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/ln(f)

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \[ \frac {f^{a+b x^n}}{b n \log (f)} \]

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])

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fricas [A]  time = 0.42, size = 24, normalized size = 1.20 \[ \frac {e^{\left (b x^{n} \log \relax (f) + a \log \relax (f)\right )}}{b n \log \relax (f)} \]

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))

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giac [A]  time = 0.22, size = 20, normalized size = 1.00 \[ \frac {f^{b x^{n} + a}}{b n \log \relax (f)} \]

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))

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maple [A]  time = 0.04, size = 25, normalized size = 1.25 \[ \frac {{\mathrm e}^{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right ) \ln \relax (f )}}{b n \ln \relax (f )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.66, size = 20, normalized size = 1.00 \[ \frac {f^{b x^{n} + a}}{b n \log \relax (f)} \]

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))

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mupad [B]  time = 3.50, size = 20, normalized size = 1.00 \[ \frac {f^{a+b\,x^n}}{b\,n\,\ln \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n)*x^(n - 1),x)

[Out]

f^(a + b*x^n)/(b*n*log(f))

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sympy [A]  time = 67.28, size = 39, normalized size = 1.95 \[ \begin {cases} \log {\relax (x )} & \text {for}\: b = 0 \wedge f = 1 \wedge n = 0 \\f^{a + b} \log {\relax (x )} & \text {for}\: n = 0 \\\frac {f^{a} x^{n}}{n} & \text {for}\: b = 0 \\\frac {x^{n}}{n} & \text {for}\: f = 1 \\\frac {f^{a} f^{b x^{n}}}{b n \log {\relax (f )}} & \text {otherwise} \end {cases} \]

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))

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