \log ^4(F) \left (3 e^2+8 e f x+5 f^2 x^2\right )+2 b^3 d^3 x \log ^3(F) \left (3 e^2+12 e f x+10 f^2 x^2\right )-6 b^2 d^2 \log ^2(F) \left (e^2+8 e f x+10 f^2 x^2\right )+24 b d f \log (F) (2 e+5 f x)-120 f^2\right )}{b^6 d^6 \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a + b*(c + d*x))*(-120*f^2 + 24*b*d*f*(2*e + 5*f*x)*Log[F] - 6*b^2*d^2*(e^2 + 8*e*f*x + 10*f^2*x^2)*Log[F]
^2 + 2*b^3*d^3*x*(3*e^2 + 12*e*f*x + 10*f^2*x^2)*Log[F]^3 - b^4*d^4*x^2*(3*e^2 + 8*e*f*x + 5*f^2*x^2)*Log[F]^4
+ b^5*d^5*x^3*(e + f*x)^2*Log[F]^5))/(b^6*d^6*Log[F]^6)

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fricas [A]  time = 0.42, size = 228, normalized size = 0.55 \[ \frac {{\left ({\left (b^{5} d^{5} f^{2} x^{5} + 2 \, b^{5} d^{5} e f x^{4} + b^{5} d^{5} e^{2} x^{3}\right )} \log \relax (F)^{5} - {\left (5 \, b^{4} d^{4} f^{2} x^{4} + 8 \, b^{4} d^{4} e f x^{3} + 3 \, b^{4} d^{4} e^{2} x^{2}\right )} \log \relax (F)^{4} + 2 \, {\left (10 \, b^{3} d^{3} f^{2} x^{3} + 12 \, b^{3} d^{3} e f x^{2} + 3 \, b^{3} d^{3} e^{2} x\right )} \log \relax (F)^{3} - 6 \, {\left (10 \, b^{2} d^{2} f^{2} x^{2} + 8 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \relax (F)^{2} - 120 \, f^{2} + 24 \, {\left (5 \, b d f^{2} x + 2 \, b d e f\right )} \log \relax (F)\right )} F^{b d x + b c + a}}{b^{6} d^{6} \log \relax (F)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^5*d^5*f^2*x^5 + 2*b^5*d^5*e*f*x^4 + b^5*d^5*e^2*x^3)*log(F)^5 - (5*b^4*d^4*f^2*x^4 + 8*b^4*d^4*e*f*x^3 + 3
*b^4*d^4*e^2*x^2)*log(F)^4 + 2*(10*b^3*d^3*f^2*x^3 + 12*b^3*d^3*e*f*x^2 + 3*b^3*d^3*e^2*x)*log(F)^3 - 6*(10*b^
2*d^2*f^2*x^2 + 8*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 - 120*f^2 + 24*(5*b*d*f^2*x + 2*b*d*e*f)*log(F))*F^(b*
d*x + b*c + a)/(b^6*d^6*log(F)^6)

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giac [C]  time = 1.52, size = 9953, normalized size = 24.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((4*(pi^3*b^3*d^3*x^3*sgn(F) - 3*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*x^3 + 3*pi*b^3*d^3*x^3*log
(abs(F))^2 + 6*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*d^2*x^2*log(abs(F)) - 6*pi*b*d*x*sgn(F) + 6*pi*b*d
*x)*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4
*log(abs(F))^3)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*lo
g(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(
F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^2) - (pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F
))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)*(3*pi^2*b^3*d^3*x^3*log(a
bs(F))*sgn(F) - 3*pi^2*b^3*d^3*x^3*log(abs(F)) + 2*b^3*d^3*x^3*log(abs(F))^3 - 3*pi^2*b^2*d^2*x^2*sgn(F) + 3*p
i^2*b^2*d^2*x^2 - 6*b^2*d^2*x^2*log(abs(F))^2 + 12*b*d*x*log(abs(F)) - 12)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*
d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3
*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(
F))^3)^2))*cos(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi
*a) + ((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^
2 - 2*b^4*d^4*log(abs(F))^4)*(pi^3*b^3*d^3*x^3*sgn(F) - 3*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*x
^3 + 3*pi*b^3*d^3*x^3*log(abs(F))^2 + 6*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*d^2*x^2*log(abs(F)) - 6*p
i*b*d*x*sgn(F) + 6*pi*b*d*x)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi
^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(a
