3.33 \(\int (a+b (F^{g (e+f x)})^n)^2 (c+d x)^2 \, dx\)
Optimal. Leaf size=239 \[ \frac {a^2 (c+d x)^3}{3 d}-\frac {4 a b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)} \]
[Out]
1/3*a^2*(d*x+c)^3/d+4*a*b*d^2*(F^(f*g*x+e*g))^n/f^3/g^3/n^3/ln(F)^3+1/4*b^2*d^2*(F^(f*g*x+e*g))^(2*n)/f^3/g^3/
n^3/ln(F)^3-4*a*b*d*(F^(f*g*x+e*g))^n*(d*x+c)/f^2/g^2/n^2/ln(F)^2-1/2*b^2*d*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f^2/
g^2/n^2/ln(F)^2+2*a*b*(F^(f*g*x+e*g))^n*(d*x+c)^2/f/g/n/ln(F)+1/2*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)^2/f/g/n/ln
(F)
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 239, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used =
{2183, 2176, 2194} \[ \frac {a^2 (c+d x)^3}{3 d}-\frac {4 a b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2,x]
[Out]
(a^2*(c + d*x)^3)/(3*d) + (4*a*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log[F]^3) + (b^2*d^2*(F^(e*g + f*g*x))^
(2*n))/(4*f^3*g^3*n^3*Log[F]^3) - (4*a*b*d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) - (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*a*b*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F
]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/(2*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 )^2 (c+d x)^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b \left (F^{e g+f g x}\right )^n (c+d x)^2+b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2\right ) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+(2 a b) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx+b^2 \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2 \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}-\frac {(4 a b d) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f g n \log (F)}-\frac {\left (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 {a^2 (c+d x)^3}{3 d}-\frac {4 a 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^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 a b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {\left (4 a 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 (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 {a^2 (c+d x)^3}{3 d}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {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 {4 a 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^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 a b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.37, size = 171, normalized size = 0.72 \[ a^2 c^2 x+a^2 c d x^2+\frac {1}{3} a^2 d^2 x^3+\frac {2 a b \left (F^{g (e+f x)}\right )^n \left (f^2 g^2 n^2 \log ^2(F) (c+d x)^2-2 d f g n \log (F) (c+d x)+2 d^2\right )}{f^3 g^3 n^3 \log ^3(F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (2 f^2 g^2 n^2 \log ^2(F) (c+d x)^2-2 d f g n \log (F) (c+d x)+d^2\right )}{4 f^3 g^3 n^3 \log ^3(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2,x]
[Out]
a^2*c^2*x + a^2*c*d*x^2 + (a^2*d^2*x^3)/3 + (2*a*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) + (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)
________________________________________________________________________________________
fricas [A] time = 0.43, size = 287, normalized size = 1.20 \[ \frac {4 \, {\left (a^{2} d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{2} c d f^{3} g^{3} n^{3} x^{2} + 3 \, a^{2} c^{2} f^{3} g^{3} n^{3} x\right )} \log \relax (F)^{3} + 3 \, {\left (b^{2} d^{2} + 2 \, {\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + b^{2} c^{2} f^{2} g^{2} n^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (b^{2} d^{2} f g n x + b^{2} c d f g n\right )} \log \relax (F)\right )} F^{2 \, f g n x + 2 \, e g n} + 24 \, {\left (2 \, a b d^{2} + {\left (a b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d f^{2} g^{2} n^{2} x + a b c^{2} f^{2} g^{2} n^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (a b d^{2} f g n x + a b c d f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n}}{12 \, f^{3} g^{3} n^{3} \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x, algorithm="fricas")
