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

Optimal. Leaf size=23 \[ \frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2246, 31} \[ \frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

[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 2246

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*} \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx &=\frac {\operatorname {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]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

[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])

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 23, normalized size = 1.00 \[ \frac {\log \left (F^{d x + c} b + a\right )}{b d \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [A]  time = 0.41, size = 24, normalized size = 1.04 \[ \frac {\log \left ({\left | F^{d x + c} b + a \right |}\right )}{b d \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 24, normalized size = 1.04 \[ \frac {\ln \left (b \,F^{d x +c}+a \right )}{b d \ln \relax (F )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 23, normalized size = 1.00 \[ \frac {\log \left (F^{d x + c} b + a\right )}{b d \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

mupad [B]  time = 0.00, size = 23, normalized size = 1.00 \[ \frac {\ln \left (a+F^{c+d\,x}\,b\right )}{b\,d\,\ln \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 17, normalized size = 0.74 \[ \frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{b d \log {\relax (F )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________