3.188 \(\int f^{a+b x^n} x^{-1-2 n} \, dx\)

Optimal. Leaf size=71 \[ \frac{b^2 f^a \log ^2(f) \text{Ei}\left (b x^n \log (f)\right )}{2 n}-\frac{x^{-2 n} f^{a+b x^n}}{2 n}-\frac{b \log (f) x^{-n} f^{a+b x^n}}{2 n} \]

[Out]

-f^(a + b*x^n)/(2*n*x^(2*n)) - (b*f^(a + b*x^n)*Log[f])/(2*n*x^n) + (b^2*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f
]^2)/(2*n)

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Rubi [A]  time = 0.0752343, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2215, 2210} \[ \frac{b^2 f^a \log ^2(f) \text{Ei}\left (b x^n \log (f)\right )}{2 n}-\frac{x^{-2 n} f^{a+b x^n}}{2 n}-\frac{b \log (f) x^{-n} f^{a+b x^n}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 - 2*n),x]

[Out]

-f^(a + b*x^n)/(2*n*x^(2*n)) - (b*f^(a + b*x^n)*Log[f])/(2*n*x^n) + (b^2*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f
]^2)/(2*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^{-1-2 n} \, dx &=-\frac{f^{a+b x^n} x^{-2 n}}{2 n}+\frac{1}{2} (b \log (f)) \int f^{a+b x^n} x^{-1-n} \, dx\\ &=-\frac{f^{a+b x^n} x^{-2 n}}{2 n}-\frac{b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac{1}{2} \left (b^2 \log ^2(f)\right ) \int \frac{f^{a+b x^n}}{x} \, dx\\ &=-\frac{f^{a+b x^n} x^{-2 n}}{2 n}-\frac{b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac{b^2 f^a \text{Ei}\left (b x^n \log (f)\right ) \log ^2(f)}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.0042671, size = 25, normalized size = 0.35 \[ -\frac{b^2 f^a \log ^2(f) \text{Gamma}\left (-2,-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^(-1 - 2*n),x]

[Out]

-((b^2*f^a*Gamma[-2, -(b*x^n*Log[f])]*Log[f]^2)/n)

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Maple [A]  time = 0.098, size = 70, normalized size = 1. \begin{align*} -{\frac{{f}^{b{x}^{n}}{f}^{a}}{2\,n \left ({x}^{n} \right ) ^{2}}}-{\frac{b\ln \left ( f \right ){f}^{b{x}^{n}}{f}^{a}}{2\,n{x}^{n}}}-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{a}{\it Ei} \left ( 1,-b{x}^{n}\ln \left ( f \right ) \right ) }{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b*x^n)*x^(-1-2*n),x)

[Out]

-1/2/n*f^(b*x^n)*f^a/(x^n)^2-1/2/n*ln(f)*b*f^(b*x^n)*f^a/(x^n)-1/2/n*ln(f)^2*b^2*f^a*Ei(1,-b*x^n*ln(f))

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Maxima [A]  time = 1.23282, size = 34, normalized size = 0.48 \begin{align*} -\frac{b^{2} f^{a} \Gamma \left (-2, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )^{2}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1-2*n),x, algorithm="maxima")

[Out]

-b^2*f^a*gamma(-2, -b*x^n*log(f))*log(f)^2/n

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Fricas [A]  time = 1.49937, size = 149, normalized size = 2.1 \begin{align*} \frac{b^{2} f^{a} x^{2 \, n}{\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right )^{2} -{\left (b x^{n} \log \left (f\right ) + 1\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1-2*n),x, algorithm="fricas")

[Out]

1/2*(b^2*f^a*x^(2*n)*Ei(b*x^n*log(f))*log(f)^2 - (b*x^n*log(f) + 1)*e^(b*x^n*log(f) + a*log(f)))/(n*x^(2*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**(-1-2*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^{-2 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b*x^n)*x^(-1-2*n),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^(-2*n - 1), x)