3.1.39 \(\int (a+b (F^{g (e+f x)})^n)^3 (c+d x)^3 \, dx\) [39]

Optimal. Leaf size=496 \[ \frac {a^3 (c+d x)^4}{4 d}-\frac {18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}-\frac {2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)}+\frac {18 a^2 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac {9 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^3 g^3 n^3 \log ^3(F)}-\frac {9 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac {9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)} \]

[Out]

1/4*a^3*(d*x+c)^4/d-18*a^2*b*d^3*(F^(f*g*x+e*g))^n/f^4/g^4/n^4/ln(F)^4-9/8*a*b^2*d^3*(F^(f*g*x+e*g))^(2*n)/f^4
/g^4/n^4/ln(F)^4-2/27*b^3*d^3*(F^(f*g*x+e*g))^(3*n)/f^4/g^4/n^4/ln(F)^4+18*a^2*b*d^2*(F^(f*g*x+e*g))^n*(d*x+c)
/f^3/g^3/n^3/ln(F)^3+9/4*a*b^2*d^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f^3/g^3/n^3/ln(F)^3+2/9*b^3*d^2*(F^(f*g*x+e*g
))^(3*n)*(d*x+c)/f^3/g^3/n^3/ln(F)^3-9*a^2*b*d*(F^(f*g*x+e*g))^n*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2-9/4*a*b^2*d*(F^
(f*g*x+e*g))^(2*n)*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2-1/3*b^3*d*(F^(f*g*x+e*g))^(3*n)*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2
+3*a^2*b*(F^(f*g*x+e*g))^n*(d*x+c)^3/f/g/n/ln(F)+3/2*a*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)^3/f/g/n/ln(F)+1/3*b^3
*(F^(f*g*x+e*g))^(3*n)*(d*x+c)^3/f/g/n/ln(F)

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Rubi [A]
time = 0.49, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2214, 2207, 2225} \begin {gather*} \frac {a^3 (c+d x)^4}{4 d}+\frac {18 a^2 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 {9 a^2 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 {3 a^2 b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac {9 a b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {9 a b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac {9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {2 b^3 d^2 (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^3 g^3 n^3 \log ^3(F)}-\frac {b^3 d (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x)^3 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}-\frac {2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(c + d*x)^4)/(4*d) - (18*a^2*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) - (9*a*b^2*d^3*(F^(e*g + f
*g*x))^(2*n))/(8*f^4*g^4*n^4*Log[F]^4) - (2*b^3*d^3*(F^(e*g + f*g*x))^(3*n))/(27*f^4*g^4*n^4*Log[F]^4) + (18*a
^2*b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) + (9*a*b^2*d^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x
))/(4*f^3*g^3*n^3*Log[F]^3) + (2*b^3*d^2*(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(9*f^3*g^3*n^3*Log[F]^3) - (9*a^2*
b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) - (9*a*b^2*d*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/
(4*f^2*g^2*n^2*Log[F]^2) - (b^3*d*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^2)/(3*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^
(e*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^3)/(2*f*g*n*Log[F])
