3.1.32 \(\int (a+b (F^{g (e+f x)})^n)^2 (c+d x)^3 \, dx\) [32]

Optimal. Leaf size=322 \[ \frac {a^2 (c+d x)^4}{4 d}-\frac {12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {12 a b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)} \]

[Out]

1/4*a^2*(d*x+c)^4/d-12*a*b*d^3*(F^(f*g*x+e*g))^n/f^4/g^4/n^4/ln(F)^4-3/8*b^2*d^3*(F^(f*g*x+e*g))^(2*n)/f^4/g^4
/n^4/ln(F)^4+12*a*b*d^2*(F^(f*g*x+e*g))^n*(d*x+c)/f^3/g^3/n^3/ln(F)^3+3/4*b^2*d^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c
)/f^3/g^3/n^3/ln(F)^3-6*a*b*d*(F^(f*g*x+e*g))^n*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2-3/4*b^2*d*(F^(f*g*x+e*g))^(2*n)*
(d*x+c)^2/f^2/g^2/n^2/ln(F)^2+2*a*b*(F^(f*g*x+e*g))^n*(d*x+c)^3/f/g/n/ln(F)+1/2*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x
+c)^3/f/g/n/ln(F)

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Rubi [A]
time = 0.34, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 10, 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^2 (c+d x)^4}{4 d}+\frac {12 a 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 {6 a 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 {2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac {3 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 {3 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 {b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac {3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 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)^2*(c + d*x)^3,x]

[Out]

(a^2*(c + d*x)^4)/(4*d) - (12*a*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) - (3*b^2*d^3*(F^(e*g + f*g*x
))^(2*n))/(8*f^4*g^4*n^4*Log[F]^4) + (12*a*b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) + (3*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) - (6*a*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/
(f^2*g^2*n^2*Log[F]^2) - (3*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) + (2*a*b*(F^(e
*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^3)/(2*f*g*n*Log[F])

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2214

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n},
x] && IGtQ[p, 0]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx &=\int \left (a^2 (c+d x)^3+2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3+b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int \left (F^{e g+f g x}\right )^n (c+d x)^3 \, dx+b^2 \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3 \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}-\frac {(6 a b d) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx}{f g n \log (F)}-\frac {\left (3 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 {a^2 (c+d x)^4}{4 d}-\frac {6 a 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 {3 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 {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {\left (12 a 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 (3 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 {a^2 (c+d x)^4}{4 d}+\frac {12 a 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^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 {6 a 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 {3 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 {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}-\frac {\left (12 a 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 (3 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 {a^2 (c+d x)^4}{4 d}-\frac {12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {3 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 {12 a 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^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 {6 a 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 {3 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 {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 239, normalized size = 0.74 \begin {gather*} a^2 c^3 x+\frac {3}{2} a^2 c^2 d x^2+a^2 c d^2 x^3+\frac {1}{4} a^2 d^3 x^4+\frac {2 a 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 {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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^3,x]

[Out]

a^2*c^3*x + (3*a^2*c^2*d*x^2)/2 + a^2*c*d^2*x^3 + (a^2*d^3*x^4)/4 + (2*a*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) + (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)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2} \left (d x +c \right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^3,x)

[Out]

int((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^3,x)

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Maxima [A]
time = 0.33, size = 588, normalized size = 1.83 \begin {gather*} \frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x + \frac {2 \, F^{f g n x + g n e} a b c^{3}}{f g n \log \left (F\right )} + \frac {F^{2 \, f g n x + 2 \, g n e} b^{2} c^{3}}{2 \, f g n \log \left (F\right )} + \frac {6 \, {\left (F^{g n e} f g n x \log \left (F\right ) - F^{g n e}\right )} F^{f g n x} a b c^{2} d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\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} b^{2} c^{2} d}{4 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {6 \, {\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 b c d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\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} b^{2} c d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, {\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 b d^{3}}{f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {{\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} b^{2} d^{3}}{8 \, 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)^2*(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.44, size = 484, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (a^{2} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{2} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{2} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{2} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - {\left (3 \, b^{2} d^{3} - 4 \, {\left (b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} g^{3} n^{3} x + b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \, {\left (b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d^{2} f^{2} g^{2} n^{2} x + b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b^{2} d^{3} f g n x + b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} - 16 \, {\left (6 \, a b d^{3} - {\left (a b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b c^{2} d f^{3} g^{3} n^{3} x + a b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (a b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d^{2} f^{2} g^{2} n^{2} x + a b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (a b d^{3} f g n x + a b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + g n e}}{8 \, 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)^2*(d*x+c)^3,x, algorithm="fricas")

