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]

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

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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]

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]))

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.43, size = 35, normalized size = 1.00 \[ -\frac {1}{2} \, b f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x^{2}}\right ) \log \relax (f) + \frac {1}{2} \, f^{\frac {a x^{2} + b}{x^{2}}} x^{2} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{2}}} x\,{d x} \]

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)

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maple [A]  time = 0.04, size = 35, normalized size = 1.00 \[ \frac {b \,f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x^{2}}\right ) \ln \relax (f )}{2}+\frac {x^{2} f^{a} f^{\frac {b}{x^{2}}}}{2} \]

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/x^2*ln(f))

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maxima [A]  time = 1.22, size = 18, normalized size = 0.51 \[ -\frac {1}{2} \, b f^{a} \Gamma \left (-1, -\frac {b \log \relax (f)}{x^{2}}\right ) \log \relax (f) \]

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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mupad [B]  time = 3.57, size = 33, normalized size = 0.94 \[ \frac {f^a\,f^{\frac {b}{x^2}}\,x^2}{2}+\frac {b\,f^a\,\ln \relax (f)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2)*x,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{2}}} x\, dx \]

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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