+ (b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^3)/(3*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*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx &=\int \left (a^3 (c+d x)^3+3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3+3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3+b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3\right ) \, dx\\ &=\frac {a^3 (c+d x)^4}{4 d}+\left (3 a^2 b\right ) \int \left (F^{e g+f g x}\right )^n (c+d x)^3 \, dx+\left (3 a b^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3 \, dx+b^3 \int \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3 \, dx\\ &=\frac {a^3 (c+d x)^4}{4 d}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}-\frac {\left (9 a^2 b d\right ) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx}{f g n \log (F)}-\frac {\left (9 a b^2 d\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2 \, dx}{2 f g n \log (F)}-\frac {\left (b^3 d\right ) \int \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2 \, dx}{f g n \log (F)}\\ &=\frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 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 {9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 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)^3}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}+\frac {\left (18 a^2 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 {\left (9 a b^2 d^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x) \, dx}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {\left (2 b^3 d^2\right ) \int \left (F^{e g+f g x}\right )^{3 n} (c+d x) \, dx}{3 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {a^3 (c+d x)^4}{4 d}+\frac {18 a^2 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 {9 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^3 g^3 n^3 \log ^3(F)}-\frac {9 a^2 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 {9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 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)^3}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}-\frac {\left (18 a^2 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 {\left (9 a b^2 d^3\right ) \int \left (F^{e g+f g x}\right )^{2 n} \, dx}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {\left (2 b^3 d^3\right ) \int \left (F^{e g+f g x}\right )^{3 n} \, dx}{9 f^3 g^3 n^3 \log ^3(F)}\\ &=\frac {a^3 (c+d x)^4}{4 d}-\frac {18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}-\frac {2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)}+\frac {18 a^2 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 {9 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^3 g^3 n^3 \log ^3(F)}-\frac {9 a^2 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 {9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 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)^3}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 341, normalized size = 0.69 \begin {gather*} a^3 c^3 x+\frac {3}{2} a^3 c^2 d x^2+a^3 c d^2 x^3+\frac {1}{4} a^3 d^3 x^4+\frac {3 a^2 b \left (F^{g (e+f x)}\right )^n \left (-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 ^2(F)+f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (-3 d^3+6 d^2 f g n (c+d x) \log (F)-6 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+4 f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n} \left (-2 d^3+6 d^2 f g n (c+d x) \log (F)-9 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+9 f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{27 f^4 g^4 n^4 \log ^4(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*c^3*x + (3*a^3*c^2*d*x^2)/2 + a^3*c*d^2*x^3 + (a^3*d^3*x^4)/4 + (3*a^2*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) + (3*a*b^2*(F^(g*(e + f*x)))^(2*n)*(-3*d^3 + 6*d^2*f*g*n*(c + d*x)*Log[F] - 6*d*f^2*g^2*n^2*(c
