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

Optimal. Leaf size=56 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{2 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(2*Sqr
t[c]*Sqrt[Log[f]])

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Rubi [A]  time = 0.0313241, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{2 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x + c*x^2),x]

[Out]

(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(2*Sqr
t[c]*Sqrt[Log[f]])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ f^{a - \frac{b^{2}}{4 c}} \int f^{\frac{\left (b + 2 c x\right )^{2}}{4 c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**(a - b**2/(4*c))*Integral(f**((b + 2*c*x)**2/(4*c)), x)

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Mathematica [A]  time = 0.0123187, size = 56, normalized size = 1. \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{2 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x + c*x^2),x]

[Out]

(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(2*Sqr
t[c]*Sqrt[Log[f]])

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Maple [A]  time = 0.029, size = 54, normalized size = 1. \[ -{\frac{\sqrt{\pi }}{2}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*x^2+b*x+a),x)

[Out]

-1/2*Pi^(1/2)*f^(1/4*(4*a*c-b^2)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2
*ln(f)*b/(-c*ln(f))^(1/2))

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Maxima [A]  time = 0.808017, size = 68, normalized size = 1.21 \[ \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right )}{2 \, \sqrt{-c \log \left (f\right )}}\right )}{2 \, \sqrt{-c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*
log(f))*f^(1/4*b^2/c))

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Fricas [A]  time = 0.262614, size = 65, normalized size = 1.16 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{2 \, \sqrt{-c \log \left (f\right )} f^{\frac{b^{2} - 4 \, a c}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*erf(1/2*(2*c*x + b)*sqrt(-c*log(f))/c)/(sqrt(-c*log(f))*f^(1/4*(b^2
- 4*a*c)/c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c*x**2+b*x+a),x)

[Out]

Integral(f**(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.254613, size = 68, normalized size = 1.21 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c{\rm ln}\left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2}{\rm ln}\left (f\right ) - 4 \, a c{\rm ln}\left (f\right )}{4 \, c}\right )}}{2 \, \sqrt{-c{\rm ln}\left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(pi)*erf(-1/2*sqrt(-c*ln(f))*(2*x + b/c))*e^(-1/4*(b^2*ln(f) - 4*a*c*ln
(f))/c)/sqrt(-c*ln(f))