3.25 \(\int (a+b (F^{g (e+f x)})^n) (c+d x)^3 \, dx\)
Optimal. Leaf size=153 \[ \frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)} \]
[Out]
1/4*a*(d*x+c)^4/d-6*b*d^3*(F^(f*g*x+e*g))^n/f^4/g^4/n^4/ln(F)^4+6*b*d^2*(F^(f*g*x+e*g))^n*(d*x+c)/f^3/g^3/n^3/
ln(F)^3-3*b*d*(F^(f*g*x+e*g))^n*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2+b*(F^(f*g*x+e*g))^n*(d*x+c)^3/f/g/n/ln(F)
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Rubi [A] time = 0.24, antiderivative size = 153, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used =
{2183, 2176, 2194} \[ \frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^3,x]
[Out]
(a*(c + d*x)^4)/(4*d) - (6*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) + (6*b*d^2*(F^(e*g + f*g*x))^n*(c
+ d*x))/(f^3*g^3*n^3*Log[F]^3) - (3*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g
+ f*g*x))^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 ) (c+d x)^3 \, dx &=\int \left (a (c+d x)^3+b \left (F^{e g+f g x}\right )^n (c+d x)^3\right ) \, dx\\ &=\frac {a (c+d x)^4}{4 d}+b \int \left (F^{e g+f g x}\right )^n (c+d x)^3 \, dx\\ &=\frac {a (c+d x)^4}{4 d}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}-\frac {(3 b d) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx}{f g n \log (F)}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {3 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {\left (6 b d^2\right ) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}-\frac {\left (6 b d^3\right ) \int \left (F^{e g+f g x}\right )^n \, dx}{f^3 g^3 n^3 \log ^3(F)}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac {6 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 130, normalized size = 0.85 \[ a c^3 x+\frac {3}{2} a c^2 d x^2+a c d^2 x^3+\frac {1}{4} a d^3 x^4+\frac {b \left (F^{g (e+f x)}\right )^n \left (6 d^2 f g n \log (F) (c+d x)+f^3 g^3 n^3 \log ^3(F) (c+d x)^3-3 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2-6 d^3\right )}{f^4 g^4 n^4 \log ^4(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^3,x]
[Out]
a*c^3*x + (3*a*c^2*d*x^2)/2 + a*c*d^2*x^3 + (a*d^3*x^4)/4 + (b*(F^(g*(e + f*x)))^n*(-6*d^3 + 6*d^2*f*g*n*(c +
d*x)*Log[F] - 3*d*f^2*g^2*n^2*(c + d*x)^2*Log[F]^2 + f^3*g^3*n^3*(c + d*x)^3*Log[F]^3))/(f^4*g^4*n^4*Log[F]^4)
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fricas [A] time = 0.45, size = 267, normalized size = 1.75 \[ \frac {{\left (a d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a c^{3} f^{4} g^{4} n^{4} x\right )} \log \relax (F)^{4} - 4 \, {\left (6 \, b d^{3} - {\left (b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b c^{2} d f^{3} g^{3} n^{3} x + b c^{3} f^{3} g^{3} n^{3}\right )} \log \relax (F)^{3} + 3 \, {\left (b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d^{2} f^{2} g^{2} n^{2} x + b c^{2} d f^{2} g^{2} n^{2}\right )} \log \relax (F)^{2} - 6 \, {\left (b d^{3} f g n x + b c d^{2} f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n}}{4 \, f^{4} g^{4} n^{4} \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x, algorithm="fricas")
[Out]
1/4*((a*d^3*f^4*g^4*n^4*x^4 + 4*a*c*d^2*f^4*g^4*n^4*x^3 + 6*a*c^2*d*f^4*g^4*n^4*x^2 + 4*a*c^3*f^4*g^4*n^4*x)*l
og(F)^4 - 4*(6*b*d^3 - (b*d^3*f^3*g^3*n^3*x^3 + 3*b*c*d^2*f^3*g^3*n^3*x^2 + 3*b*c^2*d*f^3*g^3*n^3*x + b*c^3*f^
3*g^3*n^3)*log(F)^3 + 3*(b*d^3*f^2*g^2*n^2*x^2 + 2*b*c*d^2*f^2*g^2*n^2*x + b*c^2*d*f^2*g^2*n^2)*log(F)^2 - 6*(
b*d^3*f*g*n*x + b*c*d^2*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(f^4*g^4*n^4*log(F)^4)
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giac [C] time = 0.58, size = 5732, normalized size = 37.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x, algorithm="giac")
[Out]
1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x - (((3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) -
3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 9*pi^2*b*c*d^2*f^3*g^3*n^3
