3.185 \(\int f^{a+b x^n} x^{-1+n} \, dx\)

Optimal. Leaf size=20 \[ \frac{f^{a+b x^n}}{b n \log (f)} \]

[Out]

f^(a + b*x^n)/(b*n*Log[f])

________________________________________________________________________________________

Rubi [A]  time = 0.0231736, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2209} \[ \frac{f^{a+b x^n}}{b n \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 + n),x]

[Out]

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

Mathematica [A]  time = 0.0042465, size = 20, normalized size = 1. \[ \frac{f^{a+b x^n}}{b n \log (f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^(-1 + n),x]

[Out]

f^(a + b*x^n)/(b*n*Log[f])

________________________________________________________________________________________

Maple [A]  time = 0.022, size = 25, normalized size = 1.3 \begin{align*}{\frac{{{\rm e}^{ \left ( b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) \ln \left ( f \right ) }}}{\ln \left ( f \right ) bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b*x^n)*x^(-1+n),x)

[Out]

1/ln(f)/b/n*exp((b*exp(n*ln(x))+a)*ln(f))

________________________________________________________________________________________

Maxima [A]  time = 1.16261, size = 27, normalized size = 1.35 \begin{align*} \frac{f^{b x^{n} + a}}{b n \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(b*x^n + a)/(b*n*log(f))

________________________________________________________________________________________

Fricas [A]  time = 1.56783, size = 57, normalized size = 2.85 \begin{align*} \frac{e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b n \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(b*x^n*log(f) + a*log(f))/(b*n*log(f))

________________________________________________________________________________________

Sympy [A]  time = 161.134, size = 39, normalized size = 1.95 \begin{align*} \begin{cases} \log{\left (x \right )} & \text{for}\: b = 0 \wedge f = 1 \wedge n = 0 \\f^{a + b} \log{\left (x \right )} & \text{for}\: n = 0 \\\frac{f^{a} x^{n}}{n} & \text{for}\: b = 0 \\\frac{x^{n}}{n} & \text{for}\: f = 1 \\\frac{f^{a} f^{b x^{n}}}{b n \log{\left (f \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b*x**n)*x**(-1+n),x)

[Out]

Piecewise((log(x), Eq(b, 0) & Eq(f, 1) & Eq(n, 0)), (f**(a + b)*log(x), Eq(n, 0)), (f**a*x**n/n, Eq(b, 0)), (x
**n/n, Eq(f, 1)), (f**a*f**(b*x**n)/(b*n*log(f)), True))

________________________________________________________________________________________

Giac [A]  time = 1.25439, size = 27, normalized size = 1.35 \begin{align*} \frac{f^{b x^{n} + a}}{b n \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(b*x^n + a)/(b*n*log(f))