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

Optimal. Leaf size=38 \[ \frac{b f^a \log (f) \text{Ei}\left (b x^n \log (f)\right )}{n}-\frac{x^{-n} f^{a+b x^n}}{n} \]

[Out]

-(f^(a + b*x^n)/(n*x^n)) + (b*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f])/n

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Rubi [A]  time = 0.0486405, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2215, 2210} \[ \frac{b f^a \log (f) \text{Ei}\left (b x^n \log (f)\right )}{n}-\frac{x^{-n} f^{a+b x^n}}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 - n),x]

[Out]

-(f^(a + b*x^n)/(n*x^n)) + (b*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f])/n

Rule 2215

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^Simplify[m + n]*F^(a +
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simpl
ify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, n]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && 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} x^{-n}}{n}+(b \log (f)) \int \frac{f^{a+b x^n}}{x} \, dx\\ &=-\frac{f^{a+b x^n} x^{-n}}{n}+\frac{b f^a \text{Ei}\left (b x^n \log (f)\right ) \log (f)}{n}\\ \end{align*}

Mathematica [A]  time = 0.0035926, size = 20, normalized size = 0.53 \[ \frac{b f^a \log (f) \text{Gamma}\left (-1,-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^(-1 - n),x]

[Out]

(b*f^a*Gamma[-1, -(b*x^n*Log[f])]*Log[f])/n

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Maple [A]  time = 0.09, size = 43, normalized size = 1.1 \begin{align*} -{\frac{{f}^{b{x}^{n}}{f}^{a}}{n{x}^{n}}}-{\frac{b\ln \left ( f \right ){f}^{a}{\it Ei} \left ( 1,-b{x}^{n}\ln \left ( f \right ) \right ) }{n}} \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/n*f^(b*x^n)*f^a/(x^n)-1/n*ln(f)*b*f^a*Ei(1,-b*x^n*ln(f))

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Maxima [A]  time = 1.22318, size = 27, normalized size = 0.71 \begin{align*} \frac{b f^{a} \Gamma \left (-1, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )}{n} \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]

b*f^a*gamma(-1, -b*x^n*log(f))*log(f)/n

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Fricas [A]  time = 1.57912, size = 101, normalized size = 2.66 \begin{align*} \frac{b f^{a} x^{n}{\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right ) - e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{n x^{n}} \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]

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

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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*x**n)*x**(-1-n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{-n - 1}\,{d x} \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]

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