3.135 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^3} \, dx\)

Optimal. Leaf size=20 \[ -\frac{f^{a+\frac{b}{x^2}}}{2 b \log (f)} \]

[Out]

-f^(a + b/x^2)/(2*b*Log[f])

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Rubi [A]  time = 0.0214927, antiderivative size = 20, 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 = {2209} \[ -\frac{f^{a+\frac{b}{x^2}}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^3,x]

[Out]

-f^(a + b/x^2)/(2*b*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 \frac{f^{a+\frac{b}{x^2}}}{x^3} \, dx &=-\frac{f^{a+\frac{b}{x^2}}}{2 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0037513, size = 20, normalized size = 1. \[ -\frac{f^{a+\frac{b}{x^2}}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^3,x]

[Out]

-f^(a + b/x^2)/(2*b*Log[f])

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{2\,b\ln \left ( f \right ) }{f}^{a+{\frac{b}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^3,x)

[Out]

-1/2*f^(a+b/x^2)/b/ln(f)

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Maxima [A]  time = 1.00888, size = 24, normalized size = 1.2 \begin{align*} -\frac{f^{a + \frac{b}{x^{2}}}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^3,x, algorithm="maxima")

[Out]

-1/2*f^(a + b/x^2)/(b*log(f))

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Fricas [A]  time = 1.76488, size = 50, normalized size = 2.5 \begin{align*} -\frac{f^{\frac{a x^{2} + b}{x^{2}}}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^3,x, algorithm="fricas")

[Out]

-1/2*f^((a*x^2 + b)/x^2)/(b*log(f))

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Sympy [A]  time = 0.11768, size = 29, normalized size = 1.45 \begin{align*} \begin{cases} - \frac{f^{a + \frac{b}{x^{2}}}}{2 b \log{\left (f \right )}} & \text{for}\: 2 b \log{\left (f \right )} \neq 0 \\- \frac{1}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)/x**3,x)

[Out]

Piecewise((-f**(a + b/x**2)/(2*b*log(f)), Ne(2*b*log(f), 0)), (-1/(2*x**2), True))

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Giac [A]  time = 1.73057, size = 24, normalized size = 1.2 \begin{align*} -\frac{f^{a + \frac{b}{x^{2}}}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^3,x, algorithm="giac")

[Out]

-1/2*f^(a + b/x^2)/(b*log(f))