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]

Log[a + b*F^(c + d*x)]/(b*d*Log[F])

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Rubi [A]  time = 0.0358378, antiderivative size = 23, normalized size of antiderivative = 1., 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 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]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 = 0.0052869, size = 23, normalized size = 1. \[ \frac{\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

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])

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Maple [A]  time = 0., size = 24, normalized size = 1. \begin{align*}{\frac{\ln \left ( a+b{F}^{dx+c} \right ) }{bd\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03403, size = 31, normalized size = 1.35 \begin{align*} \frac{\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \end{align*}

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))

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Fricas [A]  time = 1.50509, size = 50, normalized size = 2.17 \begin{align*} \frac{\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \end{align*}

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))

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Sympy [A]  time = 0.181558, size = 17, normalized size = 0.74 \begin{align*} \frac{\log{\left (F^{c + d x} + \frac{a}{b} \right )}}{b d \log{\left (F \right )}} \end{align*}

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))

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Giac [A]  time = 1.22573, size = 32, normalized size = 1.39 \begin{align*} \frac{\log \left ({\left | F^{d x + c} b + a \right |}\right )}{b d \log \left (F\right )} \end{align*}

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))