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

Optimal. Leaf size=82 \[ \frac {6 f^{a+\frac {b}{x}}}{b^4 \log ^4(f)}-\frac {6 f^{a+\frac {b}{x}}}{b^3 x \log ^3(f)}+\frac {3 f^{a+\frac {b}{x}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)} \]

[Out]

6*f^(a+b/x)/b^4/ln(f)^4-6*f^(a+b/x)/b^3/x/ln(f)^3+3*f^(a+b/x)/b^2/x^2/ln(f)^2-f^(a+b/x)/b/x^3/ln(f)

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Rubi [A]  time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac {3 f^{a+\frac {b}{x}}}{b^2 x^2 \log ^2(f)}-\frac {6 f^{a+\frac {b}{x}}}{b^3 x \log ^3(f)}+\frac {6 f^{a+\frac {b}{x}}}{b^4 \log ^4(f)}-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x^5,x]

[Out]

(6*f^(a + b/x))/(b^4*Log[f]^4) - (6*f^(a + b/x))/(b^3*x*Log[f]^3) + (3*f^(a + b/x))/(b^2*x^2*Log[f]^2) - f^(a
+ b/x)/(b*x^3*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]

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])

Rubi steps

\begin {align*} \int \frac {f^{a+\frac {b}{x}}}{x^5} \, dx &=-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)}-\frac {3 \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx}{b \log (f)}\\ &=\frac {3 f^{a+\frac {b}{x}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)}+\frac {6 \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {6 f^{a+\frac {b}{x}}}{b^3 x \log ^3(f)}+\frac {3 f^{a+\frac {b}{x}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)}-\frac {6 \int \frac {f^{a+\frac {b}{x}}}{x^2} \, dx}{b^3 \log ^3(f)}\\ &=\frac {6 f^{a+\frac {b}{x}}}{b^4 \log ^4(f)}-\frac {6 f^{a+\frac {b}{x}}}{b^3 x \log ^3(f)}+\frac {3 f^{a+\frac {b}{x}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 0.65 \[ \frac {f^{a+\frac {b}{x}} \left (-b^3 \log ^3(f)+3 b^2 x \log ^2(f)-6 b x^2 \log (f)+6 x^3\right )}{b^4 x^3 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)/x^5,x]

[Out]

(f^(a + b/x)*(6*x^3 - 6*b*x^2*Log[f] + 3*b^2*x*Log[f]^2 - b^3*Log[f]^3))/(b^4*x^3*Log[f]^4)

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fricas [A]  time = 0.42, size = 55, normalized size = 0.67 \[ -\frac {{\left (b^{3} \log \relax (f)^{3} - 3 \, b^{2} x \log \relax (f)^{2} + 6 \, b x^{2} \log \relax (f) - 6 \, x^{3}\right )} f^{\frac {a x + b}{x}}}{b^{4} x^{3} \log \relax (f)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^3*log(f)^3 - 3*b^2*x*log(f)^2 + 6*b*x^2*log(f) - 6*x^3)*f^((a*x + b)/x)/(b^4*x^3*log(f)^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x)/x^5, x)

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maple [A]  time = 0.03, size = 96, normalized size = 1.17 \[ \frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b \ln \relax (f )}+\frac {3 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {6 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {6 x^{4} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}}{x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6/b^4/ln(f)^4*x^4*exp((a+b/x)*ln(f))-6/b^3*x^3*exp((a+b/x)*ln(f))/ln(f)^3+3/b^2*x^2*exp((a+b/x)*ln(f))/ln(f)^
2-1/b*x*exp((a+b/x)*ln(f))/ln(f))/x^4

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maxima [C]  time = 1.30, size = 21, normalized size = 0.26 \[ \frac {f^{a} \Gamma \left (4, -\frac {b \log \relax (f)}{x}\right )}{b^{4} \log \relax (f)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^a*gamma(4, -b*log(f)/x)/(b^4*log(f)^4)

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mupad [B]  time = 3.55, size = 57, normalized size = 0.70 \[ -\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \relax (f)}+\frac {6\,x^2}{b^3\,{\ln \relax (f)}^3}-\frac {6\,x^3}{b^4\,{\ln \relax (f)}^4}-\frac {3\,x}{b^2\,{\ln \relax (f)}^2}\right )}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x)/x^5,x)

[Out]

-(f^(a + b/x)*(1/(b*log(f)) + (6*x^2)/(b^3*log(f)^3) - (6*x^3)/(b^4*log(f)^4) - (3*x)/(b^2*log(f)^2)))/x^3

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sympy [A]  time = 0.15, size = 53, normalized size = 0.65 \[ \frac {f^{a + \frac {b}{x}} \left (- b^{3} \log {\relax (f )}^{3} + 3 b^{2} x \log {\relax (f )}^{2} - 6 b x^{2} \log {\relax (f )} + 6 x^{3}\right )}{b^{4} x^{3} \log {\relax (f )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f**(a + b/x)*(-b**3*log(f)**3 + 3*b**2*x*log(f)**2 - 6*b*x**2*log(f) + 6*x**3)/(b**4*x**3*log(f)**4)

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