\(\int (a+b (F^{g (e+f x)})^n) (c+d x)^2 \, dx\) [26]
Optimal result
Integrand size = 23, antiderivative size = 115 \[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=\frac {a (c+d x)^3}{3 d}+\frac {2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}
\]
[Out]
1/3*a*(d*x+c)^3/d+2*b*d^2*(F^(f*g*x+e*g))^n/f^3/g^3/n^3/ln(F)^3-2*b*d*(F^(f*g*x+e*g))^n*(d*x+c)/f^2/g^2/n^2/ln
(F)^2+b*(F^(f*g*x+e*g))^n*(d*x+c)^2/f/g/n/ln(F)
Rubi [A] (verified)
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2214, 2207, 2225}
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=\frac {a (c+d x)^3}{3 d}-\frac {2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}
\]
[In]
Int[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^2,x]
[Out]
(a*(c + d*x)^3)/(3*d) + (2*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log[F]^3) - (2*b*d*(F^(e*g + f*g*x))^n*(c +
d*x))/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F])
Rule 2207
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] && !TrueQ[$UseGamma]
Rule 2214
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 2225
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*}
\text {integral}& = \int \left (a (c+d x)^2+b \left (F^{e g+f g x}\right )^n (c+d x)^2\right ) \, dx \\ & = \frac {a (c+d x)^3}{3 d}+b \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx \\ & = \frac {a (c+d x)^3}{3 d}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}-\frac {(2 b d) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f g n \log (F)} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {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 {b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {\left (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 {a (c+d x)^3}{3 d}+\frac {2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {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 {b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.79
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=a c^2 x+a c d x^2+\frac {1}{3} a d^2 x^3+\frac {b \left (F^{g (e+f x)}\right )^n \left (2 d^2-2 d f g n (c+d x) \log (F)+f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{f^3 g^3 n^3 \log ^3(F)}
\]
[In]
Integrate[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^2,x]
[Out]
a*c^2*x + a*c*d*x^2 + (a*d^2*x^3)/3 + (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)
Maple [A] (verified)
Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.42
| | |
| method | result | size |
| | |
| norman |
\(a \,c^{2} x +a c d \,x^{2}+\frac {b \left (\ln \left (F \right )^{2} c^{2} f^{2} g^{2} n^{2}-2 \ln \left (F \right ) c d f g n +2 d^{2}\right ) {\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{\ln \left (F \right )^{3} f^{3} g^{3} n^{3}}+\frac {b \,d^{2} x^{2} {\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{n g f \ln \left (F \right )}+\frac {a \,d^{2} x^{3}}{3}+\frac {2 b d \left (\ln \left (F \right ) c f g n -d \right ) x \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2}}\) |
\(163\) |
| parallelrisch |
\(\frac {a \,d^{2} x^{3} \ln \left (F \right )^{3} f^{3} g^{3} n^{3}+3 a c d \,x^{2} \ln \left (F \right )^{3} f^{3} g^{3} n^{3}+3 a \,c^{2} x \ln \left (F \right )^{3} f^{3} g^{3} n^{3}+3 x^{2} \left (F^{g \left (f x +e \right )}\right )^{n} b \,d^{2} \ln \left (F \right )^{2} f^{2} g^{2} n^{2}+6 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{n} b c d \,f^{2} g^{2} n^{2}+3 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{n} b \,c^{2} f^{2} g^{2} n^{2}-6 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{n} b \,d^{2} f g n -6 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{n} b c d f g n +6 \left (F^{g \left (f x +e \right )}\right )^{n} b \,d^{2}}{3 \ln \left (F \right )^{3} f^{3} g^{3} n^{3}}\) |
\(233\) |
| | |
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[In]
int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^2,x,method=_RETURNVERBOSE)
[Out]
a*c^2*x+a*c*d*x^2+b*(ln(F)^2*c^2*f^2*g^2*n^2-2*ln(F)*c*d*f*g*n+2*d^2)/ln(F)^3/f^3/g^3/n^3*exp(n*ln(exp(g*(f*x+
e)*ln(F))))+1/n/g/f/ln(F)*b*d^2*x^2*exp(n*ln(exp(g*(f*x+e)*ln(F))))+1/3*a*d^2*x^3+2*b*d*(ln(F)*c*f*g*n-d)/ln(F
)^2/f^2/g^2/n^2*x*exp(n*ln(exp(g*(f*x+e)*ln(F))))
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.44
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=\frac {{\left (a d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a c d f^{3} g^{3} n^{3} x^{2} + 3 \, a c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \, {\left (2 \, b d^{2} + {\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x + b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b d^{2} f g n x + b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{3 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}}
\]
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^2,x, algorithm="fricas")
[Out]
1/3*((a*d^2*f^3*g^3*n^3*x^3 + 3*a*c*d*f^3*g^3*n^3*x^2 + 3*a*c^2*f^3*g^3*n^3*x)*log(F)^3 + 3*(2*b*d^2 + (b*d^2*
f^2*g^2*n^2*x^2 + 2*b*c*d*f^2*g^2*n^2*x + b*c^2*f^2*g^2*n^2)*log(F)^2 - 2*(b*d^2*f*g*n*x + 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)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (110) = 220\).
