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 = 0.0574331, 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 \[ \sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{a+b x^2}}{x} \]

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]]

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Rubi in Sympy [A]  time = 5.25911, size = 44, normalized size = 0.9 \[ \sqrt{\pi } \sqrt{b} f^{a} \sqrt{\log{\left (f \right )}} \operatorname{erfi}{\left (\sqrt{b} x \sqrt{\log{\left (f \right )}} \right )} - \frac{f^{a + b x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**2+a)/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0234826, 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 = 0.026, size = 44, normalized size = 0.9 \[ -{\frac{{f}^{a}{f}^{b{x}^{2}}}{x}}+{{f}^{a}\ln \left ( f \right ) b\sqrt{\pi }{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)/x^2,x)

[Out]

-f^a*f^(b*x^2)/x+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 = 0.82241, size = 38, normalized size = 0.78 \[ -\frac{\sqrt{-b x^{2} \log \left (f\right )} f^{a} \Gamma \left (-\frac{1}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x} \]

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 = 0.257619, size = 69, normalized size = 1.41 \[ \frac{\sqrt{\pi } b f^{a} x \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) \log \left (f\right ) - \sqrt{-b \log \left (f\right )} f^{b x^{2} + a}}{\sqrt{-b \log \left (f\right )} x} \]

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

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

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

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)