bs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^2) + 4*(pi^3*b^4*d^4*log(abs(F))*sgn(F)
- pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)*(3*pi^2*b^3*d^3*x^3*
log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*x^3*log(abs(F)) + 2*b^3*d^3*x^3*log(abs(F))^3 - 3*pi^2*b^2*d^2*x^2*sgn(F)
+ 3*pi^2*b^2*d^2*x^2 - 6*b^2*d^2*x^2*log(abs(F))^2 + 12*b*d*x*log(abs(F)) - 12)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2
*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*
(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log
(abs(F))^3)^2))*sin(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1
/2*pi*a))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) - 1/2*I*((8*pi^3*b^3*d^3*x^3*sgn(F) + 24
*I*pi^2*b^3*d^3*x^3*log(abs(F))*sgn(F) - 24*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*d^3*x^3 - 24*I*pi
^2*b^3*d^3*x^3*log(abs(F)) + 24*pi*b^3*d^3*x^3*log(abs(F))^2 + 16*I*b^3*d^3*x^3*log(abs(F))^3 - 24*I*pi^2*b^2*
d^2*x^2*sgn(F) + 48*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) + 24*I*pi^2*b^2*d^2*x^2 - 48*pi*b^2*d^2*x^2*log(abs(F))
- 48*I*b^2*d^2*x^2*log(abs(F))^2 - 48*pi*b*d*x*sgn(F) + 48*pi*b*d*x + 96*I*b*d*x*log(abs(F)) - 96*I)*e^(1/2*I*
pi*b*d*x*sgn(F) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F) - 1/2*I*pi*a)/(8*pi^
4*b^4*d^4*sgn(F) + 32*I*pi^3*b^4*d^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - 32*I*pi*b^4*d
^4*log(abs(F))^3*sgn(F) - 8*pi^4*b^4*d^4 - 32*I*pi^3*b^4*d^4*log(abs(F)) + 48*pi^2*b^4*d^4*log(abs(F))^2 + 32*
I*pi*b^4*d^4*log(abs(F))^3 - 16*b^4*d^4*log(abs(F))^4) + (8*pi^3*b^3*d^3*x^3*sgn(F) - 24*I*pi^2*b^3*d^3*x^3*lo
g(abs(F))*sgn(F) - 24*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*d^3*x^3 + 24*I*pi^2*b^3*d^3*x^3*log(abs
(F)) + 24*pi*b^3*d^3*x^3*log(abs(F))^2 - 16*I*b^3*d^3*x^3*log(abs(F))^3 + 24*I*pi^2*b^2*d^2*x^2*sgn(F) + 48*pi
*b^2*d^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b^2*d^2*x^2 - 48*pi*b^2*d^2*x^2*log(abs(F)) + 48*I*b^2*d^2*x^2*log
(abs(F))^2 - 48*pi*b*d*x*sgn(F) + 48*pi*b*d*x - 96*I*b*d*x*log(abs(F)) + 96*I)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2
*I*pi*b*d*x - 1/2*I*pi*b*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*a*sgn(F) + 1/2*I*pi*a)/(8*pi^4*b^4*d^4*sgn(F) - 32
*I*pi^3*b^4*d^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) + 32*I*pi*b^4*d^4*log(abs(F))^3*sgn(
F) - 8*pi^4*b^4*d^4 + 32*I*pi^3*b^4*d^4*log(abs(F)) + 48*pi^2*b^4*d^4*log(abs(F))^2 - 32*I*pi*b^4*d^4*log(abs(
F))^3 - 16*b^4*d^4*log(abs(F))^4))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) - 2*((4*(pi^3*b
^4*d^4*f*x^4*log(abs(F))*sgn(F) - pi*b^4*d^4*f*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*f*x^4*log(abs(F)) + pi*
b^4*d^4*f*x^4*log(abs(F))^3 - pi^3*b^3*d^3*f*x^3*sgn(F) + 3*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) + pi^3*b^3*d
^3*f*x^3 - 3*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 6*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*d^2*f*x^2*log(a
bs(F)) + 6*pi*b*d*f*x*sgn(F) - 6*pi*b*d*f*x)*(pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*p
i*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)/((
pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5
+ 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2
*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs
(F))^5)^2) - (pi^4*b^4*d^4*f*x^4*sgn(F) - 6*pi^2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*f*x^4 + 6*p