[Out]

1/8*(2*(a^2*d^3*f^4*g^4*n^4*x^4 + 4*a^2*c*d^2*f^4*g^4*n^4*x^3 + 6*a^2*c^2*d*f^4*g^4*n^4*x^2 + 4*a^2*c^3*f^4*g^
4*n^4*x)*log(F)^4 - (3*b^2*d^3 - 4*(b^2*d^3*f^3*g^3*n^3*x^3 + 3*b^2*c*d^2*f^3*g^3*n^3*x^2 + 3*b^2*c^2*d*f^3*g^
3*n^3*x + b^2*c^3*f^3*g^3*n^3)*log(F)^3 + 6*(b^2*d^3*f^2*g^2*n^2*x^2 + 2*b^2*c*d^2*f^2*g^2*n^2*x + b^2*c^2*d*f
^2*g^2*n^2)*log(F)^2 - 6*(b^2*d^3*f*g*n*x + b^2*c*d^2*f*g*n)*log(F))*F^(2*f*g*n*x + 2*g*n*e) - 16*(6*a*b*d^3 -
(a*b*d^3*f^3*g^3*n^3*x^3 + 3*a*b*c*d^2*f^3*g^3*n^3*x^2 + 3*a*b*c^2*d*f^3*g^3*n^3*x + a*b*c^3*f^3*g^3*n^3)*log
(F)^3 + 3*(a*b*d^3*f^2*g^2*n^2*x^2 + 2*a*b*c*d^2*f^2*g^2*n^2*x + a*b*c^2*d*f^2*g^2*n^2)*log(F)^2 - 6*(a*b*d^3*
f*g*n*x + a*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 = 0.20, size = 707, normalized size = 2.20 \begin {gather*} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac {a^{2} d^{3} x^{4}}{4} + \begin {cases} \frac {\left (4 b^{2} c^{3} f^{7} g^{7} n^{7} \log {\left (F \right )}^{7} + 12 b^{2} c^{2} d f^{7} g^{7} n^{7} x \log {\left (F \right )}^{7} - 6 b^{2} c^{2} d f^{6} g^{6} n^{6} \log {\left (F \right )}^{6} + 12 b^{2} c d^{2} f^{7} g^{7} n^{7} x^{2} \log {\left (F \right )}^{7} - 12 b^{2} c d^{2} f^{6} g^{6} n^{6} x \log {\left (F \right )}^{6} + 6 b^{2} c d^{2} f^{5} g^{5} n^{5} \log {\left (F \right )}^{5} + 4 b^{2} d^{3} f^{7} g^{7} n^{7} x^{3} \log {\left (F \right )}^{7} - 6 b^{2} d^{3} f^{6} g^{6} n^{6} x^{2} \log {\left (F \right )}^{6} + 6 b^{2} d^{3} f^{5} g^{5} n^{5} x \log {\left (F \right )}^{5} - 3 b^{2} d^{3} f^{4} g^{4} n^{4} \log {\left (F \right )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (16 a b c^{3} f^{7} g^{7} n^{7} \log {\left (F \right )}^{7} + 48 a b c^{2} d f^{7} g^{7} n^{7} x \log {\left (F \right )}^{7} - 48 a b c^{2} d f^{6} g^{6} n^{6} \log {\left (F \right )}^{6} + 48 a b c d^{2} f^{7} g^{7} n^{7} x^{2} \log {\left (F \right )}^{7} - 96 a b c d^{2} f^{6} g^{6} n^{6} x \log {\left (F \right )}^{6} + 96 a b c d^{2} f^{5} g^{5} n^{5} \log {\left (F \right )}^{5} + 16 a b d^{3} f^{7} g^{7} n^{7} x^{3} \log {\left (F \right )}^{7} - 48 a b d^{3} f^{6} g^{6} n^{6} x^{2} \log {\left (F \right )}^{6} + 96 a b d^{3} f^{5} g^{5} n^{5} x \log {\left (F \right )}^{5} - 96 a b d^{3} f^{4} g^{4} n^{4} \log {\left (F \right )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{8 f^{8} g^{8} n^{8} \log {\left (F \right )}^{8}} & \text {for}\: f^{8} g^{8} n^{8} \log {\left (F \right )}^{8} \neq 0 \\x^{4} \left (\frac {a b d^{3}}{2} + \frac {b^{2} d^{3}}{4}\right ) + x^{3} \cdot \left (2 a b c d^{2} + b^{2} c d^{2}\right ) + x^{2} \cdot \left (3 a b c^{2} d + \frac {3 b^{2} c^{2} d}{2}\right ) + x \left (2 a b c^{3} + b^{2} 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)**2*(d*x+c)**3,x)