+ d*x)^2*Log[F]^2 + 4*f^3*g^3*n^3*(c + d*x)^3*Log[F]^3))/(8*f^4*g^4*n^4*Log[F]^4) + (b^3*(F^(g*(e + f*x)))^(3*
n)*(-2*d^3 + 6*d^2*f*g*n*(c + d*x)*Log[F] - 9*d*f^2*g^2*n^2*(c + d*x)^2*Log[F]^2 + 9*f^3*g^3*n^3*(c + d*x)^3*L
og[F]^3))/(27*f^4*g^4*n^4*Log[F]^4)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{3} \left (d x +c \right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 884, normalized size = 1.78 \begin {gather*} \frac {1}{4} \, a^{3} d^{3} x^{4} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x + \frac {3 \, F^{f g n x + g n e} a^{2} b c^{3}}{f g n \log \left (F\right )} + \frac {3 \, F^{2 \, f g n x + 2 \, g n e} a b^{2} c^{3}}{2 \, f g n \log \left (F\right )} + \frac {F^{3 \, f g n x + 3 \, g n e} b^{3} c^{3}}{3 \, f g n \log \left (F\right )} + \frac {9 \, {\left (F^{g n e} f g n x \log \left (F\right ) - F^{g n e}\right )} F^{f g n x} a^{2} b c^{2} d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {9 \, {\left (2 \, F^{2 \, g n e} f g n x \log \left (F\right ) - F^{2 \, g n e}\right )} F^{2 \, f g n x} a b^{2} c^{2} d}{4 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {{\left (3 \, F^{3 \, g n e} f g n x \log \left (F\right ) - F^{3 \, g n e}\right )} F^{3 \, f g n x} b^{3} c^{2} d}{3 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {9 \, {\left (F^{g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{g n e} f g n x \log \left (F\right ) + 2 \, F^{g n e}\right )} F^{f g n x} a^{2} b c d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {9 \, {\left (2 \, F^{2 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, g n e} f g n x \log \left (F\right ) + F^{2 \, g n e}\right )} F^{2 \, f g n x} a b^{2} c d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (9 \, F^{3 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 6 \, F^{3 \, g n e} f g n x \log \left (F\right ) + 2 \, F^{3 \, g n e}\right )} F^{3 \, f g n x} b^{3} c d^{2}}{9 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\left (F^{g n e} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{g n e} f g n x \log \left (F\right ) - 6 \, F^{g n e}\right )} F^{f g n x} a^{2} b d^{3}}{f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {3 \, {\left (4 \, F^{2 \, g n e} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 6 \, F^{2 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{2 \, g n e} f g n x \log \left (F\right ) - 3 \, F^{2 \, g n e}\right )} F^{2 \, f g n x} a b^{2} d^{3}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {{\left (9 \, F^{3 \, g n e} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 9 \, F^{3 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{3 \, g n e} f g n x \log \left (F\right ) - 2 \, F^{3 \, g n e}\right )} F^{3 \, f g n x} b^{3} d^{3}}{27 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x + 3*F^(f*g*n*x + g*n*e)*a^2*b*c^3/(f*g*n*log(F
