3.41 \(\int (a+b (F^{g (e+f x)})^n)^3 (c+d x) \, dx\)
Optimal. Leaf size=236 \[ \frac {a^3 (c+d x)^2}{2 d}+\frac {3 a^2 b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {3 a^2 b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {3 a b^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac {3 a b^2 d \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n}}{9 f^2 g^2 n^2 \log ^2(F)} \]
[Out]
1/2*a^3*(d*x+c)^2/d-3*a^2*b*d*(F^(f*g*x+e*g))^n/f^2/g^2/n^2/ln(F)^2-3/4*a*b^2*d*(F^(f*g*x+e*g))^(2*n)/f^2/g^2/
n^2/ln(F)^2-1/9*b^3*d*(F^(f*g*x+e*g))^(3*n)/f^2/g^2/n^2/ln(F)^2+3*a^2*b*(F^(f*g*x+e*g))^n*(d*x+c)/f/g/n/ln(F)+
3/2*a*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f/g/n/ln(F)+1/3*b^3*(F^(f*g*x+e*g))^(3*n)*(d*x+c)/f/g/n/ln(F)
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Rubi [A] time = 0.24, antiderivative size = 236, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used =
{2183, 2176, 2194} \[ \frac {3 a^2 b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {3 a^2 b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {a^3 (c+d x)^2}{2 d}+\frac {3 a b^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac {3 a b^2 d \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n}}{9 f^2 g^2 n^2 \log ^2(F)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x),x]
[Out]
(a^3*(c + d*x)^2)/(2*d) - (3*a^2*b*d*(F^(e*g + f*g*x))^n)/(f^2*g^2*n^2*Log[F]^2) - (3*a*b^2*d*(F^(e*g + f*g*x)
)^(2*n))/(4*f^2*g^2*n^2*Log[F]^2) - (b^3*d*(F^(e*g + f*g*x))^(3*n))/(9*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*
g + f*g*x))^n*(c + d*x))/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(2*f*g*n*Log[F]) + (b^3*
(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(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) \, dx &=\int \left (a^3 (c+d x)+3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)+3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)+b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^2}{2 d}+\left (3 a^2 b\right ) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx+\left (3 a b^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x) \, dx+b^3 \int \left (F^{e g+f g x}\right )^{3 n} (c+d x) \, dx\\ &=\frac {a^3 (c+d x)^2}{2 d}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{3 f g n \log (F)}-\frac {\left (3 a^2 b d\right ) \int \left (F^{e g+f g x}\right )^n \, 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} \, dx}{2 f g n \log (F)}-\frac {\left (b^3 d\right ) \int \left (F^{e g+f g x}\right )^{3 n} \, dx}{3 f g n \log (F)}\\ &=\frac {a^3 (c+d x)^2}{2 d}-\frac {3 a^2 b d \left (F^{e g+f g x}\right )^n}{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}}{4 f^2 g^2 n^2 \log ^2(F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n}}{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)}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{3 f g n \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 161, normalized size = 0.68 \[ \frac {18 a^3 f^2 g^2 n^2 x \log ^2(F) (2 c+d x)+6 b f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^n \left (18 a^2+9 a b \left (F^{g (e+f x)}\right )^n+2 b^2 \left (F^{g (e+f x)}\right )^{2 n}\right )-b d \left (F^{g (e+f x)}\right )^n \left (108 a^2+27 a b \left (F^{g (e+f x)}\right )^n+4 b^2 \left (F^{g (e+f x)}\right )^{2 n}\right )}{36 f^2 g^2 n^2 \log ^2(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x),x]