Time = 0.74 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.56
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=\begin {cases} \left (a + b\right ) \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: F = 1 \wedge f = 0 \wedge g = 0 \wedge n = 0 \\\left (a + b \left (F^{e g}\right )^{n}\right ) \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: f = 0 \\\left (a + b\right ) \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: F = 1 \vee g = 0 \vee n = 0 \\a c^{2} x + a c d x^{2} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} + \frac {2 b c d x \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {2 b c d \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {b d^{2} x^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {2 b d^{2} x \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {2 b d^{2} \left (F^{e g + f g x}\right )^{n}}{f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} & \text {otherwise} \end {cases}
\]
[In]
integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c)**2,x)
[Out]
Piecewise(((a + b)*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(F, 1) & Eq(f, 0) & Eq(g, 0) & Eq(n, 0)), ((a + b*(F**
(e*g))**n)*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(f, 0)), ((a + b)*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(F, 1)
| Eq(g, 0) | Eq(n, 0)), (a*c**2*x + a*c*d*x**2 + a*d**2*x**3/3 + b*c**2*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) +
2*b*c*d*x*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) - 2*b*c*d*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F)**2) + b
*d**2*x**2*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) - 2*b*d**2*x*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F)**2)
+ 2*b*d**2*(F**(e*g + f*g*x))**n/(f**3*g**3*n**3*log(F)**3), True))
Maxima [A] (verification not implemented)
none
Time = 0.19 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.49
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=\frac {1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + \frac {F^{f g n x + e g n} b c^{2}}{f g n \log \left (F\right )} + \frac {2 \, {\left (F^{e g n} f g n x \log \left (F\right ) - F^{e g n}\right )} F^{f g n x} b c d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {{\left (F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{e g n} f g n x \log \left (F\right ) + 2 \, F^{e g n}\right )} F^{f g n x} b d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}}
\]
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^2,x, algorithm="maxima")
[Out]
1/3*a*d^2*x^3 + a*c*d*x^2 + a*c^2*x + F^(f*g*n*x + e*g*n)*b*c^2/(f*g*n*log(F)) + 2*(F^(e*g*n)*f*g*n*x*log(F) -
F^(e*g*n))*F^(f*g*n*x)*b*c*d/(f^2*g^2*n^2*log(F)^2) + (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*n*x)*b*d^2/(f^3*g^3*n^3*log(F)^3)
Giac [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 2716, normalized size of antiderivative = 23.62
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx=\text {Too large to display}
\]
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^2,x, algorithm="giac")
[Out]
1/3*a*d^2*x^3 + a*c*d*x^2 + a*c^2*x - ((2*(pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*b*d^2*f^2*g^2*n^2*
x^2*log(abs(F)) + 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*b*c^
2*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*b*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b*d^2*f*g*n*x*sgn(F) + pi*b*d^2*f*g*n
*x - pi*b*c*d*f*g*n*sgn(F) + pi*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*p
i^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2) - (pi^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*b*d^
2*f^2*g^2*n^2*x^2 + 2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*b*c*d*f
^2*g^2*n^2*x + 4*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*b*c^2*f^2*g^2*n^2 +
2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*b*d^2*f*g*n*x*log(abs(F)) - 4*b*c*d*f*g*n*log(abs(F)) + 4*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(abs(F))^3)^2))*cos(