)) + 3/2*F^(2*f*g*n*x + 2*g*n*e)*a*b^2*c^3/(f*g*n*log(F)) + 1/3*F^(3*f*g*n*x + 3*g*n*e)*b^3*c^3/(f*g*n*log(F))
+ 9*(F^(g*n*e)*f*g*n*x*log(F) - F^(g*n*e))*F^(f*g*n*x)*a^2*b*c^2*d/(f^2*g^2*n^2*log(F)^2) + 9/4*(2*F^(2*g*n*e
)*f*g*n*x*log(F) - F^(2*g*n*e))*F^(2*f*g*n*x)*a*b^2*c^2*d/(f^2*g^2*n^2*log(F)^2) + 1/3*(3*F^(3*g*n*e)*f*g*n*x*
log(F) - F^(3*g*n*e))*F^(3*f*g*n*x)*b^3*c^2*d/(f^2*g^2*n^2*log(F)^2) + 9*(F^(g*n*e)*f^2*g^2*n^2*x^2*log(F)^2 -
2*F^(g*n*e)*f*g*n*x*log(F) + 2*F^(g*n*e))*F^(f*g*n*x)*a^2*b*c*d^2/(f^3*g^3*n^3*log(F)^3) + 9/4*(2*F^(2*g*n*e)
*f^2*g^2*n^2*x^2*log(F)^2 - 2*F^(2*g*n*e)*f*g*n*x*log(F) + F^(2*g*n*e))*F^(2*f*g*n*x)*a*b^2*c*d^2/(f^3*g^3*n^3
*log(F)^3) + 1/9*(9*F^(3*g*n*e)*f^2*g^2*n^2*x^2*log(F)^2 - 6*F^(3*g*n*e)*f*g*n*x*log(F) + 2*F^(3*g*n*e))*F^(3*
f*g*n*x)*b^3*c*d^2/(f^3*g^3*n^3*log(F)^3) + 3*(F^(g*n*e)*f^3*g^3*n^3*x^3*log(F)^3 - 3*F^(g*n*e)*f^2*g^2*n^2*x^
2*log(F)^2 + 6*F^(g*n*e)*f*g*n*x*log(F) - 6*F^(g*n*e))*F^(f*g*n*x)*a^2*b*d^3/(f^4*g^4*n^4*log(F)^4) + 3/8*(4*F
^(2*g*n*e)*f^3*g^3*n^3*x^3*log(F)^3 - 6*F^(2*g*n*e)*f^2*g^2*n^2*x^2*log(F)^2 + 6*F^(2*g*n*e)*f*g*n*x*log(F) -
3*F^(2*g*n*e))*F^(2*f*g*n*x)*a*b^2*d^3/(f^4*g^4*n^4*log(F)^4) + 1/27*(9*F^(3*g*n*e)*f^3*g^3*n^3*x^3*log(F)^3 -
9*F^(3*g*n*e)*f^2*g^2*n^2*x^2*log(F)^2 + 6*F^(3*g*n*e)*f*g*n*x*log(F) - 2*F^(3*g*n*e))*F^(3*f*g*n*x)*b^3*d^3/
(f^4*g^4*n^4*log(F)^4)

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Fricas [A]
time = 0.42, size = 711, normalized size = 1.43 \begin {gather*} \frac {54 \, {\left (a^{3} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{3} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{3} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{3} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 8 \, {\left (2 \, b^{3} d^{3} - 9 \, {\left (b^{3} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{3} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{3} c^{2} d f^{3} g^{3} n^{3} x + b^{3} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 9 \, {\left (b^{3} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d^{2} f^{2} g^{2} n^{2} x + b^{3} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b^{3} d^{3} f g n x + b^{3} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{3 \, f g n x + 3 \, g n e} - 81 \, {\left (3 \, a b^{2} d^{3} - 4 \, {\left (a b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b^{2} c^{2} d f^{3} g^{3} n^{3} x + a b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \, {\left (a b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d^{2} f^{2} g^{2} n^{2} x + a b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (a b^{2} d^{3} f g n x + a b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} - 648 \, {\left (6 \, a^{2} b d^{3} - {\left (a^{2} b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{2} b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a^{2} b c^{2} d f^{3} g^{3} n^{3} x + a^{2} b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (a^{2} b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d^{2} f^{2} g^{2} n^{2} x + a^{2} b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (a^{2} b d^{3} f g n x + a^{2} b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + g n e}}{216 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(54*(a^3*d^3*f^4*g^4*n^4*x^4 + 4*a^3*c*d^2*f^4*g^4*n^4*x^3 + 6*a^3*c^2*d*f^4*g^4*n^4*x^2 + 4*a^3*c^3*f^4