i^2*b^4*d^4*f*x^4*log(abs(F))^2 - 2*b^4*d^4*f*x^4*log(abs(F))^4 + 12*pi^2*b^3*d^3*f*x^3*log(abs(F))*sgn(F) - 1
2*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 8*b^3*d^3*f*x^3*log(abs(F))^3 - 12*pi^2*b^2*d^2*f*x^2*sgn(F) + 12*pi^2*b^2*
d^2*f*x^2 - 24*b^2*d^2*f*x^2*log(abs(F))^2 + 48*b*d*f*x*log(abs(F)) - 48*f)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F)
- 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d
^5*log(abs(F))^5)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sg
n(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(
F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3
- 2*b^5*d^5*log(abs(F))^5)^2))*cos(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2
*pi*a*sgn(F) + 1/2*pi*a) - ((pi^4*b^4*d^4*f*x^4*sgn(F) - 6*pi^2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*
d^4*f*x^4 + 6*pi^2*b^4*d^4*f*x^4*log(abs(F))^2 - 2*b^4*d^4*f*x^4*log(abs(F))^4 + 12*pi^2*b^3*d^3*f*x^3*log(abs
(F))*sgn(F) - 12*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 8*b^3*d^3*f*x^3*log(abs(F))^3 - 12*pi^2*b^2*d^2*f*x^2*sgn(F)
+ 12*pi^2*b^2*d^2*f*x^2 - 24*b^2*d^2*f*x^2*log(abs(F))^2 + 48*b*d*f*x*log(abs(F)) - 48*f)*(pi^5*b^5*d^5*sgn(F
) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*
log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*p
i*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2
+ (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*
pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2) + 4*(pi^3*b^4*d^4*f*x^4*log(abs(F))*sgn(F) - pi*b^4*d
^4*f*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*f*x^4*log(abs(F)) + pi*b^4*d^4*f*x^4*log(abs(F))^3 - pi^3*b^3*d^3
*f*x^3*sgn(F) + 3*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) + pi^3*b^3*d^3*f*x^3 - 3*pi*b^3*d^3*f*x^3*log(abs(F))^
2 - 6*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*d^2*f*x^2*log(abs(F)) + 6*pi*b*d*f*x*sgn(F) - 6*pi*b*d*f*
x)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10
*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*s
gn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(ab
s(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(ab
s(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2))*sin(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x -
1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi*a))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(
abs(F)) + 1) + 1/2*I*((-32*I*pi^4*b^4*d^4*f*x^4*sgn(F) + 128*pi^3*b^4*d^4*f*x^4*log(abs(F))*sgn(F) + 192*I*pi^
2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) - 128*pi*b^4*d^4*f*x^4*log(abs(F))^3*sgn(F) + 32*I*pi^4*b^4*d^4*f*x^4 - 1
28*pi^3*b^4*d^4*f*x^4*log(abs(F)) - 192*I*pi^2*b^4*d^4*f*x^4*log(abs(F))^2 + 128*pi*b^4*d^4*f*x^4*log(abs(F))^
3 + 64*I*b^4*d^4*f*x^4*log(abs(F))^4 - 128*pi^3*b^3*d^3*f*x^3*sgn(F) - 384*I*pi^2*b^3*d^3*f*x^3*log(abs(F))*sg
n(F) + 384*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) + 128*pi^3*b^3*d^3*f*x^3 + 384*I*pi^2*b^3*d^3*f*x^3*log(abs(F
)) - 384*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 256*I*b^3*d^3*f*x^3*log(abs(F))^3 + 384*I*pi^2*b^2*d^2*f*x^2*sgn(F)
- 768*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) - 384*I*pi^2*b^2*d^2*f*x^2 + 768*pi*b^2*d^2*f*x^2*log(abs(F)) + 768*
I*b^2*d^2*f*x^2*log(abs(F))^2 + 768*pi*b*d*f*x*sgn(F) - 768*pi*b*d*f*x - 1536*I*b*d*f*x*log(abs(F)) + 1536*I*f
)*e^(1/2*I*pi*b*d*x*sgn(F) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F) - 1/2*I*p
