\(\int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx\) [123]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\frac {f^{a+\frac {b}{x}}}{b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x \log (f)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\frac {f^{a+\frac {b}{x}}}{b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x \log (f)} \]

[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 2240

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]

Rule 2243

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])

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\frac {f^{a+\frac {b}{x}} (x-b \log (f))}{b^2 x \log ^2(f)} \]

[In]

Integrate[f^(a + b/x)/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {\left (b \ln \left (f \right )-x \right ) f^{\frac {a x +b}{x}}}{\ln \left (f \right )^{2} b^{2} x}\)	\(32\)
meijerg	\(-\frac {f^{a} \left (1-\frac {\left (2-\frac {2 b \ln \left (f \right )}{x}\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{2}\right )}{\ln \left (f \right )^{2} b^{2}}\)	\(35\)
parallelrisch	\(\frac {-f^{a +\frac {b}{x}} b \ln \left (f \right )+f^{a +\frac {b}{x}} x}{x \ln \left (f \right )^{2} b^{2}}\)	\(38\)
norman	\(\frac {\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b \ln \left (f \right )}}{x^{2}}\)	\(49\)

[In]

int(f^(a+b/x)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=-\frac {{\left (b \log \left (f\right ) - x\right )} f^{\frac {a x + b}{x}}}{b^{2} x \log \left (f\right )^{2}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.56 \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\frac {f^{a + \frac {b}{x}} \left (- b \log {\left (f \right )} + x\right )}{b^{2} x \log {\left (f \right )}^{2}} \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.54 \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\frac {f^{a} \Gamma \left (2, -\frac {b \log \left (f\right )}{x}\right )}{b^{2} \log \left (f\right )^{2}} \]

[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)

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\int { \frac {f^{a + \frac {b}{x}}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(f^(a + b/x)/x^3, x)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx=\frac {f^{a+\frac {b}{x}}\,\left (x-b\,\ln \left (f\right )\right )}{b^2\,x\,{\ln \left (f\right )}^2} \]

[In]

int(f^(a + b/x)/x^3,x)

[Out]

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