3.124 \(\int \frac{f^{a+\frac{b}{x}}}{x^4} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 f^{a+\frac{b}{x}}}{b^2 x \log ^2(f)}-\frac{2 f^{a+\frac{b}{x}}}{b^3 \log ^3(f)}-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)} \]

[Out]

(-2*f^(a + b/x))/(b^3*Log[f]^3) + (2*f^(a + b/x))/(b^2*x*Log[f]^2) - f^(a + b/x)/(b*x^2*Log[f])

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Rubi [A]  time = 0.0604829, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac{2 f^{a+\frac{b}{x}}}{b^2 x \log ^2(f)}-\frac{2 f^{a+\frac{b}{x}}}{b^3 \log ^3(f)}-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x^4,x]

[Out]

(-2*f^(a + b/x))/(b^3*Log[f]^3) + (2*f^(a + b/x))/(b^2*x*Log[f]^2) - f^(a + b/x)/(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}}}{x^4} \, dx &=-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)}-\frac{2 \int \frac{f^{a+\frac{b}{x}}}{x^3} \, dx}{b \log (f)}\\ &=\frac{2 f^{a+\frac{b}{x}}}{b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)}+\frac{2 \int \frac{f^{a+\frac{b}{x}}}{x^2} \, dx}{b^2 \log ^2(f)}\\ &=-\frac{2 f^{a+\frac{b}{x}}}{b^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x}}}{b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0081107, size = 41, normalized size = 0.67 \[ -\frac{f^{a+\frac{b}{x}} \left (b^2 \log ^2(f)-2 b x \log (f)+2 x^2\right )}{b^3 x^2 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)/x^4,x]

[Out]

-((f^(a + b/x)*(2*x^2 - 2*b*x*Log[f] + b^2*Log[f]^2))/(b^3*x^2*Log[f]^3))

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Maple [A]  time = 0.012, size = 73, normalized size = 1.2 \begin{align*}{\frac{1}{{x}^{3}} \left ( -2\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+2\,{\frac{{x}^{2}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-{\frac{x}{b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x)/x^4,x)

[Out]

(-2/b^3/ln(f)^3*x^3*exp((a+b/x)*ln(f))+2/b^2/ln(f)^2*x^2*exp((a+b/x)*ln(f))-1/b/ln(f)*x*exp((a+b/x)*ln(f)))/x^
3

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Maxima [C]  time = 1.27157, size = 30, normalized size = 0.49 \begin{align*} -\frac{f^{a} \Gamma \left (3, -\frac{b \log \left (f\right )}{x}\right )}{b^{3} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)/x^4,x, algorithm="maxima")

[Out]

-f^a*gamma(3, -b*log(f)/x)/(b^3*log(f)^3)

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Fricas [A]  time = 1.72028, size = 101, normalized size = 1.66 \begin{align*} -\frac{{\left (b^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right ) + 2 \, x^{2}\right )} f^{\frac{a x + b}{x}}}{b^{3} x^{2} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)/x^4,x, algorithm="fricas")

[Out]

-(b^2*log(f)^2 - 2*b*x*log(f) + 2*x^2)*f^((a*x + b)/x)/(b^3*x^2*log(f)^3)

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Sympy [A]  time = 0.133591, size = 39, normalized size = 0.64 \begin{align*} \frac{f^{a + \frac{b}{x}} \left (- b^{2} \log{\left (f \right )}^{2} + 2 b x \log{\left (f \right )} - 2 x^{2}\right )}{b^{3} x^{2} \log{\left (f \right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x)/x**4,x)

[Out]

f**(a + b/x)*(-b**2*log(f)**2 + 2*b*x*log(f) - 2*x**2)/(b**3*x**2*log(f)**3)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x}}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)/x^4,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)/x^4, x)