*x^2*log(abs(F))*sgn(F) - 9*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 6*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3
+ 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F) - 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 6*b*c^2*d*f^3*
g^3*n^3*x*log(abs(F))^3 + 3*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F)) +
2*b*c^3*f^3*g^3*n^3*log(abs(F))^3 - 3*pi^2*b*d^3*f^2*g^2*n^2*x^2*sgn(F) + 3*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 6*b*
d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 6*pi^2*b*c*d^2*f^2*g^2*n^2*x*sgn(F) + 6*pi^2*b*c*d^2*f^2*g^2*n^2*x - 12*b*
c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b*c^2*d*f^2*g^2*n^2*sgn(F) + 3*pi^2*b*c^2*d*f^2*g^2*n^2 - 6*b*c^2*d
*f^2*g^2*n^2*log(abs(F))^2 + 12*b*d^3*f*g*n*x*log(abs(F)) + 12*b*c*d^2*f*g*n*log(abs(F)) - 12*b*d^3)*(pi^4*f^4
*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^
2 - 2*f^4*g^4*n^4*log(abs(F))^4)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^
4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs(F
))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)
^2) - 4*(pi^3*b*d^3*f^3*g^3*n^3*x^3*sgn(F) - 3*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b*d^3*f^3*
g^3*n^3*x^3 + 3*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2 + 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 9*pi*b*c*d^2*
f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2 + 9*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F
))^2 + 3*pi^3*b*c^2*d*f^3*g^3*n^3*x*sgn(F) - 9*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 3*pi^3*b*c^2*d*
f^3*g^3*n^3*x + 9*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + pi^3*b*c^3*f^3*g^3*n^3*sgn(F) - 3*pi*b*c^3*f^3*g^3*
n^3*log(abs(F))^2*sgn(F) - pi^3*b*c^3*f^3*g^3*n^3 + 3*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 + 6*pi*b*d^3*f^2*g^2*
n^2*x^2*log(abs(F))*sgn(F) - 6*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F)) + 12*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*
sgn(F) - 12*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F)) + 6*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*pi*b*c^2*d*
f^2*g^2*n^2*log(abs(F)) - 6*pi*b*d^3*f*g*n*x*sgn(F) + 6*pi*b*d^3*f*g*n*x - 6*pi*b*c*d^2*f*g*n*sgn(F) + 6*pi*b*
c*d^2*f*g*n)*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log
(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) -
pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*l
og(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(ab
s(F))^3)^2))*cos(-1/2*pi*f*g*n*x*sgn(F) + 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*sgn(F) + 1/2*pi*g*n*e) - ((pi^3*b*d^3*
f^3*g^3*n^3*x^3*sgn(F) - 3*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b*d^3*f^3*g^3*n^3*x^3 + 3*pi*b
*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2 + 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 9*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(
abs(F))^2*sgn(F) - 3*pi^3*b*c*d^2*f^3*g^3*n^3*x^2 + 9*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 + 3*pi^3*b*c^2*
d*f^3*g^3*n^3*x*sgn(F) - 9*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 3*pi^3*b*c^2*d*f^3*g^3*n^3*x + 9*pi
*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + pi^3*b*c^3*f^3*g^3*n^3*sgn(F) - 3*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2*sg
n(F) - pi^3*b*c^3*f^3*g^3*n^3 + 3*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 + 6*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))*
sgn(F) - 6*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F)) + 12*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 12*pi*b*c*d
^2*f^2*g^2*n^2*x*log(abs(F)) + 6*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*pi*b*c^2*d*f^2*g^2*n^2*log(abs(
F)) - 6*pi*b*d^3*f*g*n*x*sgn(F) + 6*pi*b*d^3*f*g*n*x - 6*pi*b*c*d^2*f*g*n*sgn(F) + 6*pi*b*c*d^2*f*g*n)*(pi^4*f