*g^4*n^4*x)*log(F)^4 - 8*(2*b^3*d^3 - 9*(b^3*d^3*f^3*g^3*n^3*x^3 + 3*b^3*c*d^2*f^3*g^3*n^3*x^2 + 3*b^3*c^2*d*f
^3*g^3*n^3*x + b^3*c^3*f^3*g^3*n^3)*log(F)^3 + 9*(b^3*d^3*f^2*g^2*n^2*x^2 + 2*b^3*c*d^2*f^2*g^2*n^2*x + b^3*c^
2*d*f^2*g^2*n^2)*log(F)^2 - 6*(b^3*d^3*f*g*n*x + b^3*c*d^2*f*g*n)*log(F))*F^(3*f*g*n*x + 3*g*n*e) - 81*(3*a*b^
2*d^3 - 4*(a*b^2*d^3*f^3*g^3*n^3*x^3 + 3*a*b^2*c*d^2*f^3*g^3*n^3*x^2 + 3*a*b^2*c^2*d*f^3*g^3*n^3*x + a*b^2*c^3
*f^3*g^3*n^3)*log(F)^3 + 6*(a*b^2*d^3*f^2*g^2*n^2*x^2 + 2*a*b^2*c*d^2*f^2*g^2*n^2*x + a*b^2*c^2*d*f^2*g^2*n^2)
*log(F)^2 - 6*(a*b^2*d^3*f*g*n*x + a*b^2*c*d^2*f*g*n)*log(F))*F^(2*f*g*n*x + 2*g*n*e) - 648*(6*a^2*b*d^3 - (a^
2*b*d^3*f^3*g^3*n^3*x^3 + 3*a^2*b*c*d^2*f^3*g^3*n^3*x^2 + 3*a^2*b*c^2*d*f^3*g^3*n^3*x + a^2*b*c^3*f^3*g^3*n^3)
*log(F)^3 + 3*(a^2*b*d^3*f^2*g^2*n^2*x^2 + 2*a^2*b*c*d^2*f^2*g^2*n^2*x + a^2*b*c^2*d*f^2*g^2*n^2)*log(F)^2 - 6
*(a^2*b*d^3*f*g*n*x + a^2*b*c*d^2*f*g*n)*log(F))*F^(f*g*n*x + g*n*e))/(f^4*g^4*n^4*log(F)^4)

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Sympy [A]
time = 0.29, size = 1073, normalized size = 2.16 \begin {gather*} a^{3} c^{3} x + \frac {3 a^{3} c^{2} d x^{2}}{2} + a^{3} c d^{2} x^{3} + \frac {a^{3} d^{3} x^{4}}{4} + \begin {cases} \frac {\left (72 b^{3} c^{3} f^{11} g^{11} n^{11} \log {\left (F \right )}^{11} + 216 b^{3} c^{2} d f^{11} g^{11} n^{11} x \log {\left (F \right )}^{11} - 72 b^{3} c^{2} d f^{10} g^{10} n^{10} \log {\left (F \right )}^{10} + 216 b^{3} c d^{2} f^{11} g^{11} n^{11} x^{2} \log {\left (F \right )}^{11} - 144 b^{3} c d^{2} f^{10} g^{10} n^{10} x \log {\left (F \right )}^{10} + 48 b^{3} c d^{2} f^{9} g^{9} n^{9} \log {\left (F \right )}^{9} + 72 b^{3} d^{3} f^{11} g^{11} n^{11} x^{3} \log {\left (F \right )}^{11} - 72 b^{3} d^{3} f^{10} g^{10} n^{10} x^{2} \log {\left (F \right )}^{10} + 48 b^{3} d^{3} f^{9} g^{9} n^{9} x \log {\left (F \right )}^{9} - 16 b^{3} d^{3} f^{8} g^{8} n^{8} \log {\left (F \right )}^{8}\right ) \left (F^{g \left (e + f x\right )}\right )^{3 n} + \left (324 a b^{2} c^{3} f^{11} g^{11} n^{11} \log {\left (F \right )}^{11} + 972 a b^{2} c^{2} d f^{11} g^{11} n^{11} x \log {\left (F \right )}^{11} - 486 a b^{2} c^{2} d f^{10} g^{10} n^{10} \log {\left (F \right )}^{10} + 972 a b^{2} c d^{2} f^{11} g^{11} n^{11} x^{2} \log {\left (F \right )}^{11} - 972 a b^{2} c d^{2} f^{10} g^{10} n^{10} x \log {\left (F \right )}^{10} + 486 a b^{2} c d^{2} f^{9} g^{9} n^{9} \log {\left (F \right )}^{9} + 324 a b^{2} d^{3} f^{11} g^{11} n^{11} x^{3} \log {\left (F \right )}^{11} - 486 a b^{2} d^{3} f^{10} g^{10} n^{10} x^{2} \log {\left (F \right )}^{10} + 486 a b^{2} d^{3} f^{9} g^{9} n^{9} x \log {\left (F \right )}^{9} - 243 a b^{2} d^{3} f^{8} g^{8} n^{8} \log {\left (F \right )}^{8}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (648 a^{2} b c^{3} f^{11} g^{11} n^{11} \log {\left (F \right )}^{11} + 1944 a^{2} b c^{2} d f^{11} g^{11} n^{11} x \log {\left (F \right )}^{11} - 1944 a^{2} b c^{2} d f^{10} g^{10} n^{10} \log {\left (F \right )}^{10} + 1944 a^{2} b c d^{2} f^{11} g^{11} n^{11} x^{2} \log {\left (F \right )}^{11} - 3888 a^{2} b c d^{2} f^{10} g^{10} n^{10} x \log {\left (F \right )}^{10} + 3888 a^{2} b c d^{2} f^{9} g^{9} n^{9} \log {\left (F \right )}^{9} + 648 a^{2} b d^{3} f^{11} g^{11} n^{11} x^{3} \log {\left (F \right )}^{11} - 1944 a^{2} b d^{3} f^{10} g^{10} n^{10} x^{2} \log {\left (F \right )}^{10} + 3888 a^{2} b d^{3} f^{9} g^{9} n^{9} x \log {\left (F \right )}^{9} - 3888 a^{2} b d^{3} f^{8} g^{8} n^{8} \log {\left (F \right )}^{8}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{216 f^{12} g^{12} n^{12} \log {\left (F \right )}^{12}} & \text {for}\: f^{12} g^{12} n^{12} \log {\left (F \right )}^{12} \neq 0 \\x^{4} \cdot \left (\frac {3 a^{2} b d^{3}}{4} + \frac {3 a b^{2} d^{3}}{4} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \cdot \left (3 a^{2} b c d^{2} + 3 a b^{2} c d^{2} + b^{3} c d^{2}\right ) + x^{2} \cdot \left (\frac {9 a^{2} b c^{2} d}{2} + \frac {9 a b^{2} c^{2} d}{2} + \frac {3 b^{3} c^{2} d}{2}\right ) + x \left (3 a^{2} b c^{3} + 3 a b^{2} c^{3} + b^{3} c^{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c**3*x + 3*a**3*c**2*d*x**2/2 + a**3*c*d**2*x**3 + a**3*d**3*x**4/4 + Piecewise((((72*b**3*c**3*f**11*g**
11*n**11*log(F)**11 + 216*b**3*c**2*d*f**11*g**11*n**11*x*log(F)**11 - 72*b**3*c**2*d*f**10*g**10*n**10*log(F)
**10 + 216*b**3*c*d**2*f**11*g**11*n**11*x**2*log(F)**11 - 144*b**3*c*d**2*f**10*g**10*n**10*x*log(F)**10 + 48
*b**3*c*d**2*f**9*g**9*n**9*log(F)**9 + 72*b**3*d**3*f**11*g**11*n**11*x**3*log(F)**11 - 72*b**3*d**3*f**10*g*
*10*n**10*x**2*log(F)**10 + 48*b**3*d**3*f**9*g**9*n**9*x*log(F)**9 - 16*b**3*d**3*f**8*g**8*n**8*log(F)**8)*(
F**(g*(e + f*x)))**(3*n) + (324*a*b**2*c**3*f**11*g**11*n**11*log(F)**11 + 972*a*b**2*c**2*d*f**11*g**11*n**11
*x*log(F)**11 - 486*a*b**2*c**2*d*f**10*g**10*n**10*log(F)**10 + 972*a*b**2*c*d**2*f**11*g**11*n**11*x**2*log(
F)**11 - 972*a*b**2*c*d**2*f**10*g**10*n**10*x*log(F)**10 + 486*a*b**2*c*d**2*f**9*g**9*n**9*log(F)**9 + 324*a
*b**2*d**3*f**11*g**11*n**11*x**3*log(F)**11 - 486*a*b**2*d**3*f**10*g**10*n**10*x**2*log(F)**10 + 486*a*b**2*
d**3*f**9*g**9*n**9*x*log(F)**9 - 243*a*b**2*d**3*f**8*g**8*n**8*log(F)**8)*(F**(g*(e + f*x)))**(2*n) + (648*a
**2*b*c**3*f**11*g**11*n**11*log(F)**11 + 1944*a**2*b*c**2*d*f**11*g**11*n**11*x*log(F)**11 - 1944*a**2*b*c**2
*d*f**10*g**10*n**10*log(F)**10 + 1944*a**2*b*c*d**2*f**11*g**11*n**11*x**2*log(F)**11 - 3888*a**2*b*c*d**2*f*
*10*g**10*n**10*x*log(F)**10 + 3888*a**2*b*c*d**2*f**9*g**9*n**9*log(F)**9 + 648*a**2*b*d**3*f**11*g**11*n**11
*x**3*log(F)**11 - 1944*a**2*b*d**3*f**10*g**10*n**10*x**2*log(F)**10 + 3888*a**2*b*d**3*f**9*g**9*n**9*x*log(
F)**9 - 3888*a**2*b*d**3*f**8*g**8*n**8*log(F)**8)*(F**(g*(e + f*x)))**n)/(216*f**12*g**12*n**12*log(F)**12),
Ne(f**12*g**12*n**12*log(F)**12, 0)), (x**4*(3*a**2*b*d**3/4 + 3*a*b**2*d**3/4 + b**3*d**3/4) + x**3*(3*a**2*b
*c*d**2 + 3*a*b**2*c*d**2 + b**3*c*d**2) + x**2*(9*a**2*b*c**2*d/2 + 9*a*b**2*c**2*d/2 + 3*b**3*c**2*d/2) + x*
(3*a**2*b*c**3 + 3*a*b**2*c**3 + b**3*c**3), True))

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Giac [C] Result contains complex when optimal does not.