[Out]
(-(b*d*(F^(g*(e + f*x)))^n*(108*a^2 + 27*a*b*(F^(g*(e + f*x)))^n + 4*b^2*(F^(g*(e + f*x)))^(2*n))) + 6*b*f*(F^
(g*(e + f*x)))^n*(18*a^2 + 9*a*b*(F^(g*(e + f*x)))^n + 2*b^2*(F^(g*(e + f*x)))^(2*n))*g*n*(c + d*x)*Log[F] + 1
8*a^3*f^2*g^2*n^2*x*(2*c + d*x)*Log[F]^2)/(36*f^2*g^2*n^2*Log[F]^2)
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fricas [A] time = 0.43, size = 192, normalized size = 0.81 \[ \frac {18 \, {\left (a^{3} d f^{2} g^{2} n^{2} x^{2} + 2 \, a^{3} c f^{2} g^{2} n^{2} x\right )} \log \relax (F)^{2} - 4 \, {\left (b^{3} d - 3 \, {\left (b^{3} d f g n x + b^{3} c f g n\right )} \log \relax (F)\right )} F^{3 \, f g n x + 3 \, e g n} - 27 \, {\left (a b^{2} d - 2 \, {\left (a b^{2} d f g n x + a b^{2} c f g n\right )} \log \relax (F)\right )} F^{2 \, f g n x + 2 \, e g n} - 108 \, {\left (a^{2} b d - {\left (a^{2} b d f g n x + a^{2} b c f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n}}{36 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)^3*(d*x+c),x, algorithm="fricas")
[Out]
1/36*(18*(a^3*d*f^2*g^2*n^2*x^2 + 2*a^3*c*f^2*g^2*n^2*x)*log(F)^2 - 4*(b^3*d - 3*(b^3*d*f*g*n*x + b^3*c*f*g*n)
*log(F))*F^(3*f*g*n*x + 3*e*g*n) - 27*(a*b^2*d - 2*(a*b^2*d*f*g*n*x + a*b^2*c*f*g*n)*log(F))*F^(2*f*g*n*x + 2*
e*g*n) - 108*(a^2*b*d - (a^2*b*d*f*g*n*x + a^2*b*c*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(f^2*g^2*n^2*log(F)^2)
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giac [C] time = 1.33, size = 3572, normalized size = 15.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)^3*(d*x+c),x, algorithm="giac")
[Out]
1/2*a^3*d*x^2 + a^3*c*x + 1/9*(2*((3*b^3*d*f*g*n*x*log(abs(F)) + 3*b^3*c*f*g*n*log(abs(F)) - b^3*d)*(pi^2*f^2*
g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2
+ 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^2) + 3*(
pi*b^3*d*f*g*n*x*sgn(F) - pi*b^3*d*f*g*n*x + pi*b^3*c*f*g*n*sgn(F) - pi*b^3*c*f*g*n)*(pi*f^2*g^2*n^2*log(abs(F
))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F
))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^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) + (3*(pi*b^3*d*f*g*n*x*sgn(F) - pi*b^3*d*f*g*n*x + pi*b^3
*c*f*g*n*sgn(F) - pi*b^3*c*f*g*n)*(pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)/(
(pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*s
gn(F) - pi*f^2*g^2*n^2*log(abs(F)))^2) - 4*(3*b^3*d*f*g*n*x*log(abs(F)) + 3*b^3*c*f*g*n*log(abs(F)) - b^3*d)*(
pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 +
2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^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*((6*pi*b^3*d*f*g*n*x*sgn(F) - 6*pi*b^3*d*f*g*n*x - 12*I*b^3*d*f*g*n*x*log(abs(F)) + 6*
pi*b^3*c*f*g*n*sgn(F) - 6*pi*b^3*c*f*g*n - 12*I*b^3*c*f*g*n*log(abs(F)) + 4*I*b^3*d)*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)/(18*pi^2*f^2*g^2*n^2*sgn(F) + 36*I*pi*f^2*g^2*n