^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F)
)^2 - 2*f^4*g^4*n^4*log(abs(F))^4)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*
f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs
(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^
3)^2) + 4*(3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 3*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 2*b*d^
3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 9*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 9*pi^2*b*c*d^2*f^3*g^3*n
^3*x^2*log(abs(F)) + 6*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F)
- 9*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 6*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^3 + 3*pi^2*b*c^3*f^3*g^3*n^3
*log(abs(F))*sgn(F) - 3*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F)) + 2*b*c^3*f^3*g^3*n^3*log(abs(F))^3 - 3*pi^2*b*d^3*
f^2*g^2*n^2*x^2*sgn(F) + 3*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 6*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 6*pi^2*b*c*d^2
*f^2*g^2*n^2*x*sgn(F) + 6*pi^2*b*c*d^2*f^2*g^2*n^2*x - 12*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b*c^2*d
*f^2*g^2*n^2*sgn(F) + 3*pi^2*b*c^2*d*f^2*g^2*n^2 - 6*b*c^2*d*f^2*g^2*n^2*log(abs(F))^2 + 12*b*d^3*f*g*n*x*log(
abs(F)) + 12*b*c*d^2*f*g*n*log(abs(F)) - 12*b*d^3)*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(a
bs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + pi*f^4*g^4*n^4*log(abs(F))^3)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*p
i^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log
(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n
^4*log(abs(F)) + pi*f^4*g^4*n^4*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*pi^3*b*d^3*f^3*g^3*n^3*x^3*sgn
(F) + 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) -
8*pi^3*b*d^3*f^3*g^3*n^3*x^3 - 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(a
bs(F))^2 + 16*I*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 24*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) + 72*I*pi^2*b*c*d
^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 24*pi^3*b*c*d^2*f
^3*g^3*n^3*x^2 - 72*I*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 +
48*I*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 24*pi^3*b*c^2*d*f^3*g^3*n^3*x*sgn(F) + 72*I*pi^2*b*c^2*d*f^3*g^3
*n^3*x*log(abs(F))*sgn(F) - 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 24*pi^3*b*c^2*d*f^3*g^3*n^3*x -
72*I*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + 48*I*b*c^2*d*f^3*g^
3*n^3*x*log(abs(F))^3 + 8*pi^3*b*c^3*f^3*g^3*n^3*sgn(F) + 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 24*
pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - 8*pi^3*b*c^3*f^3*g^3*n^3 - 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))
+ 24*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 + 16*I*b*c^3*f^3*g^3*n^3*log(abs(F))^3 - 24*I*pi^2*b*d^3*f^2*g^2*n^2*
x^2*sgn(F) + 48*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) + 24*I*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 48*pi*b*d^3*f^
2*g^2*n^2*x^2*log(abs(F)) - 48*I*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x*sgn(F)
+ 96*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) + 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x - 96*pi*b*c*d^2*f^2*g^2*n^2
*x*log(abs(F)) - 96*I*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 24*I*pi^2*b*c^2*d*f^2*g^2*n^2*sgn(F) + 48*pi*b*c^2
*d*f^2*g^2*n^2*log(abs(F))*sgn(F) + 24*I*pi^2*b*c^2*d*f^2*g^2*n^2 - 48*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F)) - 48
*I*b*c^2*d*f^2*g^2*n^2*log(abs(F))^2 - 48*pi*b*d^3*f*g*n*x*sgn(F) + 48*pi*b*d^3*f*g*n*x + 96*I*b*d^3*f*g*n*x*l
og(abs(F)) - 48*pi*b*c*d^2*f*g*n*sgn(F) + 48*pi*b*c*d^2*f*g*n + 96*I*b*c*d^2*f*g*n*log(abs(F)) - 96*I*b*d^3)*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)/(8*pi^4*f^4*g^4*n^4*sgn
