3.180 \(\int \frac{f^{a+b x^n}}{x^2} \, dx\)

Optimal. Leaf size=37 \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]

[Out]

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

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Rubi [A]  time = 0.0238832, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)/x^2,x]

[Out]

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

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{f^{a+b x^n}}{x^2} \, dx &=-\frac{f^a \Gamma \left (-\frac{1}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{\frac{1}{n}}}{n x}\\ \end{align*}

Mathematica [A]  time = 0.0038872, size = 37, normalized size = 1. \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)/x^2,x]

[Out]

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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{a+b{x}^{n}}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20209, size = 50, normalized size = 1.35 \begin{align*} -\frac{\left (-b x^{n} \log \left (f\right )\right )^{\left (\frac{1}{n}\right )} f^{a} \Gamma \left (-\frac{1}{n}, -b x^{n} \log \left (f\right )\right )}{n x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{f^{b x^{n} + a}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b*x^n + a)/x^2, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b*x**n)/x**2,x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{n} + a}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a)/x^2, x)