[Out]
1/12*(4*(a^2*d^2*f^3*g^3*n^3*x^3 + 3*a^2*c*d*f^3*g^3*n^3*x^2 + 3*a^2*c^2*f^3*g^3*n^3*x)*log(F)^3 + 3*(b^2*d^2
+ 2*(b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*c*d*f^2*g^2*n^2*x + b^2*c^2*f^2*g^2*n^2)*log(F)^2 - 2*(b^2*d^2*f*g*n*x +
b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 24*(2*a*b*d^2 + (a*b*d^2*f^2*g^2*n^2*x^2 + 2*a*b*c*d*f^2*g^2*
n^2*x + a*b*c^2*f^2*g^2*n^2)*log(F)^2 - 2*(a*b*d^2*f*g*n*x + a*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 = 1.23, size = 5699, normalized size = 23.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x, algorithm="giac")
[Out]
1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + a^2*c^2*x - 1/2*(((2*pi*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 2*pi*b^2*
d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*pi*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 4*pi*b^2*c*d*f^2*g^2*n^2*x*l
og(abs(F)) + 2*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b^2*d^2*f
*g*n*x*sgn(F) + pi*b^2*d^2*f*g*n*x - pi*b^2*c*d*f*g*n*sgn(F) + pi*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*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) - (pi^2*b^2*
d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*b
^2*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*b^2*c*d*f^2*g^2*n^2*x + 4*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*b^2*
c^2*f^2*g^2*n^2*sgn(F) - pi^2*b^2*c^2*f^2*g^2*n^2 + 2*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2 - 2*b^2*d^2*f*g*n*x*lo
g(abs(F)) - 2*b^2*c*d*f*g*n*log(abs(F)) + 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*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(-pi*f*g*n*x*sgn(F) + pi*f*g*n*x - pi*g*n*e*sgn(F) +
pi*g*n*e) - ((pi^2*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*d^2*f^2*g^2*n^2*x^2*l
og(abs(F))^2 + 2*pi^2*b^2*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*b^2*c*d*f^2*g^2*n^2*x + 4*b^2*c*d*f^2*g^2*n^2*x*lo
g(abs(F))^2 + pi^2*b^2*c^2*f^2*g^2*n^2*sgn(F) - pi^2*b^2*c^2*f^2*g^2*n^2 + 2*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2
- 2*b^2*d^2*f*g*n*x*log(abs(F)) - 2*b^2*c*d*f*g*n*log(abs(F)) + 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) - 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^2*d^2*f^2
*g^2*n^2*x^2*log(abs(F))*sgn(F) - 2*pi*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*pi*b^2*c*d*f^2*g^2*n^2*x*log(ab
s(F))*sgn(F) - 4*pi*b^2*c*d*f^2*g^2*n^2*x*log(abs(F)) + 2*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*pi*b^2
*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b^2*d^2*f*g*n*x*sgn(F) + pi*b^2*d^2*f*g*n*x - pi*b^2*c*d*f*g*n*sgn(F) + pi*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*lo
g(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*((2*I*pi^2*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) - 4*pi*b^2*d^2*f^2*g^2*n^2*x^2*log(abs
(F))*sgn(F) - 2*I*pi^2*b^2*d^2*f^2*g^2*n^2*x^2 + 4*pi*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*I*b^2*d^2*f^2*g^
2*n^2*x^2*log(abs(F))^2 + 4*I*pi^2*b^2*c*d*f^2*g^2*n^2*x*sgn(F) - 8*pi*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F
) - 4*I*pi^2*b^2*c*d*f^2*g^2*n^2*x + 8*pi*b^2*c*d*f^2*g^2*n^2*x*log(abs(F)) + 8*I*b^2*c*d*f^2*g^2*n^2*x*log(ab
s(F))^2 + 2*I*pi^2*b^2*c^2*f^2*g^2*n^2*sgn(F) - 4*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*I*pi^2*b^2*c^2
