∫ (a + b(Fg(e+fx))n)(c + dx) dx [27]

Optimal. Leaf size=77 \[ \frac {a (c+d x)^2}{2 d}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)} \]

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

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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.143, Rules used = {2214, 2207, 2225} \begin {gather*} \frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \end {gather*}

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

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]

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},

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

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

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Mathematica [A]
time = 0.20, size = 73, normalized size = 0.95 \begin {gather*} \frac {1}{2} a x (2 c+d x)-\frac {b d \left (F^{g (e+f x)}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{g (e+f x)}\right )^n (c+d x)}{f g n \log (F)} \end {gather*}

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

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method	result	size

norman	$c a x +\frac {b \left (\ln \left (F \right ) c f g n -d \right ) {\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{n^{2} g^{2} f^{2} \ln \left (F \right )^{2}}+\frac {b d x \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{n g f \ln \left (F \right )}+\frac {a d \,x^{2}}{2}$	$84$

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

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Maxima [A]
time = 0.30, size = 88, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, a d x^{2} + a c x + \frac {F^{f g n x + g n e} b c}{f g n \log \left (F\right )} + \frac {{\left (F^{g n e} f g n x \log \left (F\right ) - F^{g n e}\right )} F^{f g n x} b d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \end {gather*}

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

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Fricas [A]
time = 0.35, size = 88, normalized size = 1.14 \begin {gather*} \frac {{\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} - 2 \, {\left (b d - {\left (b d f g n x + b c f g n\right )} \log \left (F\right )\right )} F^{f g n x + g n e}}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \end {gather*}

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

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Sympy [A]
time = 0.06, size = 94, normalized size = 1.22 \begin {gather*} a c x + \frac {a d x^{2}}{2} + \begin {cases} \frac {\left (b c f g n \log {\left (F \right )} + b d f g n x \log {\left (F \right )} - b d\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} & \text {for}\: f^{2} g^{2} n^{2} \log {\left (F \right )}^{2} \neq 0 \\b c x + \frac {b d x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [C] Result contains complex when optimal does not.
time = 3.43, size = 1111, normalized size = 14.43 \begin {gather*} \frac {1}{2} \, a d x^{2} + a c x + {\left (2 \, {\left (\frac {{\left (\pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )\right )} {\left (\pi b d f g n x \mathrm {sgn}\left (F\right ) - \pi b d f g n x + \pi b c f g n \mathrm {sgn}\left (F\right ) - \pi b c f g n\right )}}{{\left (\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}\right )}^{2} + 4 \, {\left (\pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )\right )}^{2}} + \frac {{\left (\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}\right )} {\left (b d f g n x \log \left ({\left | F \right |}\right ) + b c f g n \log \left ({\left | F \right |}\right ) - b d\right )}}{{\left (\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}\right )}^{2} + 4 \, {\left (\pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )\right )}^{2}}\right )} \cos \left (-\frac {1}{2} \, \pi f g n x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi f g n x - \frac {1}{2} \, \pi g n e \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi g n e\right ) + {\left (\frac {{\left (\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}\right )} {\left (\pi b d f g n x \mathrm {sgn}\left (F\right ) - \pi b d f g n x + \pi b c f g n \mathrm {sgn}\left (F\right ) - \pi b c f g n\right )}}{{\left (\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}\right )}^{2} + 4 \, {\left (\pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )\right )}^{2}} - \frac {4 \, {\left (\pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )\right )} {\left (b d f g n x \log \left ({\left | F \right |}\right ) + b c f g n \log \left ({\left | F \right |}\right ) - b d\right )}}{{\left (\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}\right )}^{2} + 4 \, {\left (\pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )\right )}^{2}}\right )} \sin \left (-\frac {1}{2} \, \pi f g n x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi f g n x - \frac {1}{2} \, \pi g n e \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi g n e\right )\right )} e^{\left (f g n x \log \left ({\left | F \right |}\right ) + g n e \log \left ({\left | F \right |}\right )\right )} - \frac {1}{2} i \, {\left (\frac {{\left (\pi b d f g n x \mathrm {sgn}\left (F\right ) - \pi b d f g n x - 2 i \, b d f g n x \log \left ({\left | F \right |}\right ) + \pi b c f g n \mathrm {sgn}\left (F\right ) - \pi b c f g n - 2 i \, b c f g n \log \left ({\left | F \right |}\right ) + 2 i \, b d\right )} e^{\left (\frac {1}{2} i \, \pi f g n x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi f g n x + \frac {1}{2} i \, \pi g n e \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi g n e\right )}}{\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) + 2 i \, \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} - 2 i \, \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}} + \frac {{\left (\pi b d f g n x \mathrm {sgn}\left (F\right ) - \pi b d f g n x + 2 i \, b d f g n x \log \left ({\left | F \right |}\right ) + \pi b c f g n \mathrm {sgn}\left (F\right ) - \pi b c f g n + 2 i \, b c f g n \log \left ({\left | F \right |}\right ) - 2 i \, b d\right )} e^{\left (-\frac {1}{2} i \, \pi f g n x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi f g n x - \frac {1}{2} i \, \pi g n e \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi g n e\right )}}{\pi ^{2} f^{2} g^{2} n^{2} \mathrm {sgn}\left (F\right ) - 2 i \, \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) \mathrm {sgn}\left (F\right ) - \pi ^{2} f^{2} g^{2} n^{2} + 2 i \, \pi f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right ) + 2 \, f^{2} g^{2} n^{2} \log \left ({\left | F \right |}\right )^{2}}\right )} e^{\left (f g n x \log \left ({\left | F \right |}\right ) + g n e \log \left ({\left | F \right |}\right )\right )} \end {gather*}

