3.373 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx\)

Optimal. Leaf size=27 \[ \frac{F^{a+b (c+d x)^n}}{b d n \log (F)} \]

[Out]

F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])

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Rubi [A]  time = 0.037157, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2209} \[ \frac{F^{a+b (c+d x)^n}}{b d n \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + n),x]

[Out]

F^(a + b*(c + d*x)^n)/(b*d*n*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 F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx &=\frac{F^{a+b (c+d x)^n}}{b d n \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0072523, size = 27, normalized size = 1. \[ \frac{F^{a+b (c+d x)^n}}{b d n \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + n),x]

[Out]

F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])

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Maple [A]  time = 0.034, size = 32, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{ \left ( a+b{{\rm e}^{n\ln \left ( dx+c \right ) }} \right ) \ln \left ( F \right ) }}}{\ln \left ( F \right ) bdn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x)

[Out]

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

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Maxima [A]  time = 0.968427, size = 36, normalized size = 1.33 \begin{align*} \frac{F^{{\left (d x + c\right )}^{n} b + a}}{b d n \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x, algorithm="maxima")

[Out]

F^((d*x + c)^n*b + a)/(b*d*n*log(F))

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Fricas [A]  time = 1.53083, size = 70, normalized size = 2.59 \begin{align*} \frac{e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b d n \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1+n),x)

[Out]

Timed out

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Giac [A]  time = 1.40768, size = 36, normalized size = 1.33 \begin{align*} \frac{F^{{\left (d x + c\right )}^{n} b + a}}{b d n \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+n),x, algorithm="giac")

[Out]

F^((d*x + c)^n*b + a)/(b*d*n*log(F))