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

Optimal. Leaf size=115 \[ \frac{a (c+d x)^3}{3 d}-\frac{2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac{2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)} \]

[Out]

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

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Rubi [A]  time = 0.154442, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2183, 2176, 2194} \[ \frac{a (c+d x)^3}{3 d}-\frac{2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac{2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.166963, size = 91, normalized size = 0.79 \[ a c^2 x+a c d x^2+\frac{1}{3} a d^2 x^3+\frac{b \left (F^{g (e+f x)}\right )^n \left (f^2 g^2 n^2 \log ^2(F) (c+d x)^2-2 d f g n \log (F) (c+d x)+2 d^2\right )}{f^3 g^3 n^3 \log ^3(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 199, normalized size = 1.7 \begin{align*}{c}^{2}ax+acd{x}^{2}+{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}{c}^{2}}{ngf\ln \left ( F \right ) }}-2\,{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}cd}{ \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}}+2\,{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}{d}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{f}^{3}{g}^{3}{n}^{3}}}+{\frac{b{d}^{2}{x}^{2}{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}}{ngf\ln \left ( F \right ) }}+{\frac{a{d}^{2}{x}^{3}}{3}}+2\,{\frac{bd \left ( \ln \left ( F \right ) cfgn-d \right ) x{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*a*x+a*c*d*x^2+b/ln(F)/f/g/n*exp(n*ln(exp(g*(f*x+e)*ln(F))))*c^2-2*b/ln(F)^2/f^2/g^2/n^2*exp(n*ln(exp(g*(f*
x+e)*ln(F))))*c*d+2*b/ln(F)^3/f^3/g^3/n^3*exp(n*ln(exp(g*(f*x+e)*ln(F))))*d^2+1/n/g/f/ln(F)*b*d^2*x^2*exp(n*ln
(exp(g*(f*x+e)*ln(F))))+1/3*a*d^2*x^3+2*b*d*(ln(F)*c*f*g*n-d)/ln(F)^2/f^2/g^2/n^2*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)*(d*x+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79352, size = 360, normalized size = 3.13 \begin{align*} \frac{{\left (a d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a c d f^{3} g^{3} n^{3} x^{2} + 3 \, a c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \,{\left (2 \, b d^{2} +{\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x + b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b d^{2} f g n x + b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{3 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.212459, size = 196, normalized size = 1.7 \begin{align*} a c^{2} x + a c d x^{2} + \frac{a d^{2} x^{3}}{3} + \begin{cases} \frac{\left (b c^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} + 2 b c d f^{2} g^{2} n^{2} x \log{\left (F \right )}^{2} - 2 b c d f g n \log{\left (F \right )} + b d^{2} f^{2} g^{2} n^{2} x^{2} \log{\left (F \right )}^{2} - 2 b d^{2} f g n x \log{\left (F \right )} + 2 b d^{2}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} & \text{for}\: f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} \neq 0 \\b c^{2} x + b c d x^{2} + \frac{b d^{2} x^{3}}{3} & \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)*(d*x+c)**2,x)

[Out]

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

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Giac [C]  time = 1.33238, size = 3680, normalized size = 32. \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)*(d*x+c)^2,x, algorithm="giac")

[Out]

