3.85 \(\int \frac{F^{c+d x}}{(a+b F^{c+d x})^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*Log[F]))

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Rubi [A]  time = 0.0352445, antiderivative size = 25, 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, 32} \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[F^(c + d*x)/(a + b*F^(c + d*x))^2,x]

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+b x)^2} \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=-\frac{1}{b d \left (a+b F^{c+d x}\right ) \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0198392, size = 25, normalized size = 1. \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c + d*x)/(a + b*F^(c + d*x))^2,x]

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*Log[F]))

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Maple [A]  time = 0.003, size = 26, normalized size = 1. \begin{align*} -{\frac{1}{bd \left ( a+b{F}^{dx+c} \right ) \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.1812, size = 34, normalized size = 1.36 \begin{align*} -\frac{1}{{\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))^2,x, algorithm="maxima")

[Out]

-1/((F^(d*x + c)*b + a)*b*d*log(F))

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Fricas [A]  time = 1.47666, size = 62, normalized size = 2.48 \begin{align*} -\frac{1}{F^{d x + c} b^{2} d \log \left (F\right ) + a 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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.115061, size = 26, normalized size = 1.04 \begin{align*} - \frac{1}{F^{c + d x} b^{2} d \log{\left (F \right )} + a 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))**2,x)

[Out]

-1/(F**(c + d*x)*b**2*d*log(F) + a*b*d*log(F))

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Giac [A]  time = 1.27317, size = 34, normalized size = 1.36 \begin{align*} -\frac{1}{{\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))^2,x, algorithm="giac")

[Out]

-1/((F^(d*x + c)*b + a)*b*d*log(F))