-1/2*pi*f*g*n*x*sgn(F) + 1/2*pi*f*g*n*x - 1/2*pi*e*g*n*sgn(F) + 1/2*pi*e*g*n) - ((pi^2*b*d^2*f^2*g^2*n^2*x^2*s
gn(F) - pi^2*b*d^2*f^2*g^2*n^2*x^2 + 2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F)
- 2*pi^2*b*c*d*f^2*g^2*n^2*x + 4*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*b*c
^2*f^2*g^2*n^2 + 2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*b*d^2*f*g*n*x*log(abs(F)) - 4*b*c*d*f*g*n*log(abs(F)) +
4*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*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*b*d^2*f^2*g^2*n^2*x^2*log(abs
(F)) + 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*b*c^2*f^2*g^2*n
^2*log(abs(F))*sgn(F) - pi*b*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b*d^2*f*g*n*x*sgn(F) + pi*b*d^2*f*g*n*x - pi*b*c
*d*f*g*n*sgn(F) + pi*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*e*g*n*sgn(F) + 1/2*p
i*e*g*n))*e^(f*g*n*x*log(abs(F)) + e*g*n*log(abs(F))) - 2*I*((-I*pi^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) + 2*pi*b*d^
2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) + I*pi^2*b*d^2*f^2*g^2*n^2*x^2 - 2*pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) -
2*I*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 - 2*I*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F) + 4*pi*b*c*d*f^2*g^2*n^2*x*log(
abs(F))*sgn(F) + 2*I*pi^2*b*c*d*f^2*g^2*n^2*x - 4*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) - 4*I*b*c*d*f^2*g^2*n^2*x
*log(abs(F))^2 - I*pi^2*b*c^2*f^2*g^2*n^2*sgn(F) + 2*pi*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) + I*pi^2*b*c^2*f^
2*g^2*n^2 - 2*pi*b*c^2*f^2*g^2*n^2*log(abs(F)) - 2*I*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 2*pi*b*d^2*f*g*n*x*sgn(
F) + 2*pi*b*d^2*f*g*n*x + 4*I*b*d^2*f*g*n*x*log(abs(F)) - 2*pi*b*c*d*f*g*n*sgn(F) + 2*pi*b*c*d*f*g*n + 4*I*b*c
*d*f*g*n*log(abs(F)) - 4*I*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*e*g*n*sgn(F) - 1/2*
I*pi*e*g*n)/(-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(a
bs(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) - (-I*pi^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - 2*pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))
*sgn(F) + I*pi^2*b*d^2*f^2*g^2*n^2*x^2 + 2*pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) - 2*I*b*d^2*f^2*g^2*n^2*x^2*lo
g(abs(F))^2 - 2*I*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F) - 4*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) + 2*I*pi^2*b*c
*d*f^2*g^2*n^2*x + 4*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) - 4*I*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 - I*pi^2*b*c^2
*f^2*g^2*n^2*sgn(F) - 2*pi*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) + I*pi^2*b*c^2*f^2*g^2*n^2 + 2*pi*b*c^2*f^2*g^
2*n^2*log(abs(F)) - 2*I*b*c^2*f^2*g^2*n^2*log(abs(F))^2 + 2*pi*b*d^2*f*g*n*x*sgn(F) - 2*pi*b*d^2*f*g*n*x + 4*I
*b*d^2*f*g*n*x*log(abs(F)) + 2*pi*b*c*d*f*g*n*sgn(F) - 2*pi*b*c*d*f*g*n + 4*I*b*c*d*f*g*n*log(abs(F)) - 4*I*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*e*g*n*sgn(F) + 1/2*I*pi*e*g*n)/(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)) + e*g*n*log(abs(F)))
Mupad [B] (verification not implemented)
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.17
\[
\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^2 \, dx={\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {b\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b\,d^2\,x^2}{f\,g\,n\,\ln \left (F\right )}-\frac {2\,b\,d\,x\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {a\,d^2\,x^3}{3}+a\,c^2\,x+a\,c\,d\,x^2
\]
[In]
int((a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^2,x)
[Out]
(F^(f*g*x)*F^(e*g))^n*((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) + (b
*d^2*x^2)/(f*g*n*log(F)) - (2*b*d*x*(d - c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F)^2)) + (a*d^2*x^3)/3 + a*c^2*x +
a*c*d*x^2