^2*log(abs(F))*sgn(F) - 18*pi^2*f^2*g^2*n^2 - 36*I*pi*f^2*g^2*n^2*log(abs(F)) + 36*f^2*g^2*n^2*log(abs(F))^2)
+ (6*pi*b^3*d*f*g*n*x*sgn(F) - 6*pi*b^3*d*f*g*n*x + 12*I*b^3*d*f*g*n*x*log(abs(F)) + 6*pi*b^3*c*f*g*n*sgn(F) -
6*pi*b^3*c*f*g*n + 12*I*b^3*c*f*g*n*log(abs(F)) - 4*I*b^3*d)*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)/(18*pi^2*f^2*g^2*n^2*sgn(F) - 36*I*pi*f^2*g^2*n^2*log(abs(F))*sgn(F)
- 18*pi^2*f^2*g^2*n^2 + 36*I*pi*f^2*g^2*n^2*log(abs(F)) + 36*f^2*g^2*n^2*log(abs(F))^2))*e^(3*f*g*n*x*log(abs(
F)) + 3*g*n*e*log(abs(F))) + 3/2*(((2*a*b^2*d*f*g*n*x*log(abs(F)) + 2*a*b^2*c*f*g*n*log(abs(F)) - a*b^2*d)*(pi
^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g
^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^2
) + 2*(pi*a*b^2*d*f*g*n*x*sgn(F) - pi*a*b^2*d*f*g*n*x + pi*a*b^2*c*f*g*n*sgn(F) - pi*a*b^2*c*f*g*n)*(pi*f^2*g^
2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^
2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^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*a*b^2*d*f*g*n*x*sgn(F) - pi*a*b^2*d*f*g*n*x + pi*a*
b^2*c*f*g*n*sgn(F) - pi*a*b^2*c*f*g*n)*(pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))
^2)/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(
F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^2) - 2*(2*a*b^2*d*f*g*n*x*log(abs(F)) + 2*a*b^2*c*f*g*n*log(abs(F)) -
a*b^2*d)*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^
2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F))
)^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*lo
g(abs(F))) - 1/2*I*((3*pi*a*b^2*d*f*g*n*x*sgn(F) - 3*pi*a*b^2*d*f*g*n*x - 6*I*a*b^2*d*f*g*n*x*log(abs(F)) + 3*
pi*a*b^2*c*f*g*n*sgn(F) - 3*pi*a*b^2*c*f*g*n - 6*I*a*b^2*c*f*g*n*log(abs(F)) + 3*I*a*b^2*d)*e^(I*pi*f*g*n*x*sg
n(F) - I*pi*f*g*n*x + I*pi*g*n*e*sgn(F) - I*pi*g*n*e)/(2*pi^2*f^2*g^2*n^2*sgn(F) + 4*I*pi*f^2*g^2*n^2*log(abs(
F))*sgn(F) - 2*pi^2*f^2*g^2*n^2 - 4*I*pi*f^2*g^2*n^2*log(abs(F)) + 4*f^2*g^2*n^2*log(abs(F))^2) + (3*pi*a*b^2*
d*f*g*n*x*sgn(F) - 3*pi*a*b^2*d*f*g*n*x + 6*I*a*b^2*d*f*g*n*x*log(abs(F)) + 3*pi*a*b^2*c*f*g*n*sgn(F) - 3*pi*a
*b^2*c*f*g*n + 6*I*a*b^2*c*f*g*n*log(abs(F)) - 3*I*a*b^2*d)*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)/(2*pi^2*f^2*g^2*n^2*sgn(F) - 4*I*pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*pi^2*f^2*g^2*n^2
+ 4*I*pi*f^2*g^2*n^2*log(abs(F)) + 4*f^2*g^2*n^2*log(abs(F))^2))*e^(2*f*g*n*x*log(abs(F)) + 2*g*n*e*log(abs(F
))) + 3*(2*((a^2*b*d*f*g*n*x*log(abs(F)) + a^2*b*c*f*g*n*log(abs(F)) - a^2*b*d)*(pi^2*f^2*g^2*n^2*sgn(F) - pi^
2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(
abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^2) + (pi*a^2*b*d*f*g*n*x*sgn
(F) - pi*a^2*b*d*f*g*n*x + pi*a^2*b*c*f*g*n*sgn(F) - pi*a^2*b*c*f*g*n)*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi
*f^2*g^2*n^2*log(abs(F)))/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(p
i*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^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*a^2*b*d*f*g*n*x*sgn(F) - pi*a^2*b*d*f*g*n*x + pi*a^2*b*c*f*g*n*s