(F) + 32*I*pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - 48*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) - 32*I*pi*f^4*g^4*n^
4*log(abs(F))^3*sgn(F) - 8*pi^4*f^4*g^4*n^4 - 32*I*pi^3*f^4*g^4*n^4*log(abs(F)) + 48*pi^2*f^4*g^4*n^4*log(abs(
F))^2 + 32*I*pi*f^4*g^4*n^4*log(abs(F))^3 - 16*f^4*g^4*n^4*log(abs(F))^4) + (8*pi^3*b*d^3*f^3*g^3*n^3*x^3*sgn(
F) - 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - 8
*pi^3*b*d^3*f^3*g^3*n^3*x^3 + 24*I*pi^2*b*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 24*pi*b*d^3*f^3*g^3*n^3*x^3*log(ab
s(F))^2 - 16*I*b*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 24*pi^3*b*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 72*I*pi^2*b*c*d^
2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 24*pi^3*b*c*d^2*f^
3*g^3*n^3*x^2 + 72*I*pi^2*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 72*pi*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 -
48*I*b*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 24*pi^3*b*c^2*d*f^3*g^3*n^3*x*sgn(F) - 72*I*pi^2*b*c^2*d*f^3*g^3*
n^3*x*log(abs(F))*sgn(F) - 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 24*pi^3*b*c^2*d*f^3*g^3*n^3*x +
72*I*pi^2*b*c^2*d*f^3*g^3*n^3*x*log(abs(F)) + 72*pi*b*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 - 48*I*b*c^2*d*f^3*g^3
*n^3*x*log(abs(F))^3 + 8*pi^3*b*c^3*f^3*g^3*n^3*sgn(F) - 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 24*p
i*b*c^3*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - 8*pi^3*b*c^3*f^3*g^3*n^3 + 24*I*pi^2*b*c^3*f^3*g^3*n^3*log(abs(F))
+ 24*pi*b*c^3*f^3*g^3*n^3*log(abs(F))^2 - 16*I*b*c^3*f^3*g^3*n^3*log(abs(F))^3 + 24*I*pi^2*b*d^3*f^2*g^2*n^2*x
^2*sgn(F) + 48*pi*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b*d^3*f^2*g^2*n^2*x^2 - 48*pi*b*d^3*f^2
*g^2*n^2*x^2*log(abs(F)) + 48*I*b*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 + 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x*sgn(F) +
96*pi*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 48*I*pi^2*b*c*d^2*f^2*g^2*n^2*x - 96*pi*b*c*d^2*f^2*g^2*n^2*
x*log(abs(F)) + 96*I*b*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 + 24*I*pi^2*b*c^2*d*f^2*g^2*n^2*sgn(F) + 48*pi*b*c^2*
d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 24*I*pi^2*b*c^2*d*f^2*g^2*n^2 - 48*pi*b*c^2*d*f^2*g^2*n^2*log(abs(F)) + 48*
I*b*c^2*d*f^2*g^2*n^2*log(abs(F))^2 - 48*pi*b*d^3*f*g*n*x*sgn(F) + 48*pi*b*d^3*f*g*n*x - 96*I*b*d^3*f*g*n*x*lo
g(abs(F)) - 48*pi*b*c*d^2*f*g*n*sgn(F) + 48*pi*b*c*d^2*f*g*n - 96*I*b*c*d^2*f*g*n*log(abs(F)) + 96*I*b*d^3)*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)/(8*pi^4*f^4*g^4*n^4*sgn
(F) - 32*I*pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - 48*pi^2*f^4*g^4*n^4*log(abs(F))^2*sgn(F) + 32*I*pi*f^4*g^4*n^
4*log(abs(F))^3*sgn(F) - 8*pi^4*f^4*g^4*n^4 + 32*I*pi^3*f^4*g^4*n^4*log(abs(F)) + 48*pi^2*f^4*g^4*n^4*log(abs(
F))^2 - 32*I*pi*f^4*g^4*n^4*log(abs(F))^3 - 16*f^4*g^4*n^4*log(abs(F))^4))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(
abs(F)))
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right ) \left (d x +c \right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x)
[Out]
int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x)
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maxima [A] time = 1.35, size = 294, normalized size = 1.92 \[ \frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {{\left (F^{f g x + e g}\right )}^{n} b c^{3}}{f g n \log \relax (F)} + \frac {3 \, {\left ({\left (F^{e g}\right )}^{n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} b c^{2} d}{f^{2} g^{2} n^{2} \log \relax (F)^{2}} + \frac {3 \, {\left ({\left (F^{e g}\right )}^{n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} - 2 \, {\left (F^{e g}\right )}^{n} f g n x \log \relax (F) + 2 \, {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} b c d^{2}}{f^{3} g^{3} n^{3} \log \relax (F)^{3}} + \frac {{\left ({\left (F^{e g}\right )}^{n} f^{3} g^{3} n^{3} x^{3} \log \relax (F)^{3} - 3 \, {\left (F^{e g}\right )}^{n} f^{2} g^{2} n^{2} x^{2} \log \relax (F)^{2} + 6 \, {\left (F^{e g}\right )}^{n} f g n x \log \relax (F) - 6 \, {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} b d^{3}}{f^{4} g^{4} n^{4} \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^3,x, algorithm="maxima")