*f^2*g^2*n^2 + 4*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F)) + 4*I*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2 + 2*pi*b^2*d^2*f*g
*n*x*sgn(F) - 2*pi*b^2*d^2*f*g*n*x - 4*I*b^2*d^2*f*g*n*x*log(abs(F)) + 2*pi*b^2*c*d*f*g*n*sgn(F) - 2*pi*b^2*c*
d*f*g*n - 4*I*b^2*c*d*f*g*n*log(abs(F)) + 2*I*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) - (2*I*pi^2*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) + 4*pi*b^2*d^2*f^2*g^2*n^2*x^2*l
og(abs(F))*sgn(F) - 2*I*pi^2*b^2*d^2*f^2*g^2*n^2*x^2 - 4*pi*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*I*b^2*d^2*
f^2*g^2*n^2*x^2*log(abs(F))^2 + 4*I*pi^2*b^2*c*d*f^2*g^2*n^2*x*sgn(F) + 8*pi*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))
*sgn(F) - 4*I*pi^2*b^2*c*d*f^2*g^2*n^2*x - 8*pi*b^2*c*d*f^2*g^2*n^2*x*log(abs(F)) + 8*I*b^2*c*d*f^2*g^2*n^2*x*
log(abs(F))^2 + 2*I*pi^2*b^2*c^2*f^2*g^2*n^2*sgn(F) + 4*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*I*pi^2*b
^2*c^2*f^2*g^2*n^2 - 4*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F)) + 4*I*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2 - 2*pi*b^2*d
^2*f*g*n*x*sgn(F) + 2*pi*b^2*d^2*f*g*n*x - 4*I*b^2*d^2*f*g*n*x*log(abs(F)) - 2*pi*b^2*c*d*f*g*n*sgn(F) + 2*pi*
b^2*c*d*f*g*n - 4*I*b^2*c*d*f*g*n*log(abs(F)) + 2*I*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))) - 2*((2*(pi*a*b*d^2*
f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 2*pi*a*b*c*d*f^2*g^2*n^2*x*log(a
bs(F))*sgn(F) - 2*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*a*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*a*b*c^
2*f^2*g^2*n^2*log(abs(F)) - pi*a*b*d^2*f*g*n*x*sgn(F) + pi*a*b*d^2*f*g*n*x - pi*a*b*c*d*f*g*n*sgn(F) + pi*a*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*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*a*b*d^2*f^2*g^2*n^2*x^2 + 2*a*b*d^2*
f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*a*b*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*a*b*c*d*f^2*g^2*n^2*x + 4*a*b*c*d
*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*a*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*a*b*c^2*f^2*g^2*n^2 + 2*a*b*c^2*f^2*g^2*
n^2*log(abs(F))^2 - 4*a*b*d^2*f*g*n*x*log(abs(F)) - 4*a*b*c*d*f*g*n*log(abs(F)) + 4*a*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*g*n*e*sgn(F) + 1/2*pi*g*n*e) - ((pi^2*a*b*d^2*f^2*g^2*n^2*x^2*sgn(F) -
pi^2*a*b*d^2*f^2*g^2*n^2*x^2 + 2*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*a*b*c*d*f^2*g^2*n^2*x*sgn(F) -
2*pi^2*a*b*c*d*f^2*g^2*n^2*x + 4*a*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*a*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2
*a*b*c^2*f^2*g^2*n^2 + 2*a*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*a*b*d^2*f*g*n*x*log(abs(F)) - 4*a*b*c*d*f*g*n*l
og(abs(F)) + 4*a*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(a
bs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2) + 2*(pi*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*a*b*d^2*f^2*g
^2*n^2*x^2*log(abs(F)) + 2*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 2*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F)
) + pi*a*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*a*b*c^2*f^2*g^2*n^2*log(abs(F)) - pi*a*b*d^2*f*g*n*x*sgn(F)
+ pi*a*b*d^2*f*g*n*x - pi*a*b*c*d*f*g*n*sgn(F) + pi*a*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*((8*I*pi^2*a*b
*d^2*f^2*g^2*n^2*x^2*sgn(F) - 16*pi*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 8*I*pi^2*a*b*d^2*f^2*g^2*n^2*
x^2 + 16*pi*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 16*I*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 16*I*pi^2*a*b*c
*d*f^2*g^2*n^2*x*sgn(F) - 32*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 16*I*pi^2*a*b*c*d*f^2*g^2*n^2*x + 3