i*a)/(16*I*pi^5*b^5*d^5*sgn(F) - 80*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 160*I*pi^3*b^5*d^5*log(abs(F))^2*sgn(F)
+ 160*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) + 80*I*pi*b^5*d^5*log(abs(F))^4*sgn(F) - 16*I*pi^5*b^5*d^5 + 80*pi^4*b
^5*d^5*log(abs(F)) + 160*I*pi^3*b^5*d^5*log(abs(F))^2 - 160*pi^2*b^5*d^5*log(abs(F))^3 - 80*I*pi*b^5*d^5*log(a
bs(F))^4 + 32*b^5*d^5*log(abs(F))^5) - (-32*I*pi^4*b^4*d^4*f*x^4*sgn(F) - 128*pi^3*b^4*d^4*f*x^4*log(abs(F))*s
gn(F) + 192*I*pi^2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) + 128*pi*b^4*d^4*f*x^4*log(abs(F))^3*sgn(F) + 32*I*pi^4*
b^4*d^4*f*x^4 + 128*pi^3*b^4*d^4*f*x^4*log(abs(F)) - 192*I*pi^2*b^4*d^4*f*x^4*log(abs(F))^2 - 128*pi*b^4*d^4*f
*x^4*log(abs(F))^3 + 64*I*b^4*d^4*f*x^4*log(abs(F))^4 + 128*pi^3*b^3*d^3*f*x^3*sgn(F) - 384*I*pi^2*b^3*d^3*f*x
^3*log(abs(F))*sgn(F) - 384*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) - 128*pi^3*b^3*d^3*f*x^3 + 384*I*pi^2*b^3*d^
3*f*x^3*log(abs(F)) + 384*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 256*I*b^3*d^3*f*x^3*log(abs(F))^3 + 384*I*pi^2*b^2*
d^2*f*x^2*sgn(F) + 768*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) - 384*I*pi^2*b^2*d^2*f*x^2 - 768*pi*b^2*d^2*f*x^2*l
og(abs(F)) + 768*I*b^2*d^2*f*x^2*log(abs(F))^2 - 768*pi*b*d*f*x*sgn(F) + 768*pi*b*d*f*x - 1536*I*b*d*f*x*log(a
bs(F)) + 1536*I*f)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2*I*pi*b*d*x - 1/2*I*pi*b*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*
a*sgn(F) + 1/2*I*pi*a)/(-16*I*pi^5*b^5*d^5*sgn(F) - 80*pi^4*b^5*d^5*log(abs(F))*sgn(F) + 160*I*pi^3*b^5*d^5*lo
g(abs(F))^2*sgn(F) + 160*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 80*I*pi*b^5*d^5*log(abs(F))^4*sgn(F) + 16*I*pi^5*
b^5*d^5 + 80*pi^4*b^5*d^5*log(abs(F)) - 160*I*pi^3*b^5*d^5*log(abs(F))^2 - 160*pi^2*b^5*d^5*log(abs(F))^3 + 80
*I*pi*b^5*d^5*log(abs(F))^4 + 32*b^5*d^5*log(abs(F))^5))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)
) + 1) - (((5*pi^4*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3*sgn(F) - 5*pi^4*
b^5*d^5*f^2*x^5*log(abs(F)) + 10*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3 - 2*b^5*d^5*f^2*x^5*log(abs(F))^5 - 5*pi^4
*b^4*d^4*f^2*x^4*sgn(F) + 30*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) + 5*pi^4*b^4*d^4*f^2*x^4 - 30*pi^2*b^4*
d^4*f^2*x^4*log(abs(F))^2 + 10*b^4*d^4*f^2*x^4*log(abs(F))^4 - 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 60
*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 40*b^3*d^3*f^2*x^3*log(abs(F))^3 + 60*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 60*pi^
2*b^2*d^2*f^2*x^2 + 120*b^2*d^2*f^2*x^2*log(abs(F))^2 - 240*b*d*f^2*x*log(abs(F)) + 240*f^2)*(pi^6*b^6*d^6*sgn
(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6
*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)/((pi^6*b^6*d^6*sgn(F) - 15*pi^4*
b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6*d^6*log(abs(F
))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*
pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b
^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(F))^5)^2) - 2*(pi^5*b^5*d^5*f^2*x^5*sgn(F) - 10*pi^3*b^5*d^5*f^2*x
^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*f^2*x^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5*f^2*x^5 + 10*pi^3*b^5*d^5*f
^2*x^5*log(abs(F))^2 - 5*pi*b^5*d^5*f^2*x^5*log(abs(F))^4 + 20*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - 20*pi
