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

Optimal. Leaf size=81 \[ \frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}} \]

[Out]

f^(a + b*x + c*x^2)/(2*c*Log[f]) - (b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*
x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Sqrt[Log[f]])

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Rubi [A]  time = 0.0675503, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x + c*x^2)*x,x]

[Out]

f^(a + b*x + c*x^2)/(2*c*Log[f]) - (b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*
x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Sqrt[Log[f]])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0708535, size = 81, normalized size = 1. \[ \frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

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

[Out]

f^(a + b*x + c*x^2)/(2*c*Log[f]) - (b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*
x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Sqrt[Log[f]])

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Maple [A]  time = 0.033, size = 80, normalized size = 1. \[{\frac{{f}^{c{x}^{2}+bx+a}}{2\,c\ln \left ( f \right ) }}+{\frac{b\sqrt{\pi }}{4\,c}{f}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*x^2+b*x+a)*x,x)

[Out]

1/2*f^(c*x^2+b*x+a)/c/ln(f)+1/4*b/c*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 = 0.812306, size = 182, normalized size = 2.25 \[ -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )} b{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{c \log \left (f\right )}}\right ) - 1\right )} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{3}{2}} \sqrt{-\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{c \log \left (f\right )}}} - \frac{2 \, c e^{\left (\frac{{\left (2 \, c x \log \left (f\right ) + b \log \left (f\right )\right )}^{2}}{4 \, c \log \left (f\right )}\right )} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{3}{2}}}\right )} f^{a}}{4 \, \sqrt{c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(sqrt(pi)*(2*c*x*log(f) + b*log(f))*b*(erf(1/2*sqrt(-(2*c*x*log(f) + b*log(
f))^2/(c*log(f)))) - 1)*log(f)/((c*log(f))^(3/2)*sqrt(-(2*c*x*log(f) + b*log(f))
^2/(c*log(f)))) - 2*c*e^(1/4*(2*c*x*log(f) + b*log(f))^2/(c*log(f)))*log(f)/(c*l
og(f))^(3/2))*f^a/(sqrt(c*log(f))*f^(1/4*b^2/c))

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Fricas [A]  time = 0.257668, size = 109, normalized size = 1.35 \[ -\frac{\frac{\sqrt{\pi } b \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right ) \log \left (f\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}} - 2 \, \sqrt{-c \log \left (f\right )} f^{c x^{2} + b x + a}}{4 \, \sqrt{-c \log \left (f\right )} c \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f**(a + b*x + c*x**2)*x, x)

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GIAC/XCAS [A]  time = 0.260188, size = 108, normalized size = 1.33 \[ \frac{\frac{\sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c{\rm ln}\left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2}{\rm ln}\left (f\right ) - 4 \, a c{\rm ln}\left (f\right )}{4 \, c}\right )}}{\sqrt{-c{\rm ln}\left (f\right )}} + \frac{2 \, e^{\left (c x^{2}{\rm ln}\left (f\right ) + b x{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{{\rm ln}\left (f\right )}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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