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

Optimal. Leaf size=49 \[ \sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{a+b x^2}}{x} \]

[Out]

-(f^(a + b*x^2)/x) + Sqrt[b]*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Sqrt[Log[f]]

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Rubi [A]  time = 0.0314357, antiderivative size = 49, 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, 2204} \[ \sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{a+b x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^2,x]

[Out]

-(f^(a + b*x^2)/x) + Sqrt[b]*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Sqrt[Log[f]]

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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{f^{a+b x^2}}{x^2} \, dx &=-\frac{f^{a+b x^2}}{x}+(2 b \log (f)) \int f^{a+b x^2} \, dx\\ &=-\frac{f^{a+b x^2}}{x}+\sqrt{b} f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right ) \sqrt{\log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0142022, size = 49, normalized size = 1. \[ \sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{a+b x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^2,x]

[Out]

-(f^(a + b*x^2)/x) + Sqrt[b]*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Sqrt[Log[f]]

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Maple [A]  time = 0.022, size = 44, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}{f}^{b{x}^{2}}}{x}}+{{f}^{a}\ln \left ( f \right ) b\sqrt{\pi }{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.23406, size = 38, normalized size = 0.78 \begin{align*} -\frac{\sqrt{-b x^{2} \log \left (f\right )} f^{a} \Gamma \left (-\frac{1}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78717, size = 103, normalized size = 2.1 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} x \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) + f^{b x^{2} + a}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(pi)*sqrt(-b*log(f))*f^a*x*erf(sqrt(-b*log(f))*x) + f^(b*x^2 + a))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**2)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^2 + a)/x^2, x)