*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - 20*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + 20*pi*b^4*d^4*f^2*x^4*log(abs(F)
)^3 - 20*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 60*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) + 20*pi^3*b^3*d^3*f^2*x^3 -
60*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 120*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 120*pi*b^2*d^2*f^2*x^2*log(a
bs(F)) + 120*pi*b*d*f^2*x*sgn(F) - 120*pi*b*d*f^2*x)*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(
abs(F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F)
)^3 - 3*pi*b^6*d^6*log(abs(F))^5)/((pi^6*b^6*d^6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d
^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6
*d^6*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6
*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(
F))^5)^2))*cos(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi
*a) - ((pi^5*b^5*d^5*f^2*x^5*sgn(F) - 10*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*f^2*x^5*log(
abs(F))^4*sgn(F) - pi^5*b^5*d^5*f^2*x^5 + 10*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2 - 5*pi*b^5*d^5*f^2*x^5*log(abs
(F))^4 + 20*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - 20*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - 20*pi^3*b^4
*d^4*f^2*x^4*log(abs(F)) + 20*pi*b^4*d^4*f^2*x^4*log(abs(F))^3 - 20*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 60*pi*b^3*d^
3*f^2*x^3*log(abs(F))^2*sgn(F) + 20*pi^3*b^3*d^3*f^2*x^3 - 60*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 120*pi*b^2*d^
2*f^2*x^2*log(abs(F))*sgn(F) + 120*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 120*pi*b*d*f^2*x*sgn(F) - 120*pi*b*d*f^2*x
)*(pi^6*b^6*d^6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^
6*d^6 + 15*pi^4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)/((pi^6*b^6*d^
6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^
4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(a
bs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(
abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(F))^5)^2) + 2*(5*pi^4*b^5*d^5*f^2*x^5*log(abs(F
))*sgn(F) - 10*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*f^2*x^5*log(abs(F)) + 10*pi^2*b^5*d^
5*f^2*x^5*log(abs(F))^3 - 2*b^5*d^5*f^2*x^5*log(abs(F))^5 - 5*pi^4*b^4*d^4*f^2*x^4*sgn(F) + 30*pi^2*b^4*d^4*f^
2*x^4*log(abs(F))^2*sgn(F) + 5*pi^4*b^4*d^4*f^2*x^4 - 30*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 10*b^4*d^4*f^2*x
^4*log(abs(F))^4 - 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 40*b^3*d
^3*f^2*x^3*log(abs(F))^3 + 60*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 60*pi^2*b^2*d^2*f^2*x^2 + 120*b^2*d^2*f^2*x^2*log(
abs(F))^2 - 240*b*d*f^2*x*log(abs(F)) + 240*f^2)*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(
F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3
- 3*pi*b^6*d^6*log(abs(F))^5)/((pi^6*b^6*d^6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*l
og(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6
*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6*d^6
*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(F))^
5)^2))*sin(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi*a))
*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F))) - 1/2*I*((32*pi^5*b^5*d^5*f^2*x^5*sgn(F) + 160*I*pi^4
*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 320*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2*sgn(F) - 320*I*pi^2*b^5*d^5*f^2*x