[Out]
1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x + (F^(f*g*x + e*g))^n*b*c^3/(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*b*c^2*d/(f^2*g^2*n^2*log(F)^2) + 3*((F^(e*g))^n*f^2*g^2*n^2*
x^2*log(F)^2 - 2*(F^(e*g))^n*f*g*n*x*log(F) + 2*(F^(e*g))^n)*(F^(f*g*x))^n*b*c*d^2/(f^3*g^3*n^3*log(F)^3) + ((
F^(e*g))^n*f^3*g^3*n^3*x^3*log(F)^3 - 3*(F^(e*g))^n*f^2*g^2*n^2*x^2*log(F)^2 + 6*(F^(e*g))^n*f*g*n*x*log(F) -
6*(F^(e*g))^n)*(F^(f*g*x))^n*b*d^3/(f^4*g^4*n^4*log(F)^4)
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mupad [B] time = 3.81, size = 225, normalized size = 1.47 \[ \frac {a\,d^3\,x^4}{4}-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {b\,\left (-c^3\,f^3\,g^3\,n^3\,{\ln \relax (F)}^3+3\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-6\,c\,d^2\,f\,g\,n\,\ln \relax (F)+6\,d^3\right )}{f^4\,g^4\,n^4\,{\ln \relax (F)}^4}-\frac {b\,d^3\,x^3}{f\,g\,n\,\ln \relax (F)}-\frac {3\,b\,d\,x\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \relax (F)}^2-2\,c\,d\,f\,g\,n\,\ln \relax (F)+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \relax (F)}^3}+\frac {3\,b\,d^2\,x^2\,\left (d-c\,f\,g\,n\,\ln \relax (F)\right )}{f^2\,g^2\,n^2\,{\ln \relax (F)}^2}\right )+a\,c^3\,x+\frac {3\,a\,c^2\,d\,x^2}{2}+a\,c\,d^2\,x^3 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^3,x)
[Out]
(a*d^3*x^4)/4 - (F^(f*g*x)*F^(e*g))^n*((b*(6*d^3 - c^3*f^3*g^3*n^3*log(F)^3 - 6*c*d^2*f*g*n*log(F) + 3*c^2*d*f
^2*g^2*n^2*log(F)^2))/(f^4*g^4*n^4*log(F)^4) - (b*d^3*x^3)/(f*g*n*log(F)) - (3*b*d*x*(2*d^2 + c^2*f^2*g^2*n^2*
log(F)^2 - 2*c*d*f*g*n*log(F)))/(f^3*g^3*n^3*log(F)^3) + (3*b*d^2*x^2*(d - c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F
)^2)) + a*c^3*x + (3*a*c^2*d*x^2)/2 + a*c*d^2*x^3
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sympy [A] time = 0.28, size = 332, normalized size = 2.17 \[ a c^{3} x + \frac {3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac {a d^{3} x^{4}}{4} + \begin {cases} \frac {\left (b c^{3} f^{3} g^{3} n^{3} \log {\relax (F )}^{3} + 3 b c^{2} d f^{3} g^{3} n^{3} x \log {\relax (F )}^{3} - 3 b c^{2} d f^{2} g^{2} n^{2} \log {\relax (F )}^{2} + 3 b c d^{2} f^{3} g^{3} n^{3} x^{2} \log {\relax (F )}^{3} - 6 b c d^{2} f^{2} g^{2} n^{2} x \log {\relax (F )}^{2} + 6 b c d^{2} f g n \log {\relax (F )} + b d^{3} f^{3} g^{3} n^{3} x^{3} \log {\relax (F )}^{3} - 3 b d^{3} f^{2} g^{2} n^{2} x^{2} \log {\relax (F )}^{2} + 6 b d^{3} f g n x \log {\relax (F )} - 6 b d^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{4} g^{4} n^{4} \log {\relax (F )}^{4}} & \text {for}\: f^{4} g^{4} n^{4} \log {\relax (F )}^{4} \neq 0 \\b c^{3} x + \frac {3 b c^{2} d x^{2}}{2} + b c d^{2} x^{3} + \frac {b d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c)**3,x)
[Out]
a*c**3*x + 3*a*c**2*d*x**2/2 + a*c*d**2*x**3 + a*d**3*x**4/4 + Piecewise(((b*c**3*f**3*g**3*n**3*log(F)**3 + 3
*b*c**2*d*f**3*g**3*n**3*x*log(F)**3 - 3*b*c**2*d*f**2*g**2*n**2*log(F)**2 + 3*b*c*d**2*f**3*g**3*n**3*x**2*lo
g(F)**3 - 6*b*c*d**2*f**2*g**2*n**2*x*log(F)**2 + 6*b*c*d**2*f*g*n*log(F) + b*d**3*f**3*g**3*n**3*x**3*log(F)*
*3 - 3*b*d**3*f**2*g**2*n**2*x**2*log(F)**2 + 6*b*d**3*f*g*n*x*log(F) - 6*b*d**3)*(F**(g*(e + f*x)))**n/(f**4*
g**4*n**4*log(F)**4), Ne(f**4*g**4*n**4*log(F)**4, 0)), (b*c**3*x + 3*b*c**2*d*x**2/2 + b*c*d**2*x**3 + b*d**3
*x**4/4, True))
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