\(\int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx\) [10]

Optimal result

Integrand size = 21, antiderivative size = 23 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

[Out]

ln(a+b*F^(d*x+c))/b/d/ln(F)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2278, 31} \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

[In]

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

[Out]

Log[a + b*F^(c + d*x)]/(b*d*Log[F])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2278

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,F^{c+d x}\right )}{d \log (F)} \\ & = \frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

[In]

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

[Out]

Log[a + b*F^(c + d*x)]/(b*d*Log[F])

Maple [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {\ln \left (a +b \,F^{d x +c}\right )}{b d \ln \left (F \right )}\)	\(24\)
default	\(\frac {\ln \left (a +b \,F^{d x +c}\right )}{b d \ln \left (F \right )}\)	\(24\)
parallelrisch	\(\frac {\ln \left (a +b \,F^{d x +c}\right )}{b d \ln \left (F \right )}\)	\(24\)
norman	\(\frac {\ln \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}{b d \ln \left (F \right )}\)	\(26\)
risch	\(-\frac {c}{d b}+\frac {\ln \left (F^{d x +c}+\frac {a}{b}\right )}{b d \ln \left (F \right )}\)	\(36\)

[In]

int(F^(d*x+c)/(a+b*F^(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

ln(a+b*F^(d*x+c))/b/d/ln(F)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \]

[In]

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

[Out]

log(F^(d*x + c)*b + a)/(b*d*log(F))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{b d \log {\left (F \right )}} \]

[In]

integrate(F**(d*x+c)/(a+b*F**(d*x+c)),x)

[Out]

log(F**(c + d*x) + a/b)/(b*d*log(F))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \]

[In]

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

[Out]

log(F^(d*x + c)*b + a)/(b*d*log(F))

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\log \left ({\left | F^{d x + c} b + a \right |}\right )}{b d \log \left (F\right )} \]

[In]

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

[Out]

log(abs(F^(d*x + c)*b + a))/(b*d*log(F))

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx=\frac {\ln \left (a+F^{c+d\,x}\,b\right )}{b\,d\,\ln \left (F\right )} \]

[In]

int(F^(c + d*x)/(a + F^(c + d*x)*b),x)

[Out]

log(a + F^(c + d*x)*b)/(b*d*log(F))