^5*log(abs(F))^3*sgn(F) + 160*pi*b^5*d^5*f^2*x^5*log(abs(F))^4*sgn(F) - 32*pi^5*b^5*d^5*f^2*x^5 - 160*I*pi^4*b
^5*d^5*f^2*x^5*log(abs(F)) + 320*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2 + 320*I*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3
- 160*pi*b^5*d^5*f^2*x^5*log(abs(F))^4 - 64*I*b^5*d^5*f^2*x^5*log(abs(F))^5 - 160*I*pi^4*b^4*d^4*f^2*x^4*sgn(
F) + 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) + 960*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - 640*pi*b^
4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) + 160*I*pi^4*b^4*d^4*f^2*x^4 - 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) - 960*I
*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 640*pi*b^4*d^4*f^2*x^4*log(abs(F))^3 + 320*I*b^4*d^4*f^2*x^4*log(abs(F))
^4 - 640*pi^3*b^3*d^3*f^2*x^3*sgn(F) - 1920*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 1920*pi*b^3*d^3*f^2*x^
3*log(abs(F))^2*sgn(F) + 640*pi^3*b^3*d^3*f^2*x^3 + 1920*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 1920*pi*b^3*d^3*
f^2*x^3*log(abs(F))^2 - 1280*I*b^3*d^3*f^2*x^3*log(abs(F))^3 + 1920*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 3840*pi*b^
2*d^2*f^2*x^2*log(abs(F))*sgn(F) - 1920*I*pi^2*b^2*d^2*f^2*x^2 + 3840*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 3840*I*
b^2*d^2*f^2*x^2*log(abs(F))^2 + 3840*pi*b*d*f^2*x*sgn(F) - 3840*pi*b*d*f^2*x - 7680*I*b*d*f^2*x*log(abs(F)) +
7680*I*f^2)*e^(1/2*I*pi*b*d*x*sgn(F) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F)
- 1/2*I*pi*a)/(32*pi^6*b^6*d^6*sgn(F) + 192*I*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 480*pi^4*b^6*d^6*log(abs(F))^
2*sgn(F) - 640*I*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 480*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) + 192*I*pi*b^6*d^6*
log(abs(F))^5*sgn(F) - 32*pi^6*b^6*d^6 - 192*I*pi^5*b^6*d^6*log(abs(F)) + 480*pi^4*b^6*d^6*log(abs(F))^2 + 640
*I*pi^3*b^6*d^6*log(abs(F))^3 - 480*pi^2*b^6*d^6*log(abs(F))^4 - 192*I*pi*b^6*d^6*log(abs(F))^5 + 64*b^6*d^6*l
og(abs(F))^6) + (32*pi^5*b^5*d^5*f^2*x^5*sgn(F) - 160*I*pi^4*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 320*pi^3*b^5
*d^5*f^2*x^5*log(abs(F))^2*sgn(F) + 320*I*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3*sgn(F) + 160*pi*b^5*d^5*f^2*x^5*l
og(abs(F))^4*sgn(F) - 32*pi^5*b^5*d^5*f^2*x^5 + 160*I*pi^4*b^5*d^5*f^2*x^5*log(abs(F)) + 320*pi^3*b^5*d^5*f^2*
x^5*log(abs(F))^2 - 320*I*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3 - 160*pi*b^5*d^5*f^2*x^5*log(abs(F))^4 + 64*I*b^5
*d^5*f^2*x^5*log(abs(F))^5 + 160*I*pi^4*b^4*d^4*f^2*x^4*sgn(F) + 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) -
960*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - 640*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - 160*I*pi^4*b^
4*d^4*f^2*x^4 - 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + 960*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 640*pi*b^4*d
^4*f^2*x^4*log(abs(F))^3 - 320*I*b^4*d^4*f^2*x^4*log(abs(F))^4 - 640*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 1920*I*pi^2
*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 1920*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) + 640*pi^3*b^3*d^3*f^2*x^3
- 1920*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 1920*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 1280*I*b^3*d^3*f^2*x^3*log
(abs(F))^3 - 1920*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 3840*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 1920*I*pi^2*b^2