gn(F) - pi*a^2*b*c*f*g*n)*(pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)/((pi^2*f^
2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) -
pi*f^2*g^2*n^2*log(abs(F)))^2) - 4*(a^2*b*d*f*g*n*x*log(abs(F)) + a^2*b*c*f*g*n*log(abs(F)) - a^2*b*d)*(pi*f^2
*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))/((pi^2*f^2*g^2*n^2*sgn(F) - pi^2*f^2*g^2*n^2 + 2*f^2
*g^2*n^2*log(abs(F))^2)^2 + 4*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))^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*((6*pi*a^2*b*d*f*g*n*x*sgn(F) - 6*pi*a^2*b*d*f*g*n*x - 12*I*a^2*b*d*f*g*n*x*log(abs(F)) + 6*pi*a
^2*b*c*f*g*n*sgn(F) - 6*pi*a^2*b*c*f*g*n - 12*I*a^2*b*c*f*g*n*log(abs(F)) + 12*I*a^2*b*d)*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)/(2*pi^2*f^2*g^2*n^2*sgn(F) + 4*I*pi*f^2*g^
2*n^2*log(abs(F))*sgn(F) - 2*pi^2*f^2*g^2*n^2 - 4*I*pi*f^2*g^2*n^2*log(abs(F)) + 4*f^2*g^2*n^2*log(abs(F))^2)
+ (6*pi*a^2*b*d*f*g*n*x*sgn(F) - 6*pi*a^2*b*d*f*g*n*x + 12*I*a^2*b*d*f*g*n*x*log(abs(F)) + 6*pi*a^2*b*c*f*g*n*
sgn(F) - 6*pi*a^2*b*c*f*g*n + 12*I*a^2*b*c*f*g*n*log(abs(F)) - 12*I*a^2*b*d)*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)/(2*pi^2*f^2*g^2*n^2*sgn(F) - 4*I*pi*f^2*g^2*n^2*log(ab
s(F))*sgn(F) - 2*pi^2*f^2*g^2*n^2 + 4*I*pi*f^2*g^2*n^2*log(abs(F)) + 4*f^2*g^2*n^2*log(abs(F))^2))*e^(f*g*n*x*
log(abs(F)) + g*n*e*log(abs(F)))
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{3} \left (d x +c \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*(F^((f*x+e)*g))^n+a)^3*(d*x+c),x)
[Out]
int((b*(F^((f*x+e)*g))^n+a)^3*(d*x+c),x)
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maxima [A] time = 1.42, size = 284, normalized size = 1.20 \[ \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x + \frac {3 \, {\left (F^{f g x + e g}\right )}^{n} a^{2} b c}{f g n \log \relax (F)} + \frac {3 \, {\left (F^{f g x + e g}\right )}^{2 \, n} a b^{2} c}{2 \, f g n \log \relax (F)} + \frac {{\left (F^{f g x + e g}\right )}^{3 \, n} b^{3} c}{3 \, f g n \log \relax (F)} + \frac {3 \, {\left ({\left (F^{e g}\right )}^{n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} a^{2} b d}{f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {3 \, {\left (2 \, {\left (F^{e g}\right )}^{2 \, n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{2 \, n}\right )} {\left (F^{f g x}\right )}^{2 \, n} a b^{2} d}{4 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {{\left (3 \, {\left (F^{e g}\right )}^{3 \, n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{3 \, n}\right )} {\left (F^{f g x}\right )}^{3 \, n} b^{3} d}{9 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)^3*(d*x+c),x, algorithm="maxima")
[Out]
1/2*a^3*d*x^2 + a^3*c*x + 3*(F^(f*g*x + e*g))^n*a^2*b*c/(f*g*n*log(F)) + 3/2*(F^(f*g*x + e*g))^(2*n)*a*b^2*c/(
f*g*n*log(F)) + 1/3*(F^(f*g*x + e*g))^(3*n)*b^3*c/(f*g*n*log(F)) + 3*((F^(e*g))^n*f*g*n*x*log(F) - (F^(e*g))^n
)*(F^(f*g*x))^n*a^2*b*d/(f^2*g^2*n^2*log(F)^2) + 3/4*(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/(f^2*g^2*n^2*log(F)^2) + 1/9*(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*d/(f^2*g^2*n^2*log(F)^2)
