3.115 \(\int f^{a+\frac{b}{x}} x^m \, dx\)

Optimal. Leaf size=35 \[ f^a x^{m+1} \left (-\frac{b \log (f)}{x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (f)}{x}\right ) \]

[Out]

f^a*x^(1 + m)*Gamma[-1 - m, -((b*Log[f])/x)]*(-((b*Log[f])/x))^(1 + m)

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Rubi [A]  time = 0.0179775, antiderivative size = 35, 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} \[ f^a x^{m+1} \left (-\frac{b \log (f)}{x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)*x^m,x]

[Out]

f^a*x^(1 + m)*Gamma[-1 - m, -((b*Log[f])/x)]*(-((b*Log[f])/x))^(1 + m)

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^m \, dx &=f^a x^{1+m} \Gamma \left (-1-m,-\frac{b \log (f)}{x}\right ) \left (-\frac{b \log (f)}{x}\right )^{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0071684, size = 35, normalized size = 1. \[ f^a x^{m+1} \left (-\frac{b \log (f)}{x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)*x^m,x]

[Out]

f^a*x^(1 + m)*Gamma[-1 - m, -((b*Log[f])/x)]*(-((b*Log[f])/x))^(1 + m)

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Maple [B]  time = 0.036, size = 136, normalized size = 3.9 \begin{align*}{f}^{a} \left ( -b \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{1+m}b \left ( -{\frac{{x}^{m} \left ( -b \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m}\Gamma \left ( -m \right ) }{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{x}} \right ) ^{m}}+{\frac{{x}^{1+m} \left ( -b \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m-1}}{ \left ( 1+m \right ) b}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{x}}}}}+{\frac{{x}^{m} \left ( -b \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m}}{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{x}} \right ) ^{m}\Gamma \left ( -m,-{\frac{b\ln \left ( f \right ) }{x}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^a*(-b)^m*ln(f)^(1+m)*b*(-1/(1+m)*x^m*(-b)^(-m)*ln(f)^(-m)*GAMMA(-m)*(-b*ln(f)/x)^m+1/(1+m)*x^(1+m)*(-b)^(-m)
*ln(f)^(-m-1)/b*exp(b*ln(f)/x)+1/(1+m)*x^m*(-b)^(-m)*ln(f)^(-m)*(-b*ln(f)/x)^m*GAMMA(-m,-b*ln(f)/x))

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Maxima [A]  time = 1.27095, size = 47, normalized size = 1.34 \begin{align*} f^{a} x^{m + 1} \left (-\frac{b \log \left (f\right )}{x}\right )^{m + 1} \Gamma \left (-m - 1, -\frac{b \log \left (f\right )}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^a*x^(m + 1)*(-b*log(f)/x)^(m + 1)*gamma(-m - 1, -b*log(f)/x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{\frac{a x + b}{x}} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^((a*x + b)/x)*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x)*x**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x)*x^m, x)