*d^2*f^2*x^2 + 3840*pi*b^2*d^2*f^2*x^2*log(abs(F)) - 3840*I*b^2*d^2*f^2*x^2*log(abs(F))^2 + 3840*pi*b*d*f^2*x*
sgn(F) - 3840*pi*b*d*f^2*x + 7680*I*b*d*f^2*x*log(abs(F)) - 7680*I*f^2)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2*I*pi*b
*d*x - 1/2*I*pi*b*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*a*sgn(F) + 1/2*I*pi*a)/(32*pi^6*b^6*d^6*sgn(F) - 192*I*pi
^5*b^6*d^6*log(abs(F))*sgn(F) - 480*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 640*I*pi^3*b^6*d^6*log(abs(F))^3*sgn(F
) + 480*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - 192*I*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 32*pi^6*b^6*d^6 + 192*I*pi
^5*b^6*d^6*log(abs(F)) + 480*pi^4*b^6*d^6*log(abs(F))^2 - 640*I*pi^3*b^6*d^6*log(abs(F))^3 - 480*pi^2*b^6*d^6*
log(abs(F))^4 + 192*I*pi*b^6*d^6*log(abs(F))^5 + 64*b^6*d^6*log(abs(F))^6))*e^(b*d*x*log(abs(F)) + b*c*log(abs
(F)) + a*log(abs(F)))

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maple [A]  time = 0.01, size = 250, normalized size = 0.60 \[ \frac {\left (b^{5} d^{5} f^{2} x^{5} \ln \relax (F )^{5}+2 b^{5} d^{5} e f \,x^{4} \ln \relax (F )^{5}+b^{5} d^{5} e^{2} x^{3} \ln \relax (F )^{5}-5 b^{4} d^{4} f^{2} x^{4} \ln \relax (F )^{4}-8 b^{4} d^{4} e f \,x^{3} \ln \relax (F )^{4}-3 b^{4} d^{4} e^{2} x^{2} \ln \relax (F )^{4}+20 b^{3} d^{3} f^{2} x^{3} \ln \relax (F )^{3}+24 b^{3} d^{3} e f \,x^{2} \ln \relax (F )^{3}+6 b^{3} d^{3} e^{2} x \ln \relax (F )^{3}-60 b^{2} d^{2} f^{2} x^{2} \ln \relax (F )^{2}-48 b^{2} d^{2} e f x \ln \relax (F )^{2}-6 b^{2} d^{2} e^{2} \ln \relax (F )^{2}+120 b d \,f^{2} x \ln \relax (F )+48 b d e f \ln \relax (F )-120 f^{2}\right ) F^{b d x +b c +a}}{b^{6} d^{6} \ln \relax (F )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(f^2*x^5*ln(F)^5*b^5*d^5+2*ln(F)^5*b^5*d^5*e*f*x^4+ln(F)^5*b^5*d^5*e^2*x^3-5*ln(F)^4*b^4*d^4*f^2*x^4-8*ln(F)^4
*b^4*d^4*e*f*x^3-3*ln(F)^4*b^4*d^4*e^2*x^2+20*ln(F)^3*b^3*d^3*f^2*x^3+24*ln(F)^3*b^3*d^3*e*f*x^2+6*ln(F)^3*b^3
*d^3*e^2*x-60*ln(F)^2*b^2*d^2*f^2*x^2-48*ln(F)^2*b^2*d^2*e*f*x-6*ln(F)^2*b^2*d^2*e^2+120*ln(F)*b*d*f^2*x+48*e*
f*ln(F)*b*d-120*f^2)*F^(b*d*x+b*c+a)/ln(F)^6/b^6/d^6

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maxima [A]  time = 0.80, size = 328, normalized size = 0.79 \[ \frac {{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \relax (F)^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} + 6 \, F^{b c + a} b d x \log \relax (F) - 6 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{4} d^{4} \log \relax (F)^{4}} + \frac {2 \, {\left (F^{b c + a} b^{4} d^{4} x^{4} \log \relax (F)^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \relax (F)^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} - 24 \, F^{b c + a} b d x \log \relax (F) + 24 \, F^{b c + a}\right )} F^{b d x} e f}{b^{5} d^{5} \log \relax (F)^{5}} + \frac {{\left (F^{b c + a} b^{5} d^{5} x^{5} \log \relax (F)^{5} - 5 \, F^{b c + a} b^{4} d^{4} x^{4} \log \relax (F)^{4} + 20 \, F^{b c + a} b^{3} d^{3} x^{3} \log \relax (F)^{3} - 60 \, F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} + 120 \, F^{b c + a} b d x \log \relax (F) - 120 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{6} d^{6} \log \relax (F)^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(b*c + a)*b^3*d^3*x^3*log(F)^3 - 3*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 + 6*F^(b*c + a)*b*d*x*log(F) - 6*F^(b*c