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mupad [B] time = 3.77, size = 219, normalized size = 0.93 \[ \frac {a^3\,d\,x^2}{2}-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {3\,a\,b^2\,\left (d-2\,c\,f\,g\,n\,\ln \relax (F)\right )}{4\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2}-\frac {3\,a\,b^2\,d\,x}{2\,f\,g\,n\,\ln \relax (F)}\right )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}\,\left (\frac {b^3\,\left (d-3\,c\,f\,g\,n\,\ln \relax (F)\right )}{9\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2}-\frac {b^3\,d\,x}{3\,f\,g\,n\,\ln \relax (F)}\right )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {3\,a^2\,b\,\left (d-c\,f\,g\,n\,\ln \relax (F)\right )}{f^2\,g^2\,n^2\,{\ln \relax (F)}^2}-\frac {3\,a^2\,b\,d\,x}{f\,g\,n\,\ln \relax (F)}\right )+a^3\,c\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x),x)
[Out]
(a^3*d*x^2)/2 - (F^(f*g*x)*F^(e*g))^(2*n)*((3*a*b^2*(d - 2*c*f*g*n*log(F)))/(4*f^2*g^2*n^2*log(F)^2) - (3*a*b^
2*d*x)/(2*f*g*n*log(F))) - (F^(f*g*x)*F^(e*g))^(3*n)*((b^3*(d - 3*c*f*g*n*log(F)))/(9*f^2*g^2*n^2*log(F)^2) -
(b^3*d*x)/(3*f*g*n*log(F))) - (F^(f*g*x)*F^(e*g))^n*((3*a^2*b*(d - c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F)^2) - (
3*a^2*b*d*x)/(f*g*n*log(F))) + a^3*c*x
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sympy [A] time = 0.34, size = 350, normalized size = 1.48 \[ a^{3} c x + \frac {a^{3} d x^{2}}{2} + \begin {cases} \frac {\left (12 b^{3} c f^{5} g^{5} n^{5} \log {\relax (F )}^{5} + 12 b^{3} d f^{5} g^{5} n^{5} x \log {\relax (F )}^{5} - 4 b^{3} d f^{4} g^{4} n^{4} \log {\relax (F )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{3 n} + \left (54 a b^{2} c f^{5} g^{5} n^{5} \log {\relax (F )}^{5} + 54 a b^{2} d f^{5} g^{5} n^{5} x \log {\relax (F )}^{5} - 27 a b^{2} d f^{4} g^{4} n^{4} \log {\relax (F )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (108 a^{2} b c f^{5} g^{5} n^{5} \log {\relax (F )}^{5} + 108 a^{2} b d f^{5} g^{5} n^{5} x \log {\relax (F )}^{5} - 108 a^{2} b d f^{4} g^{4} n^{4} \log {\relax (F )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{36 f^{6} g^{6} n^{6} \log {\relax (F )}^{6}} & \text {for}\: 36 f^{6} g^{6} n^{6} \log {\relax (F )}^{6} \neq 0 \\x^{2} \left (\frac {3 a^{2} b d}{2} + \frac {3 a b^{2} d}{2} + \frac {b^{3} d}{2}\right ) + x \left (3 a^{2} b c + 3 a b^{2} c + b^{3} c\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F**(g*(f*x+e)))**n)**3*(d*x+c),x)
[Out]
a**3*c*x + a**3*d*x**2/2 + Piecewise((((12*b**3*c*f**5*g**5*n**5*log(F)**5 + 12*b**3*d*f**5*g**5*n**5*x*log(F)
**5 - 4*b**3*d*f**4*g**4*n**4*log(F)**4)*(F**(g*(e + f*x)))**(3*n) + (54*a*b**2*c*f**5*g**5*n**5*log(F)**5 + 5
4*a*b**2*d*f**5*g**5*n**5*x*log(F)**5 - 27*a*b**2*d*f**4*g**4*n**4*log(F)**4)*(F**(g*(e + f*x)))**(2*n) + (108
*a**2*b*c*f**5*g**5*n**5*log(F)**5 + 108*a**2*b*d*f**5*g**5*n**5*x*log(F)**5 - 108*a**2*b*d*f**4*g**4*n**4*log
(F)**4)*(F**(g*(e + f*x)))**n)/(36*f**6*g**6*n**6*log(F)**6), Ne(36*f**6*g**6*n**6*log(F)**6, 0)), (x**2*(3*a*
*2*b*d/2 + 3*a*b**2*d/2 + b**3*d/2) + x*(3*a**2*b*c + 3*a*b**2*c + b**3*c), True))
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