1/2*a*d*x^2 + a*c*x + (2*((pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi*f^2*g^2*n^2*log(abs(F)))*(pi*b*d*f*g*n*x*sgn

(F) - pi*b*d*f*g*n*x + pi*b*c*f*g*n*sgn(F) - pi*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) + (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)*(b*d*f*g*n*x*log(abs(F)) + b*c*f*g*n*log(abs

(F)) - 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))*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*b*d

*f*g*n*x*sgn(F) - pi*b*d*f*g*n*x + pi*b*c*f*g*n*sgn(F) - pi*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)))*(b*d*f*g*n*x*log(abs(F)) + b*c*f*g*n*log

(abs(F)) - 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*((pi*b*d*f*g*n*x*sgn(F) - pi

*b*d*f*g*n*x - 2*I*b*d*f*g*n*x*log(abs(F)) + pi*b*c*f*g*n*sgn(F) - pi*b*c*f*g*n - 2*I*b*c*f*g*n*log(abs(F)) +

2*I*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)/(pi^2*f^2*g^2

*n^2*sgn(F) + 2*I*pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - pi^2*f^2*g^2*n^2 - 2*I*pi*f^2*g^2*n^2*log(abs(F)) + 2*f^

2*g^2*n^2*log(abs(F))^2) + (pi*b*d*f*g*n*x*sgn(F) - pi*b*d*f*g*n*x + 2*I*b*d*f*g*n*x*log(abs(F)) + pi*b*c*f*g*

n*sgn(F) - pi*b*c*f*g*n + 2*I*b*c*f*g*n*log(abs(F)) - 2*I*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)/(pi^2*f^2*g^2*n^2*sgn(F) - 2*I*pi*f^2*g^2*n^2*log(abs(F))*sgn(F) - p

i^2*f^2*g^2*n^2 + 2*I*pi*f^2*g^2*n^2*log(abs(F)) + 2*f^2*g^2*n^2*log(abs(F))^2))*e^(f*g*n*x*log(abs(F)) + g*n*

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Mupad [B]
time = 3.60, size = 72, normalized size = 0.94 \begin {gather*} a\,c\,x-\left (\frac {b\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}-\frac {b\,d\,x}{f\,g\,n\,\ln \left (F\right )}\right )\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n+\frac {a\,d\,x^2}{2} \end {gather*}

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

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3.1.27 \(\int (a+b (F^{g (e+f x)})^n) (c+d x) \, dx\) [27]