2*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + 32*I*a*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + 8*I*pi^2*a*b*c^2*f^2*g^2*n
^2*sgn(F) - 16*pi*a*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 8*I*pi^2*a*b*c^2*f^2*g^2*n^2 + 16*pi*a*b*c^2*f^2*g^
2*n^2*log(abs(F)) + 16*I*a*b*c^2*f^2*g^2*n^2*log(abs(F))^2 + 16*pi*a*b*d^2*f*g*n*x*sgn(F) - 16*pi*a*b*d^2*f*g*
n*x - 32*I*a*b*d^2*f*g*n*x*log(abs(F)) + 16*pi*a*b*c*d*f*g*n*sgn(F) - 16*pi*a*b*c*d*f*g*n - 32*I*a*b*c*d*f*g*n
*log(abs(F)) + 32*I*a*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) - (8*I*pi^2*a*b*d^2*f^2*g^2*n^2*x^2*sgn(F) + 16*pi*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)
)*sgn(F) - 8*I*pi^2*a*b*d^2*f^2*g^2*n^2*x^2 - 16*pi*a*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 16*I*a*b*d^2*f^2*g^2
*n^2*x^2*log(abs(F))^2 + 16*I*pi^2*a*b*c*d*f^2*g^2*n^2*x*sgn(F) + 32*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(
F) - 16*I*pi^2*a*b*c*d*f^2*g^2*n^2*x - 32*pi*a*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + 32*I*a*b*c*d*f^2*g^2*n^2*x*lo
g(abs(F))^2 + 8*I*pi^2*a*b*c^2*f^2*g^2*n^2*sgn(F) + 16*pi*a*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 8*I*pi^2*a*
b*c^2*f^2*g^2*n^2 - 16*pi*a*b*c^2*f^2*g^2*n^2*log(abs(F)) + 16*I*a*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 16*pi*a*b
*d^2*f*g*n*x*sgn(F) + 16*pi*a*b*d^2*f*g*n*x - 32*I*a*b*d^2*f*g*n*x*log(abs(F)) - 16*pi*a*b*c*d*f*g*n*sgn(F) +
16*pi*a*b*c*d*f*g*n - 32*I*a*b*c*d*f*g*n*log(abs(F)) + 32*I*a*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 = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{2} \left (d x +c \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*(F^((f*x+e)*g))^n+a)^2*(d*x+c)^2,x)
[Out]
int((b*(F^((f*x+e)*g))^n+a)^2*(d*x+c)^2,x)
________________________________________________________________________________________
maxima [A] time = 1.61, size = 368, normalized size = 1.54 \[ \frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + a^{2} c^{2} x + \frac {2 \, {\left (F^{f g x + e g}\right )}^{n} a b c^{2}}{f g n \log \relax (F)} + \frac {{\left (F^{f g x + e g}\right )}^{2 \, n} b^{2} c^{2}}{2 \, f g n \log \relax (F)} + \frac {4 \, {\left ({\left (F^{e g}\right )}^{n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} a b c d}{f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {{\left (2 \, {\left (F^{e g}\right )}^{2 \, n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{2 \, n}\right )} {\left (F^{f g x}\right )}^{2 \, n} b^{2} c d}{2 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {2 \, {\left ({\left (F^{e g}\right )}^{n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} - 2 \, {\left (F^{e g}\right )}^{n} f g n x \log \relax (F) + 2 \, {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} a b d^{2}}{f^{3} g^{3} n^{3} \log \relax (F)^{3}} + \frac {{\left (2 \, {\left (F^{e g}\right )}^{2 \, n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} - 2 \, {\left (F^{e g}\right )}^{2 \, n} f g n x \log \relax (F) + {\left (F^{e g}\right )}^{2 \, n}\right )} {\left (F^{f g x}\right )}^{2 \, n} b^{2} d^{2}}{4 \, f^{3} g^{3} n^{3} \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x, algorithm="maxima")
[Out]
1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + a^2*c^2*x + 2*(F^(f*g*x + e*g))^n*a*b*c^2/(f*g*n*log(F)) + 1/2*(F^(f*g*x + e*g
))^(2*n)*b^2*c^2/(f*g*n*log(F)) + 4*((F^(e*g))^n*f*g*n*x*log(F) - (F^(e*g))^n)*(F^(f*g*x))^n*a*b*c*d/(f^2*g^2*
n^2*log(F)^2) + 1/2*(2*(F^(e*g))^(2*n)*f*g*n*x*log(F) - (F^(e*g))^(2*n))*(F^(f*g*x))^(2*n)*b^2*c*d/(f^2*g^2*n^
2*log(F)^2) + 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*
x))^n*a*b*d^2/(f^3*g^3*n^3*log(F)^3) + 1/4*(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)*b^2*d^2/(f^3*g^3*n^3*log(F)^3)