1/3*a*d^2*x^3 + a*c*d*x^2 + a*c^2*x - ((2*(pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*b*d^2*f^2*g^2*n^2*
x^2*log(abs(F)) + 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*b*c^
2*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*b*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b*d^2*f*g*n*x*sgn(F) + pi*b*d^2*f*g*n
*x - pi*b*c*d*f*g*n*sgn(F) + pi*b*c*d*f*g*n)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F)
- pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^
2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*p
i^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)^2) - (pi^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*b*d^
2*f^2*g^2*n^2*x^2 + 2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F) - 2*pi^2*b*c*d*f
^2*g^2*n^2*x + 4*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*b*c^2*f^2*g^2*n^2 +
2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*b*d^2*f*g*n*x*log(abs(F)) - 4*b*c*d*f*g*n*log(abs(F)) + 4*b*d^2)*(3*pi^2
*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3
*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (
3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3*log(abs(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^2*b*d^2*f^2*g^2*n^2*x^2*s
gn(F) - pi^2*b*d^2*f^2*g^2*n^2*x^2 + 2*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F)
- 2*pi^2*b*c*d*f^2*g^2*n^2*x + 4*b*c*d*f^2*g^2*n^2*x*log(abs(F))^2 + pi^2*b*c^2*f^2*g^2*n^2*sgn(F) - pi^2*b*c
^2*f^2*g^2*n^2 + 2*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 4*b*d^2*f*g*n*x*log(abs(F)) - 4*b*c*d*f*g*n*log(abs(F)) +
4*b*d^2)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n
^3*log(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*
f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3
*g^3*n^3*log(abs(F))^3)^2) + 2*(pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - pi*b*d^2*f^2*g^2*n^2*x^2*log(abs
(F)) + 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 2*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + pi*b*c^2*f^2*g^2*n
^2*log(abs(F))*sgn(F) - pi*b*c^2*f^2*g^2*n^2*log(abs(F)) - pi*b*d^2*f*g*n*x*sgn(F) + pi*b*d^2*f*g*n*x - pi*b*c
*d*f*g*n*sgn(F) + pi*b*c*d*f*g*n)*(3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*
f^3*g^3*n^3*log(abs(F))^3)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^
3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)
) + 2*f^3*g^3*n^3*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*p
i*g*n*e))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(abs(F))) + 1/2*I*((4*I*pi^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) - 8*pi*b
*d^2*f^2*g^2*n^2*x^2*log(abs(F))*sgn(F) - 4*I*pi^2*b*d^2*f^2*g^2*n^2*x^2 + 8*pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(
F)) + 8*I*b*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 8*I*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F) - 16*pi*b*c*d*f^2*g^2*n^2*
x*log(abs(F))*sgn(F) - 8*I*pi^2*b*c*d*f^2*g^2*n^2*x + 16*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + 16*I*b*c*d*f^2*g
^2*n^2*x*log(abs(F))^2 + 4*I*pi^2*b*c^2*f^2*g^2*n^2*sgn(F) - 8*pi*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 4*I*p
i^2*b*c^2*f^2*g^2*n^2 + 8*pi*b*c^2*f^2*g^2*n^2*log(abs(F)) + 8*I*b*c^2*f^2*g^2*n^2*log(abs(F))^2 + 8*pi*b*d^2*
f*g*n*x*sgn(F) - 8*pi*b*d^2*f*g*n*x - 16*I*b*d^2*f*g*n*x*log(abs(F)) + 8*pi*b*c*d*f*g*n*sgn(F) - 8*pi*b*c*d*f*
g*n - 16*I*b*c*d*f*g*n*log(abs(F)) + 16*I*b*d^2)*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)/(-4*I*pi^3*f^3*g^3*n^3*sgn(F) + 12*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) + 12*I*pi*f^
3*g^3*n^3*log(abs(F))^2*sgn(F) + 4*I*pi^3*f^3*g^3*n^3 - 12*pi^2*f^3*g^3*n^3*log(abs(F)) - 12*I*pi*f^3*g^3*n^3*
log(abs(F))^2 + 8*f^3*g^3*n^3*log(abs(F))^3) - (4*I*pi^2*b*d^2*f^2*g^2*n^2*x^2*sgn(F) + 8*pi*b*d^2*f^2*g^2*n^2
*x^2*log(abs(F))*sgn(F) - 4*I*pi^2*b*d^2*f^2*g^2*n^2*x^2 - 8*pi*b*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 8*I*b*d^2*
f^2*g^2*n^2*x^2*log(abs(F))^2 + 8*I*pi^2*b*c*d*f^2*g^2*n^2*x*sgn(F) + 16*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F))*sg
n(F) - 8*I*pi^2*b*c*d*f^2*g^2*n^2*x - 16*pi*b*c*d*f^2*g^2*n^2*x*log(abs(F)) + 16*I*b*c*d*f^2*g^2*n^2*x*log(abs
(F))^2 + 4*I*pi^2*b*c^2*f^2*g^2*n^2*sgn(F) + 8*pi*b*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 4*I*pi^2*b*c^2*f^2*g^
2*n^2 - 8*pi*b*c^2*f^2*g^2*n^2*log(abs(F)) + 8*I*b*c^2*f^2*g^2*n^2*log(abs(F))^2 - 8*pi*b*d^2*f*g*n*x*sgn(F) +
8*pi*b*d^2*f*g*n*x - 16*I*b*d^2*f*g*n*x*log(abs(F)) - 8*pi*b*c*d*f*g*n*sgn(F) + 8*pi*b*c*d*f*g*n - 16*I*b*c*d
*f*g*n*log(abs(F)) + 16*I*b*d^2)*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)/(4*I*pi^3*f^3*g^3*n^3*sgn(F) + 12*pi^2*f^3*g^3*n^3*log(abs(F))*sgn(F) - 12*I*pi*f^3*g^3*n^3*log(ab
s(F))^2*sgn(F) - 4*I*pi^3*f^3*g^3*n^3 - 12*pi^2*f^3*g^3*n^3*log(abs(F)) + 12*I*pi*f^3*g^3*n^3*log(abs(F))^2 +
8*f^3*g^3*n^3*log(abs(F))^3))*e^(f*g*n*x*log(abs(F)) + g*n*e*log(abs(F)))