[Out]

a**2*c**3*x + 3*a**2*c**2*d*x**2/2 + a**2*c*d**2*x**3 + a**2*d**3*x**4/4 + Piecewise((((4*b**2*c**3*f**7*g**7*
n**7*log(F)**7 + 12*b**2*c**2*d*f**7*g**7*n**7*x*log(F)**7 - 6*b**2*c**2*d*f**6*g**6*n**6*log(F)**6 + 12*b**2*
c*d**2*f**7*g**7*n**7*x**2*log(F)**7 - 12*b**2*c*d**2*f**6*g**6*n**6*x*log(F)**6 + 6*b**2*c*d**2*f**5*g**5*n**
5*log(F)**5 + 4*b**2*d**3*f**7*g**7*n**7*x**3*log(F)**7 - 6*b**2*d**3*f**6*g**6*n**6*x**2*log(F)**6 + 6*b**2*d
**3*f**5*g**5*n**5*x*log(F)**5 - 3*b**2*d**3*f**4*g**4*n**4*log(F)**4)*(F**(g*(e + f*x)))**(2*n) + (16*a*b*c**
3*f**7*g**7*n**7*log(F)**7 + 48*a*b*c**2*d*f**7*g**7*n**7*x*log(F)**7 - 48*a*b*c**2*d*f**6*g**6*n**6*log(F)**6
+ 48*a*b*c*d**2*f**7*g**7*n**7*x**2*log(F)**7 - 96*a*b*c*d**2*f**6*g**6*n**6*x*log(F)**6 + 96*a*b*c*d**2*f**5
*g**5*n**5*log(F)**5 + 16*a*b*d**3*f**7*g**7*n**7*x**3*log(F)**7 - 48*a*b*d**3*f**6*g**6*n**6*x**2*log(F)**6 +
96*a*b*d**3*f**5*g**5*n**5*x*log(F)**5 - 96*a*b*d**3*f**4*g**4*n**4*log(F)**4)*(F**(g*(e + f*x)))**n)/(8*f**8
*g**8*n**8*log(F)**8), Ne(f**8*g**8*n**8*log(F)**8, 0)), (x**4*(a*b*d**3/2 + b**2*d**3/4) + x**3*(2*a*b*c*d**2
+ b**2*c*d**2) + x**2*(3*a*b*c**2*d + 3*b**2*c**2*d/2) + x*(2*a*b*c**3 + b**2*c**3), True))

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Giac [C] Result contains complex when optimal does not.
time = 4.35, size = 12033, normalized size = 37.37 \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)^2*(d*x+c)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.97, size = 438, normalized size = 1.36 \begin {gather*} a^2\,c^3\,x-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {2\,a\,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 {2\,a\,b\,d^3\,x^3}{f\,g\,n\,\ln \left (F\right )}-\frac {6\,a\,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 {6\,a\,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 {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 {b^2\,d^3\,x^3}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {3\,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 {3\,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 )+\frac {a^2\,d^3\,x^4}{4}+\frac {3\,a^2\,c^2\,d\,x^2}{2}+a^2\,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)^2*(c + d*x)^3,x)

[Out]

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

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