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]

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

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.41, size = 35, normalized size = 1.00 \[ \frac {b f^{a} x^{2} {\rm Ei}\left (b x^{2} \log \relax (f)\right ) \log \relax (f) - f^{b x^{2} + a}}{2 \, x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{2} + a}}{x^{3}}\,{d x} \]

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)

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

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 = 0.64, size = 18, normalized size = 0.51 \[ \frac {1}{2} \, b f^{a} \Gamma \left (-1, -b x^{2} \log \relax (f)\right ) \log \relax (f) \]

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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mupad [B]  time = 3.43, size = 32, normalized size = 0.91 \[ -\frac {f^a\,\left (f^{b\,x^2}+b\,x^2\,\ln \relax (f)\,\mathrm {expint}\left (-b\,x^2\,\ln \relax (f)\right )\right )}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^2)/x^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x^{2}}}{x^{3}}\, dx \]

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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