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

Optimal. Leaf size=56 \[ -\frac{1}{2} b^2 f^a \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x}\right )+\frac{1}{2} x^2 f^{a+\frac{b}{x}}+\frac{1}{2} b x \log (f) f^{a+\frac{b}{x}} \]

[Out]

(f^(a + b/x)*x^2)/2 + (b*f^(a + b/x)*x*Log[f])/2 - (b^2*f^a*ExpIntegralEi[(b*Log[f])/x]*Log[f]^2)/2

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Rubi [A]  time = 0.0349944, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2214, 2206, 2210} \[ -\frac{1}{2} b^2 f^a \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x}\right )+\frac{1}{2} x^2 f^{a+\frac{b}{x}}+\frac{1}{2} b x \log (f) f^{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)*x,x]

[Out]

(f^(a + b/x)*x^2)/2 + (b*f^(a + b/x)*x*Log[f])/2 - (b^2*f^a*ExpIntegralEi[(b*Log[f])/x]*Log[f]^2)/2

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), 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[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2210

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

Rubi steps

\begin{align*} \int f^{a+\frac{b}{x}} x \, dx &=\frac{1}{2} f^{a+\frac{b}{x}} x^2+\frac{1}{2} (b \log (f)) \int f^{a+\frac{b}{x}} \, dx\\ &=\frac{1}{2} f^{a+\frac{b}{x}} x^2+\frac{1}{2} b f^{a+\frac{b}{x}} x \log (f)+\frac{1}{2} \left (b^2 \log ^2(f)\right ) \int \frac{f^{a+\frac{b}{x}}}{x} \, dx\\ &=\frac{1}{2} f^{a+\frac{b}{x}} x^2+\frac{1}{2} b f^{a+\frac{b}{x}} x \log (f)-\frac{1}{2} b^2 f^a \text{Ei}\left (\frac{b \log (f)}{x}\right ) \log ^2(f)\\ \end{align*}

Mathematica [A]  time = 0.0174743, size = 40, normalized size = 0.71 \[ \frac{1}{2} f^a \left (x f^{b/x} (b \log (f)+x)-b^2 \log ^2(f) \text{Ei}\left (\frac{b \log (f)}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)*x,x]

[Out]

(f^a*(-(b^2*ExpIntegralEi[(b*Log[f])/x]*Log[f]^2) + f^(b/x)*x*(x + b*Log[f])))/2

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Maple [A]  time = 0.062, size = 55, normalized size = 1. \begin{align*}{\frac{{f}^{a}{x}^{2}}{2}{f}^{{\frac{b}{x}}}}+{\frac{b\ln \left ( f \right ){f}^{a}x}{2}{f}^{{\frac{b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{a}}{2}{\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,x)

[Out]

1/2*f^a*f^(b/x)*x^2+1/2*b*ln(f)*f^a*f^(b/x)*x+1/2*b^2*ln(f)^2*f^a*Ei(1,-b*ln(f)/x)

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Maxima [A]  time = 1.21554, size = 28, normalized size = 0.5 \begin{align*} b^{2} f^{a} \Gamma \left (-2, -\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*f^a*gamma(-2, -b*log(f)/x)*log(f)^2

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Fricas [A]  time = 1.75354, size = 107, normalized size = 1.91 \begin{align*} -\frac{1}{2} \, b^{2} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{2} + \frac{1}{2} \,{\left (b x \log \left (f\right ) + x^{2}\right )} f^{\frac{a x + b}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x)*x, x)