________________________________________________________________________________________
mupad [B] time = 3.82, size = 267, normalized size = 1.12 \[ {\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {b^2\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-2\,c\,d\,f\,g\,n\,\ln \relax (F)+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \relax (F)}^3}+\frac {b^2\,d^2\,x^2}{2\,f\,g\,n\,\ln \relax (F)}-\frac {b^2\,d\,x\,\left (d-2\,c\,f\,g\,n\,\ln \relax (F)\right )}{2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {2\,a\,b\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-2\,c\,d\,f\,g\,n\,\ln \relax (F)+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \relax (F)}^3}+\frac {2\,a\,b\,d^2\,x^2}{f\,g\,n\,\ln \relax (F)}-\frac {4\,a\,b\,d\,x\,\left (d-c\,f\,g\,n\,\ln \relax (F)\right )}{f^2\,g^2\,n^2\,{\ln \relax (F)}^2}\right )+a^2\,c^2\,x+\frac {a^2\,d^2\,x^3}{3}+a^2\,c\,d\,x^2 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2,x)
[Out]
(F^(f*g*x)*F^(e*g))^(2*n)*((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) + (b^2*d^2*x^2)/(2*f*g*n*log(F)) - (b^2*d*x*(d - 2*c*f*g*n*log(F)))/(2*f^2*g^2*n^2*log(F)^2)) + (F^(f*g*x)
*F^(e*g))^n*((2*a*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) + (2*a*b*d
^2*x^2)/(f*g*n*log(F)) - (4*a*b*d*x*(d - c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F)^2)) + a^2*c^2*x + (a^2*d^2*x^3)/
3 + a^2*c*d*x^2
________________________________________________________________________________________
sympy [A] time = 0.37, size = 439, normalized size = 1.84 \[ a^{2} c^{2} x + a^{2} c d x^{2} + \frac {a^{2} d^{2} x^{3}}{3} + \begin {cases} \frac {\left (2 b^{2} c^{2} f^{5} g^{5} n^{5} \log {\relax (F )}^{5} + 4 b^{2} c d f^{5} g^{5} n^{5} x \log {\relax (F )}^{5} - 2 b^{2} c d f^{4} g^{4} n^{4} \log {\relax (F )}^{4} + 2 b^{2} d^{2} f^{5} g^{5} n^{5} x^{2} \log {\relax (F )}^{5} - 2 b^{2} d^{2} f^{4} g^{4} n^{4} x \log {\relax (F )}^{4} + b^{2} d^{2} f^{3} g^{3} n^{3} \log {\relax (F )}^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (8 a b c^{2} f^{5} g^{5} n^{5} \log {\relax (F )}^{5} + 16 a b c d f^{5} g^{5} n^{5} x \log {\relax (F )}^{5} - 16 a b c d f^{4} g^{4} n^{4} \log {\relax (F )}^{4} + 8 a b d^{2} f^{5} g^{5} n^{5} x^{2} \log {\relax (F )}^{5} - 16 a b d^{2} f^{4} g^{4} n^{4} x \log {\relax (F )}^{4} + 16 a b d^{2} f^{3} g^{3} n^{3} \log {\relax (F )}^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{4 f^{6} g^{6} n^{6} \log {\relax (F )}^{6}} & \text {for}\: 4 f^{6} g^{6} n^{6} \log {\relax (F )}^{6} \neq 0 \\x^{3} \left (\frac {2 a b d^{2}}{3} + \frac {b^{2} d^{2}}{3}\right ) + x^{2} \left (2 a b c d + b^{2} c d\right ) + x \left (2 a b c^{2} + b^{2} c^{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F**(g*(f*x+e)))**n)**2*(d*x+c)**2,x)
[Out]
a**2*c**2*x + a**2*c*d*x**2 + a**2*d**2*x**3/3 + Piecewise((((2*b**2*c**2*f**5*g**5*n**5*log(F)**5 + 4*b**2*c*
d*f**5*g**5*n**5*x*log(F)**5 - 2*b**2*c*d*f**4*g**4*n**4*log(F)**4 + 2*b**2*d**2*f**5*g**5*n**5*x**2*log(F)**5
- 2*b**2*d**2*f**4*g**4*n**4*x*log(F)**4 + b**2*d**2*f**3*g**3*n**3*log(F)**3)*(F**(g*(e + f*x)))**(2*n) + (8
*a*b*c**2*f**5*g**5*n**5*log(F)**5 + 16*a*b*c*d*f**5*g**5*n**5*x*log(F)**5 - 16*a*b*c*d*f**4*g**4*n**4*log(F)*
*4 + 8*a*b*d**2*f**5*g**5*n**5*x**2*log(F)**5 - 16*a*b*d**2*f**4*g**4*n**4*x*log(F)**4 + 16*a*b*d**2*f**3*g**3
*n**3*log(F)**3)*(F**(g*(e + f*x)))**n)/(4*f**6*g**6*n**6*log(F)**6), Ne(4*f**6*g**6*n**6*log(F)**6, 0)), (x**
3*(2*a*b*d**2/3 + b**2*d**2/3) + x**2*(2*a*b*c*d + b**2*c*d) + x*(2*a*b*c**2 + b**2*c**2), True))
________________________________________________________________________________________