time = 2.83, size = 18737, normalized size = 37.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x - 1/27*(((27*pi^2*b^3*d^3*f^3*g^3*n^3*x^3*log(
abs(F))*sgn(F) - 27*pi^2*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 18*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 81*p
i^2*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 81*pi^2*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F)) + 54*b^3*c*d^
2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 81*pi^2*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F) - 81*pi^2*b^3*c^2*d*f^3*g
^3*n^3*x*log(abs(F)) + 54*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))^3 + 27*pi^2*b^3*c^3*f^3*g^3*n^3*log(abs(F))*sgn(
F) - 27*pi^2*b^3*c^3*f^3*g^3*n^3*log(abs(F)) + 18*b^3*c^3*f^3*g^3*n^3*log(abs(F))^3 - 9*pi^2*b^3*d^3*f^2*g^2*n
^2*x^2*sgn(F) + 9*pi^2*b^3*d^3*f^2*g^2*n^2*x^2 - 18*b^3*d^3*f^2*g^2*n^2*x^2*log(abs(F))^2 - 18*pi^2*b^3*c*d^2*
f^2*g^2*n^2*x*sgn(F) + 18*pi^2*b^3*c*d^2*f^2*g^2*n^2*x - 36*b^3*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 9*pi^2*b^3
*c^2*d*f^2*g^2*n^2*sgn(F) + 9*pi^2*b^3*c^2*d*f^2*g^2*n^2 - 18*b^3*c^2*d*f^2*g^2*n^2*log(abs(F))^2 + 12*b^3*d^3
*f*g*n*x*log(abs(F)) + 12*b^3*c*d^2*f*g*n*log(abs(F)) - 4*b^3*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) - 12*(3*pi^3*b^3*d^3*f^3*g^3*n
^3*x^3*sgn(F) - 9*pi*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F) - 3*pi^3*b^3*d^3*f^3*g^3*n^3*x^3 + 9*pi*b^3*
d^3*f^3*g^3*n^3*x^3*log(abs(F))^2 + 9*pi^3*b^3*c*d^2*f^3*g^3*n^3*x^2*sgn(F) - 27*pi*b^3*c*d^2*f^3*g^3*n^3*x^2*
log(abs(F))^2*sgn(F) - 9*pi^3*b^3*c*d^2*f^3*g^3*n^3*x^2 + 27*pi*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 + 9*pi
^3*b^3*c^2*d*f^3*g^3*n^3*x*sgn(F) - 27*pi*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2*sgn(F) - 9*pi^3*b^3*c^2*d*f^3*
g^3*n^3*x + 27*pi*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + 3*pi^3*b^3*c^3*f^3*g^3*n^3*sgn(F) - 9*pi*b^3*c^3*f^3
*g^3*n^3*log(abs(F))^2*sgn(F) - 3*pi^3*b^3*c^3*f^3*g^3*n^3 + 9*pi*b^3*c^3*f^3*g^3*n^3*log(abs(F))^2 + 6*pi*b^3
*d^3*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^3*d^3*f^2*g^2*n^2*x^2*log(abs(F)) + 12*pi*b^3*c*d^2*f^2*g^2*n
^2*x*log(abs(F))*sgn(F) - 12*pi*b^3*c*d^2*f^2*g^2*n^2*x*log(abs(F)) + 6*pi*b^3*c^2*d*f^2*g^2*n^2*log(abs(F))*s
gn(F) - 6*pi*b^3*c^2*d*f^2*g^2*n^2*log(abs(F)) - 2*pi*b^3*d^3*f*g*n*x*sgn(F) + 2*pi*b^3*d^3*f*g*n*x - 2*pi*b^3
*c*d^2*f*g*n*sgn(F) + 2*pi*b^3*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*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))*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*(3*pi^3*b^3*d^3*f^3*g^3*n^3*x^3*sgn(F) - 9*pi*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2*sgn(F
) - 3*pi^3*b^3*d^3*f^3*g^3*n^3*x^3 + 9*pi*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))^2 + 9*pi^3*b^3*c*d^2*f^3*g^3*n^3
*x^2*sgn(F) - 27*pi*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2*sgn(F) - 9*pi^3*b^3*c*d^2*f^3*g^3*n^3*x^2 + 27*pi*
b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^2 + 9*pi^3*b^3*c^2*d*f^3*g^3*n^3*x*sgn(F) - 27*pi*b^3*c^2*d*f^3*g^3*n^3*
x*log(abs(F))^2*sgn(F) - 9*pi^3*b^3*c^2*d*f^3*g^3*n^3*x + 27*pi*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))^2 + 3*pi^3
*b^3*c^3*f^3*g^3*n^3*sgn(F) - 9*pi*b^3*c^3*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - 3*pi^3*b^3*c^3*f^3*g^3*n^3 + 9*p
i*b^3*c^3*f^3*g^3*n^3*log(abs(F))^2 + 6*pi*b^3*d^3*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^3*d^3*f^2*g^2*n
^2*x^2*log(abs(F)) + 12*pi*b^3*c*d^2*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 12*pi*b^3*c*d^2*f^2*g^2*n^2*x*log(abs(
F)) + 6*pi*b^3*c^2*d*f^2*g^2*n^2*log(abs(F))*sgn(F) - 6*pi*b^3*c^2*d*f^2*g^2*n^2*log(abs(F)) - 2*pi*b^3*d^3*f*
g*n*x*sgn(F) + 2*pi*b^3*d^3*f*g*n*x - 2*pi*b^3*c*d^2*f*g*n*sgn(F) + 2*pi*b^3*c*d^2*f*g*n)*(pi^4*f^4*g^4*n^4*sg
n(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*(27
*pi^2*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 27*pi^2*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F)) + 18*b^3*d^3*f^
3*g^3*n^3*x^3*log(abs(F))^3 + 81*pi^2*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 81*pi^2*b^3*c*d^2*f^3*g^3
*n^3*x^2*log(abs(F)) + 54*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 81*pi^2*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F)
)*sgn(F) - 81*pi^2*b^3*c^2*d*f^3*g^3*n^3*x*log(...

