3.2.24 \(\int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx\) [124]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {2243, 2240} \begin {gather*} -\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 {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.67 \begin {gather*} -\frac {f^{a+\frac {b}{x}} \left (2 x^2-2 b x \log (f)+b^2 \log ^2(f)\right )}{b^3 x^2 \log ^3(f)} \end {gather*}

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 = 0.02, size = 44, normalized size = 0.72

method	result	size
risch	\(-\frac {\left (\ln \left (f \right )^{2} b^{2}-2 \ln \left (f \right ) b x +2 x^{2}\right ) f^{\frac {a x +b}{x}}}{\ln \left (f \right )^{3} b^{3} x^{2}}\)	\(44\)
meijerg	\(\frac {f^{a} \left (2-\frac {\left (\frac {3 b^{2} \ln \left (f \right )^{2}}{x^{2}}-\frac {6 b \ln \left (f \right )}{x}+6\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{3}\right )}{\ln \left (f \right )^{3} b^{3}}\)	\(46\)
norman	\(\frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{\ln \left (f \right ) b}+\frac {2 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {2 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}}{x^{3}}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.32, size = 22, normalized size = 0.36 \begin {gather*} -\frac {f^{a} \Gamma \left (3, -\frac {b \log \left (f\right )}{x}\right )}{b^{3} \log \left (f\right )^{3}} \end {gather*}

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 = 0.45, size = 43, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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 = 0.05, size = 39, normalized size = 0.64 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 3.55, size = 45, normalized size = 0.74 \begin {gather*} -\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \left (f\right )}+\frac {2\,x^2}{b^3\,{\ln \left (f\right )}^3}-\frac {2\,x}{b^2\,{\ln \left (f\right )}^2}\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x)/x^4,x)

[Out]

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

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