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

Optimal. Leaf size=39 \[ \frac{f^{a+\frac{b}{x}}}{b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0364967, antiderivative size = 39, 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} \[ \frac{f^{a+\frac{b}{x}}}{b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0060578, size = 27, normalized size = 0.69 \[ \frac{f^{a+\frac{b}{x}} (x-b \log (f))}{b^2 x \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)/x^3,x]

[Out]

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

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Maple [A]  time = 0.008, size = 49, normalized size = 1.3 \begin{align*}{\frac{1}{{x}^{2}} \left ({\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^3,x)

[Out]

(1/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^2

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Maxima [C]  time = 1.19299, size = 28, normalized size = 0.72 \begin{align*} \frac{f^{a} \Gamma \left (2, -\frac{b \log \left (f\right )}{x}\right )}{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)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.89046, size = 68, normalized size = 1.74 \begin{align*} -\frac{{\left (b \log \left (f\right ) - x\right )} f^{\frac{a x + b}{x}}}{b^{2} x \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.11709, size = 22, normalized size = 0.56 \begin{align*} \frac{f^{a + \frac{b}{x}} \left (- b \log{\left (f \right )} + x\right )}{b^{2} x \log{\left (f \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x)/x^3, x)