\(\int \frac {a+b (F^{g (e+f x)})^n}{c+d x} \, dx\) [29]

Optimal result

Integrand size = 23, antiderivative size = 68 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {b F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2214, 2213, 2209} \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {a \log (c+d x)}{d}+\frac {b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d} \]

[In]

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

[Out]

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2213

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*F^(g*(e +
f*x)))^n/F^(g*n*(e + f*x)), Int[(c + d*x)^m*F^(g*n*(e + f*x)), x], x] /; FreeQ[{F, b, c, d, e, f, g, m, n}, x]

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]

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {b F^{-\frac {f g n (c+d x)}{d}} \left (F^{g (e+f x)}\right )^n \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )+a \log (c+d x)}{d} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {a +b \left (F^{g \left (f x +e \right )}\right )^{n}}{d x +c}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {F^{\frac {{\left (d e - c f\right )} g n}{d}} b {\rm Ei}\left (\frac {{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + a \log \left (d x + c\right )}{d} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int \frac {a + b \left (F^{e g + f g x}\right )^{n}}{c + d x}\, dx \]

[In]

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

[Out]

Integral((a + b*(F**(e*g + f*g*x))**n)/(c + d*x), x)

Maxima [F]

\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int { \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int { \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int \frac {a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n}{c+d\,x} \,d x \]

[In]

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

[Out]

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