3.118 \(\int f^{a+\frac{b}{x}} x^2 \, dx\)

Optimal. Leaf size=79 \[ -\frac{1}{6} b^3 f^a \log ^3(f) \text{Ei}\left (\frac{b \log (f)}{x}\right )+\frac{1}{6} b^2 x \log ^2(f) f^{a+\frac{b}{x}}+\frac{1}{3} x^3 f^{a+\frac{b}{x}}+\frac{1}{6} b x^2 \log (f) f^{a+\frac{b}{x}} \]

[Out]

(f^(a + b/x)*x^3)/3 + (b*f^(a + b/x)*x^2*Log[f])/6 + (b^2*f^(a + b/x)*x*Log[f]^2)/6 - (b^3*f^a*ExpIntegralEi[(
b*Log[f])/x]*Log[f]^3)/6

________________________________________________________________________________________

Rubi [A]  time = 0.0592223, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2214, 2206, 2210} \[ -\frac{1}{6} b^3 f^a \log ^3(f) \text{Ei}\left (\frac{b \log (f)}{x}\right )+\frac{1}{6} b^2 x \log ^2(f) f^{a+\frac{b}{x}}+\frac{1}{3} x^3 f^{a+\frac{b}{x}}+\frac{1}{6} b x^2 \log (f) f^{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)*x^2,x]

[Out]

(f^(a + b/x)*x^3)/3 + (b*f^(a + b/x)*x^2*Log[f])/6 + (b^2*f^(a + b/x)*x*Log[f]^2)/6 - (b^3*f^a*ExpIntegralEi[(
b*Log[f])/x]*Log[f]^3)/6

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^2 \, dx &=\frac{1}{3} f^{a+\frac{b}{x}} x^3+\frac{1}{3} (b \log (f)) \int f^{a+\frac{b}{x}} x \, dx\\ &=\frac{1}{3} f^{a+\frac{b}{x}} x^3+\frac{1}{6} b f^{a+\frac{b}{x}} x^2 \log (f)+\frac{1}{6} \left (b^2 \log ^2(f)\right ) \int f^{a+\frac{b}{x}} \, dx\\ &=\frac{1}{3} f^{a+\frac{b}{x}} x^3+\frac{1}{6} b f^{a+\frac{b}{x}} x^2 \log (f)+\frac{1}{6} b^2 f^{a+\frac{b}{x}} x \log ^2(f)+\frac{1}{6} \left (b^3 \log ^3(f)\right ) \int \frac{f^{a+\frac{b}{x}}}{x} \, dx\\ &=\frac{1}{3} f^{a+\frac{b}{x}} x^3+\frac{1}{6} b f^{a+\frac{b}{x}} x^2 \log (f)+\frac{1}{6} b^2 f^{a+\frac{b}{x}} x \log ^2(f)-\frac{1}{6} b^3 f^a \text{Ei}\left (\frac{b \log (f)}{x}\right ) \log ^3(f)\\ \end{align*}

Mathematica [A]  time = 0.0238828, size = 53, normalized size = 0.67 \[ \frac{1}{6} f^a \left (x f^{b/x} \left (b^2 \log ^2(f)+b x \log (f)+2 x^2\right )-b^3 \log ^3(f) \text{Ei}\left (\frac{b \log (f)}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)*x^2,x]

[Out]

(f^a*(-(b^3*ExpIntegralEi[(b*Log[f])/x]*Log[f]^3) + f^(b/x)*x*(2*x^2 + b*x*Log[f] + b^2*Log[f]^2)))/6

________________________________________________________________________________________

Maple [A]  time = 0.066, size = 77, normalized size = 1. \begin{align*}{\frac{{f}^{a}{x}^{3}}{3}{f}^{{\frac{b}{x}}}}+{\frac{b\ln \left ( f \right ){f}^{a}{x}^{2}}{6}{f}^{{\frac{b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{a}x}{6}{f}^{{\frac{b}{x}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{f}^{a}}{6}{\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^2,x)

[Out]

1/3*f^a*f^(b/x)*x^3+1/6*b*ln(f)*f^a*f^(b/x)*x^2+1/6*b^2*ln(f)^2*f^a*f^(b/x)*x+1/6*b^3*ln(f)^3*f^a*Ei(1,-b*ln(f
)/x)

________________________________________________________________________________________

Maxima [A]  time = 1.16895, size = 30, normalized size = 0.38 \begin{align*} -b^{3} f^{a} \Gamma \left (-3, -\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)*x^2,x, algorithm="maxima")

[Out]

-b^3*f^a*gamma(-3, -b*log(f)/x)*log(f)^3

________________________________________________________________________________________

Fricas [A]  time = 1.77556, size = 135, normalized size = 1.71 \begin{align*} -\frac{1}{6} \, b^{3} f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right )^{3} + \frac{1}{6} \,{\left (b^{2} x \log \left (f\right )^{2} + b x^{2} \log \left (f\right ) + 2 \, x^{3}\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^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x}} x^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x)*x**2,x)

[Out]

Integral(f**(a + b/x)*x**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x)*x^2,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)*x^2, x)