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

Optimal. Leaf size=35 \[ \frac{1}{2} x^2 f^{a+\frac{b}{x^2}}-\frac{1}{2} b f^a \log (f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0359106, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2214, 2210} \[ \frac{1}{2} x^2 f^{a+\frac{b}{x^2}}-\frac{1}{2} b f^a \log (f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x,x]

[Out]

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

Mathematica [A]  time = 0.0050389, size = 32, normalized size = 0.91 \[ \frac{1}{2} f^a \left (x^2 f^{\frac{b}{x^2}}-b \log (f) \text{Ei}\left (\frac{b \log (f)}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x,x]

[Out]

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

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Maple [A]  time = 0.024, size = 35, normalized size = 1. \begin{align*}{\frac{{f}^{a}{x}^{2}}{2}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{{f}^{a}\ln \left ( f \right ) b}{2}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)*x,x)

[Out]

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

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Maxima [A]  time = 1.29685, size = 24, normalized size = 0.69 \begin{align*} -\frac{1}{2} \, b f^{a} \Gamma \left (-1, -\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*f^a*gamma(-1, -b*log(f)/x^2)*log(f)

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Fricas [A]  time = 1.69334, size = 89, normalized size = 2.54 \begin{align*} -\frac{1}{2} \, b f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right ) + \frac{1}{2} \, f^{\frac{a x^{2} + b}{x^{2}}} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**2)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^2)*x, x)