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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0189867, antiderivative size = 22, normalized size of antiderivative = 1., 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 = 0.0023878, size = 22, normalized size = 1. \[ -b^5 f^a \log ^5(f) \text{Gamma}\left (-5,-\frac{b \log (f)}{x}\right ) \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.073, size = 121, normalized size = 5.5 \begin{align*}{\frac{{f}^{a}{x}^{5}}{5}{f}^{{\frac{b}{x}}}}+{\frac{b\ln \left ( f \right ){f}^{a}{x}^{4}}{20}{f}^{{\frac{b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{a}{x}^{3}}{60}{f}^{{\frac{b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{f}^{a}{x}^{2}}{120}{f}^{{\frac{b}{x}}}}+{\frac{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}{f}^{a}x}{120}{f}^{{\frac{b}{x}}}}+{\frac{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}{f}^{a}}{120}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{x}} \right ) } \end{align*}

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*ln(f)/x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x}} x^{4}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x}} x^{4}\,{d x} \end{align*}

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)