3.136 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^5} \, dx\)

Optimal. Leaf size=44 \[ \frac{f^{a+\frac{b}{x^2}}}{2 b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^2 \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0451656, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac{f^{a+\frac{b}{x^2}}}{2 b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^2 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^5,x]

[Out]

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

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

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^5} \, dx &=-\frac{f^{a+\frac{b}{x^2}}}{2 b x^2 \log (f)}-\frac{\int \frac{f^{a+\frac{b}{x^2}}}{x^3} \, dx}{b \log (f)}\\ &=\frac{f^{a+\frac{b}{x^2}}}{2 b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^2 \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0069059, size = 32, normalized size = 0.73 \[ \frac{f^{a+\frac{b}{x^2}} \left (x^2-b \log (f)\right )}{2 b^2 x^2 \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^5,x]

[Out]

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

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Maple [A]  time = 0.011, size = 52, normalized size = 1.2 \begin{align*}{\frac{1}{{x}^{4}} \left ({\frac{{x}^{4}}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{2}}{2\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^5,x)

[Out]

(1/2/b^2/ln(f)^2*x^4*exp((a+b/x^2)*ln(f))-1/2/b/ln(f)*x^2*exp((a+b/x^2)*ln(f)))/x^4

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Maxima [C]  time = 1.17494, size = 30, normalized size = 0.68 \begin{align*} \frac{f^{a} \Gamma \left (2, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, b^{2} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^5,x, algorithm="maxima")

[Out]

1/2*f^a*gamma(2, -b*log(f)/x^2)/(b^2*log(f)^2)

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Fricas [A]  time = 1.66412, size = 82, normalized size = 1.86 \begin{align*} \frac{{\left (x^{2} - b \log \left (f\right )\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{2 \, b^{2} x^{2} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.122252, size = 29, normalized size = 0.66 \begin{align*} \frac{f^{a + \frac{b}{x^{2}}} \left (- b \log{\left (f \right )} + x^{2}\right )}{2 b^{2} x^{2} \log{\left (f \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)/x**5,x)

[Out]

f**(a + b/x**2)*(-b*log(f) + x**2)/(2*b**2*x**2*log(f)**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^5,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^5, x)