3.77 \(\int \frac{f^{a+b x^2}}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0448627, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ \frac{1}{2} b f^a \log (f) \text{Ei}\left (b x^2 \log (f)\right )-\frac{f^{a+b x^2}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0098074, size = 32, normalized size = 0.91 \[ \frac{1}{2} f^a \left (b \log (f) \text{Ei}\left (b x^2 \log (f)\right )-\frac{f^{b x^2}}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^3,x]

[Out]

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

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Maple [A]  time = 0.02, size = 35, normalized size = 1. \begin{align*} -{\frac{{f}^{a}{f}^{b{x}^{2}}}{2\,{x}^{2}}}-{\frac{{f}^{a}\ln \left ( f \right ) b{\it Ei} \left ( 1,-b{x}^{2}\ln \left ( f \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)/x^3,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 = 1.17222, size = 24, normalized size = 0.69 \begin{align*} \frac{1}{2} \, b f^{a} \Gamma \left (-1, -b x^{2} \log \left (f\right )\right ) \log \left (f\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83609, size = 82, normalized size = 2.34 \begin{align*} \frac{b f^{a} x^{2}{\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \log \left (f\right ) - f^{b x^{2} + a}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x^{2}}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)/x**3,x)

[Out]

Integral(f**(a + b*x**2)/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{2} + a}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^3,x, algorithm="giac")

[Out]

integrate(f^(b*x^2 + a)/x^3, x)