3.1.76 \(\int \frac {f^{a+b x^2}}{x} \, dx\) [76]

Optimal. Leaf size=15 \[ \frac {1}{2} f^a \text {Ei}\left (b x^2 \log (f)\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2241} \begin {gather*} \frac {1}{2} f^a \text {Ei}\left (b x^2 \log (f)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x,x]

[Out]

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

Rule 2241

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

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Mathematica [A]
time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{2} f^a \text {Ei}\left (b x^2 \log (f)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x,x]

[Out]

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

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Maple [A]
time = 0.02, size = 16, normalized size = 1.07

method	result	size
risch	\(-\frac {f^{a} \expIntegral \left (1, -b \,x^{2} \ln \left (f \right )\right )}{2}\)	\(16\)
meijerg	\(\frac {f^{a} \left (-\ln \left (-b \,x^{2} \ln \left (f \right )\right )-\expIntegral \left (1, -b \,x^{2} \ln \left (f \right )\right )+2 \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )\right )}{2}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, f^{a} {\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.42, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, f^{a} {\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a + b x^{2}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**2)/x, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^2 + a)/x, x)

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Mupad [B]
time = 3.18, size = 13, normalized size = 0.87 \begin {gather*} \frac {f^a\,\mathrm {ei}\left (b\,x^2\,\ln \left (f\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^2)/x,x)

[Out]

(f^a*ei(b*x^2*log(f)))/2

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