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

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

[Out]

-(F^(a + b/(c + d*x))/(b*d*Log[F]))

________________________________________________________________________________________

Rubi [A]  time = 0.0423626, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2209} \[ -\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(F^(a + b/(c + d*x))/(b*d*Log[F]))

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.0065504, size = 25, normalized size = 1. \[ -\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(F^(a + b/(c + d*x))/(b*d*Log[F]))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c))/(d*x+c)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.999679, size = 34, normalized size = 1.36 \begin{align*} -\frac{F^{a + \frac{b}{d x + c}}}{b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))/(d*x+c)^2,x, algorithm="maxima")

[Out]

-F^(a + b/(d*x + c))/(b*d*log(F))

________________________________________________________________________________________

Fricas [A]  time = 1.54189, size = 63, normalized size = 2.52 \begin{align*} -\frac{F^{\frac{a d x + a c + b}{d x + c}}}{b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.371855, size = 34, normalized size = 1.36 \begin{align*} \begin{cases} - \frac{F^{a + \frac{b}{c + d x}}}{b d \log{\left (F \right )}} & \text{for}\: b d \log{\left (F \right )} \neq 0 \\- \frac{1}{c d + d^{2} x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c))/(d*x+c)**2,x)

[Out]

Piecewise((-F**(a + b/(c + d*x))/(b*d*log(F)), Ne(b*d*log(F), 0)), (-1/(c*d + d**2*x), True))

________________________________________________________________________________________

Giac [A]  time = 1.34049, size = 34, normalized size = 1.36 \begin{align*} -\frac{F^{a + \frac{b}{d x + c}}}{b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))/(d*x+c)^2,x, algorithm="giac")

[Out]

-F^(a + b/(d*x + c))/(b*d*log(F))