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Mupad [B]
time = 4.37, size = 652, normalized size = 1.31 \begin {gather*} a^3\,c^3\,x-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {3\,a^2\,b\,\left (-c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+3\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+6\,d^3\right )}{f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {3\,a^2\,b\,d^3\,x^3}{f\,g\,n\,\ln \left (F\right )}-\frac {9\,a^2\,b\,d\,x\,\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 {9\,a^2\,b\,d^2\,x^2\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {3\,a\,b^2\,\left (-4\,c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+6\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+3\,d^3\right )}{8\,f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {3\,a\,b^2\,d^3\,x^3}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {9\,a\,b^2\,d\,x\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {9\,a\,b^2\,d^2\,x^2\,\left (d-2\,c\,f\,g\,n\,\ln \left (F\right )\right )}{4\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}\,\left (\frac {b^3\,\left (-9\,c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+9\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+2\,d^3\right )}{27\,f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {b^3\,d^3\,x^3}{3\,f\,g\,n\,\ln \left (F\right )}-\frac {b^3\,d\,x\,\left (9\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{9\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b^3\,d^2\,x^2\,\left (d-3\,c\,f\,g\,n\,\ln \left (F\right )\right )}{3\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {a^3\,d^3\,x^4}{4}+\frac {3\,a^3\,c^2\,d\,x^2}{2}+a^3\,c\,d^2\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*c^3*x - (F^(f*g*x)*F^(e*g))^n*((3*a^2*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) - (3*a^2*b*d^3*x^3)/(f*g*n*log(F)) - (9*a^2*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) + (9*a^2*b*d^2*x^2*(d - c*f*g*n*log(F)))/(f
^2*g^2*n^2*log(F)^2)) - (F^(f*g*x)*F^(e*g))^(2*n)*((3*a*b^2*(3*d^3 - 4*c^3*f^3*g^3*n^3*log(F)^3 - 6*c*d^2*f*g*
n*log(F) + 6*c^2*d*f^2*g^2*n^2*log(F)^2))/(8*f^4*g^4*n^4*log(F)^4) - (3*a*b^2*d^3*x^3)/(2*f*g*n*log(F)) - (9*a
*b^2*d*x*(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) + (9*a*b^2*d^2*x^2*
(d - 2*c*f*g*n*log(F)))/(4*f^2*g^2*n^2*log(F)^2)) - (F^(f*g*x)*F^(e*g))^(3*n)*((b^3*(2*d^3 - 9*c^3*f^3*g^3*n^3
*log(F)^3 - 6*c*d^2*f*g*n*log(F) + 9*c^2*d*f^2*g^2*n^2*log(F)^2))/(27*f^4*g^4*n^4*log(F)^4) - (b^3*d^3*x^3)/(3
*f*g*n*log(F)) - (b^3*d*x*(2*d^2 + 9*c^2*f^2*g^2*n^2*log(F)^2 - 6*c*d*f*g*n*log(F)))/(9*f^3*g^3*n^3*log(F)^3)
+ (b^3*d^2*x^2*(d - 3*c*f*g*n*log(F)))/(3*f^2*g^2*n^2*log(F)^2)) + (a^3*d^3*x^4)/4 + (3*a^3*c^2*d*x^2)/2 + a^3
*c*d^2*x^3

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