3.34 \(\int (a+b (F^{g (e+f x)})^n)^2 (c+d x) \, dx\)

Optimal. Leaf size=156 \[ \frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{2 a b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{b^2 d \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)} \]

[Out]

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

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Rubi [A]  time = 0.156936, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2183, 2176, 2194} \[ \frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{2 a b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{b^2 d \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2194

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

Rubi steps

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

Mathematica [A]  time = 0.255795, size = 117, normalized size = 0.75 \[ \frac{2 a^2 f^2 g^2 n^2 x \log ^2(F) (2 c+d x)+2 b f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^n \left (4 a+b \left (F^{g (e+f x)}\right )^n\right )-b d \left (F^{g (e+f x)}\right )^n \left (8 a+b \left (F^{g (e+f x)}\right )^n\right )}{4 f^2 g^2 n^2 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 220, normalized size = 1.4 \begin{align*}{a}^{2}cx+{\frac{{a}^{2}d{x}^{2}}{2}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }} \right ) ^{2}c}{2\,ngf\ln \left ( F \right ) }}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }} \right ) ^{2}d}{4\, \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}}+2\,{\frac{ab{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}c}{ngf\ln \left ( F \right ) }}-2\,{\frac{ab{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}d}{ \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}}+{\frac{{b}^{2}dx \left ({{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }} \right ) ^{2}}{2\,ngf\ln \left ( F \right ) }}+2\,{\frac{abdx{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}}{ngf\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54302, size = 317, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (a^{2} d f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} -{\left (b^{2} d - 2 \,{\left (b^{2} d f g n x + b^{2} c f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 8 \,{\left (a b d -{\left (a b d f g n x + a b c f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{4 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c),x, algorithm="fricas")

[Out]

1/4*(2*(a^2*d*f^2*g^2*n^2*x^2 + 2*a^2*c*f^2*g^2*n^2*x)*log(F)^2 - (b^2*d - 2*(b^2*d*f*g*n*x + b^2*c*f*g*n)*log
(F))*F^(2*f*g*n*x + 2*e*g*n) - 8*(a*b*d - (a*b*d*f*g*n*x + a*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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Sympy [A]  time = 0.319764, size = 233, normalized size = 1.49 \begin{align*} a^{2} c x + \frac{a^{2} d x^{2}}{2} + \begin{cases} \frac{\left (2 b^{2} c f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} + 2 b^{2} d f^{3} g^{3} n^{3} x \log{\left (F \right )}^{3} - b^{2} d f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (8 a b c f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} + 8 a b d f^{3} g^{3} n^{3} x \log{\left (F \right )}^{3} - 8 a b d f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{4 f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} & \text{for}\: 4 f^{4} g^{4} n^{4} \log{\left (F \right )}^{4} \neq 0 \\x^{2} \left (a b d + \frac{b^{2} d}{2}\right ) + x \left (2 a b c + b^{2} c\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*x + a**2*d*x**2/2 + Piecewise((((2*b**2*c*f**3*g**3*n**3*log(F)**3 + 2*b**2*d*f**3*g**3*n**3*x*log(F)**
3 - b**2*d*f**2*g**2*n**2*log(F)**2)*(F**(g*(e + f*x)))**(2*n) + (8*a*b*c*f**3*g**3*n**3*log(F)**3 + 8*a*b*d*f
**3*g**3*n**3*x*log(F)**3 - 8*a*b*d*f**2*g**2*n**2*log(F)**2)*(F**(g*(e + f*x)))**n)/(4*f**4*g**4*n**4*log(F)*
*4), Ne(4*f**4*g**4*n**4*log(F)**4, 0)), (x**2*(a*b*d + b**2*d/2) + x*(2*a*b*c + b**2*c), True))

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Giac [C]  time = 1.5752, size = 3121, normalized size = 20.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c),x, algorithm="giac")

[Out]

1/2*a^2*d*x^2 + a^2*c*x + 1/2*((2*(pi*b^2*d*f*g*n*x*sgn(F) - pi*b^2*d*f*g*n*x + pi*b^2*c*f*g*n*sgn(F) - pi*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) + (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*b^2*d*f*g*n*x*log(abs(F))
+ 2*b^2*c*f*g*n*log(abs(F)) - 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)^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^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*b^2*d*f*g*n*x*sgn(F) - pi*b^2*d*f*g*n*x + pi*b^2*c*f*g*n*sgn(F) - pi*b^2*c*f*g*n)/((pi^2*f^2*g^2*n^2*sg
n(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*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))*(2*b^2*d*f*g*n*x*log(a
bs(F)) + 2*b^2*c*f*g*n*log(abs(F)) - 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(a
bs(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*log(abs(F))) - 1/2*I*((pi*b^2*d*f
*g*n*x*sgn(F) - pi*b^2*d*f*g*n*x - 2*I*b^2*d*f*g*n*x*log(abs(F)) + pi*b^2*c*f*g*n*sgn(F) - pi*b^2*c*f*g*n - 2*
I*b^2*c*f*g*n*log(abs(F)) + I*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*lo
g(abs(F)) + 4*f^2*g^2*n^2*log(abs(F))^2) + (pi*b^2*d*f*g*n*x*sgn(F) - pi*b^2*d*f*g*n*x + 2*I*b^2*d*f*g*n*x*log
(abs(F)) + pi*b^2*c*f*g*n*sgn(F) - pi*b^2*c*f*g*n + 2*I*b^2*c*f*g*n*log(abs(F)) - I*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))*e^(2*f*g*n*x*
log(abs(F)) + 2*g*n*e*log(abs(F))) + 2*(2*((pi*a*b*d*f*g*n*x*sgn(F) - pi*a*b*d*f*g*n*x + pi*a*b*c*f*g*n*sgn(F)
- pi*a*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*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*lo
g(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)*(a*b*d*f*g*n*x*log(
abs(F)) + a*b*c*f*g*n*log(abs(F)) - a*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(ab
s(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(-1/2*pi*f*g*n*x*sgn(F)
+ 1/2*pi*f*g*n*x - 1/2*pi*g*n*e*sgn(F) + 1/2*pi*g*n*e) + ((pi^2*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*a*b*d*f*g*n*x*sgn(F) - pi*a*b*d*f*g*n*x + pi*a*b*c*f*g*n*sgn(F) - pi*a*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)^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*(pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))*(
a*b*d*f*g*n*x*log(abs(F)) + a*b*c*f*g*n*log(abs(F)) - a*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)^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*((4*pi*a*b*d*f*g*n*x*sgn(F) - 4*pi*a*b*d*f*g*n*x - 8*I*a*b*d*f*g*n*x*log(abs(F)) + 4*pi*a*b*c
*f*g*n*sgn(F) - 4*pi*a*b*c*f*g*n - 8*I*a*b*c*f*g*n*log(abs(F)) + 8*I*a*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) + (4*pi*a*b*d*
f*g*n*x*sgn(F) - 4*pi*a*b*d*f*g*n*x + 8*I*a*b*d*f*g*n*x*log(abs(F)) + 4*pi*a*b*c*f*g*n*sgn(F) - 4*pi*a*b*c*f*g
*n + 8*I*a*b*c*f*g*n*log(abs(F)) - 8*I*a*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))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(abs(F))
)