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

Optimal. Leaf size=22 \[ -b^5 f^a \log ^5(f) \Gamma \left (-5,-\frac {b \log (f)}{x}\right ) \]

[Out]

f^a*x^5*Ei(6,-b*ln(f)/x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ -b^5 f^a \log ^5(f) \text {Gamma}\left (-5,-\frac {b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)*x^4,x]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x)]*Log[f]^5)

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ -b^5 f^a \log ^5(f) \Gamma \left (-5,-\frac {b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)*x^4,x]

[Out]

-(b^5*f^a*Gamma[-5, -((b*Log[f])/x)]*Log[f]^5)

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 80, normalized size = 3.64 \[ -\frac {1}{120} \, b^{5} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{5} + \frac {1}{120} \, {\left (b^{4} x \log \relax (f)^{4} + b^{3} x^{2} \log \relax (f)^{3} + 2 \, b^{2} x^{3} \log \relax (f)^{2} + 6 \, b x^{4} \log \relax (f) + 24 \, x^{5}\right )} f^{\frac {a x + b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*b^5*f^a*Ei(b*log(f)/x)*log(f)^5 + 1/120*(b^4*x*log(f)^4 + b^3*x^2*log(f)^3 + 2*b^2*x^3*log(f)^2 + 6*b*x
^4*log(f) + 24*x^5)*f^((a*x + b)/x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}} x^{4}\,{d x} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.11, size = 121, normalized size = 5.50 \[ \frac {b^{5} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )^{5}}{120}+\frac {b^{4} x \,f^{a} f^{\frac {b}{x}} \ln \relax (f )^{4}}{120}+\frac {b^{3} x^{2} f^{a} f^{\frac {b}{x}} \ln \relax (f )^{3}}{120}+\frac {b^{2} x^{3} f^{a} f^{\frac {b}{x}} \ln \relax (f )^{2}}{60}+\frac {b \,x^{4} f^{a} f^{\frac {b}{x}} \ln \relax (f )}{20}+\frac {x^{5} f^{a} f^{\frac {b}{x}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*f^a*f^(b/x)*x^5+1/20*b*ln(f)*f^a*f^(b/x)*x^4+1/60*b^2*ln(f)^2*f^a*f^(b/x)*x^3+1/120*b^3*ln(f)^3*f^a*f^(b/x
)*x^2+1/120*b^4*ln(f)^4*f^a*f^(b/x)*x+1/120*b^5*ln(f)^5*f^a*Ei(1,-b/x*ln(f))

________________________________________________________________________________________

maxima [B]  time = 1.52, size = 22, normalized size = 1.00 \[ -b^{5} f^{a} \Gamma \left (-5, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^5*f^a*gamma(-5, -b*log(f)/x)*log(f)^5

________________________________________________________________________________________

mupad [B]  time = 3.69, size = 99, normalized size = 4.50 \[ \frac {b^5\,f^a\,{\ln \relax (f)}^5\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )}{120}+b^5\,f^a\,f^{b/x}\,{\ln \relax (f)}^5\,\left (\frac {x^2}{120\,b^2\,{\ln \relax (f)}^2}+\frac {x^3}{60\,b^3\,{\ln \relax (f)}^3}+\frac {x^4}{20\,b^4\,{\ln \relax (f)}^4}+\frac {x^5}{5\,b^5\,{\ln \relax (f)}^5}+\frac {x}{120\,b\,\ln \relax (f)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x)*x^4,x)

[Out]

(b^5*f^a*log(f)^5*expint(-(b*log(f))/x))/120 + b^5*f^a*f^(b/x)*log(f)^5*(x^2/(120*b^2*log(f)^2) + x^3/(60*b^3*
log(f)^3) + x^4/(20*b^4*log(f)^4) + x^5/(5*b^5*log(f)^5) + x/(120*b*log(f)))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}} x^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x)*x**4, x)

________________________________________________________________________________________