+ a))*F^(b*d*x)*e^2/(b^4*d^4*log(F)^4) + 2*(F^(b*c + a)*b^4*d^4*x^4*log(F)^4 - 4*F^(b*c + a)*b^3*d^3*x^3*log(
F)^3 + 12*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 - 24*F^(b*c + a)*b*d*x*log(F) + 24*F^(b*c + a))*F^(b*d*x)*e*f/(b^5*
d^5*log(F)^5) + (F^(b*c + a)*b^5*d^5*x^5*log(F)^5 - 5*F^(b*c + a)*b^4*d^4*x^4*log(F)^4 + 20*F^(b*c + a)*b^3*d^
3*x^3*log(F)^3 - 60*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 + 120*F^(b*c + a)*b*d*x*log(F) - 120*F^(b*c + a))*F^(b*d*
x)*f^2/(b^6*d^6*log(F)^6)

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mupad [B]  time = 3.65, size = 249, normalized size = 0.60 \[ \frac {F^{a+b\,c+b\,d\,x}\,\left (b^5\,d^5\,e^2\,x^3\,{\ln \relax (F)}^5+2\,b^5\,d^5\,e\,f\,x^4\,{\ln \relax (F)}^5+b^5\,d^5\,f^2\,x^5\,{\ln \relax (F)}^5-3\,b^4\,d^4\,e^2\,x^2\,{\ln \relax (F)}^4-8\,b^4\,d^4\,e\,f\,x^3\,{\ln \relax (F)}^4-5\,b^4\,d^4\,f^2\,x^4\,{\ln \relax (F)}^4+6\,b^3\,d^3\,e^2\,x\,{\ln \relax (F)}^3+24\,b^3\,d^3\,e\,f\,x^2\,{\ln \relax (F)}^3+20\,b^3\,d^3\,f^2\,x^3\,{\ln \relax (F)}^3-6\,b^2\,d^2\,e^2\,{\ln \relax (F)}^2-48\,b^2\,d^2\,e\,f\,x\,{\ln \relax (F)}^2-60\,b^2\,d^2\,f^2\,x^2\,{\ln \relax (F)}^2+48\,b\,d\,e\,f\,\ln \relax (F)+120\,b\,d\,f^2\,x\,\ln \relax (F)-120\,f^2\right )}{b^6\,d^6\,{\ln \relax (F)}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(a + b*c + b*d*x)*(120*b*d*f^2*x*log(F) - 6*b^2*d^2*e^2*log(F)^2 - 120*f^2 + 6*b^3*d^3*e^2*x*log(F)^3 - 3*b
^4*d^4*e^2*x^2*log(F)^4 + b^5*d^5*e^2*x^3*log(F)^5 - 60*b^2*d^2*f^2*x^2*log(F)^2 + 20*b^3*d^3*f^2*x^3*log(F)^3
- 5*b^4*d^4*f^2*x^4*log(F)^4 + b^5*d^5*f^2*x^5*log(F)^5 + 48*b*d*e*f*log(F) - 48*b^2*d^2*e*f*x*log(F)^2 + 24*
b^3*d^3*e*f*x^2*log(F)^3 - 8*b^4*d^4*e*f*x^3*log(F)^4 + 2*b^5*d^5*e*f*x^4*log(F)^5))/(b^6*d^6*log(F)^6)

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sympy [A]  time = 0.28, size = 323, normalized size = 0.78 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{5} d^{5} e^{2} x^{3} \log {\relax (F )}^{5} + 2 b^{5} d^{5} e f x^{4} \log {\relax (F )}^{5} + b^{5} d^{5} f^{2} x^{5} \log {\relax (F )}^{5} - 3 b^{4} d^{4} e^{2} x^{2} \log {\relax (F )}^{4} - 8 b^{4} d^{4} e f x^{3} \log {\relax (F )}^{4} - 5 b^{4} d^{4} f^{2} x^{4} \log {\relax (F )}^{4} + 6 b^{3} d^{3} e^{2} x \log {\relax (F )}^{3} + 24 b^{3} d^{3} e f x^{2} \log {\relax (F )}^{3} + 20 b^{3} d^{3} f^{2} x^{3} \log {\relax (F )}^{3} - 6 b^{2} d^{2} e^{2} \log {\relax (F )}^{2} - 48 b^{2} d^{2} e f x \log {\relax (F )}^{2} - 60 b^{2} d^{2} f^{2} x^{2} \log {\relax (F )}^{2} + 48 b d e f \log {\relax (F )} + 120 b d f^{2} x \log {\relax (F )} - 120 f^{2}\right )}{b^{6} d^{6} \log {\relax (F )}^{6}} & \text {for}\: b^{6} d^{6} \log {\relax (F )}^{6} \neq 0 \\\frac {e^{2} x^{4}}{4} + \frac {2 e f x^{5}}{5} + \frac {f^{2} x^{6}}{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((F**(a + b*(c + d*x))*(b**5*d**5*e**2*x**3*log(F)**5 + 2*b**5*d**5*e*f*x**4*log(F)**5 + b**5*d**5*f*
*2*x**5*log(F)**5 - 3*b**4*d**4*e**2*x**2*log(F)**4 - 8*b**4*d**4*e*f*x**3*log(F)**4 - 5*b**4*d**4*f**2*x**4*l
og(F)**4 + 6*b**3*d**3*e**2*x*log(F)**3 + 24*b**3*d**3*e*f*x**2*log(F)**3 + 20*b**3*d**3*f**2*x**3*log(F)**3 -
6*b**2*d**2*e**2*log(F)**2 - 48*b**2*d**2*e*f*x*log(F)**2 - 60*b**2*d**2*f**2*x**2*log(F)**2 + 48*b*d*e*f*log
(F) + 120*b*d*f**2*x*log(F) - 120*f**2)/(b**6*d**6*log(F)**6), Ne(b**6*d**6*log(F)**6, 0)), (e**2*x**4/4 + 2*e
*f*x**5/5 + f**2*x**6/6, True))

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