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

Optimal. Leaf size=37 \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.0182608, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.03141, size = 36, normalized size = 0.97 \[ \frac{\sqrt{\pi } f^{a} \operatorname{erfi}{\left (\sqrt{b} x \sqrt{\log{\left (f \right )}} \right )}}{2 \sqrt{b} \sqrt{\log{\left (f \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.00635998, size = 37, normalized size = 1. \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.019, size = 26, normalized size = 0.7 \[{\frac{\sqrt{\pi }{f}^{a}}{2}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 0.778535, size = 34, normalized size = 0.92 \[ \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{2 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/sqrt(-b*log(f))

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Fricas [A]  time = 0.251391, size = 34, normalized size = 0.92 \[ \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{2 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/sqrt(-b*log(f))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**2+a),x)

[Out]

Integral(f**(a + b*x**2), x)

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GIAC/XCAS [A]  time = 0.249239, size = 38, normalized size = 1.03 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (f\right )} x\right ) e^{\left (a{\rm ln}\left (f\right )\right )}}{2 \, \sqrt{-b{\rm ln}\left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(pi)*erf(-sqrt(-b*ln(f))*x)*e^(a*ln(f))/sqrt(-b*ln(f))