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

Optimal. Leaf size=164 \[ -\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{\sqrt{\pi } b^2 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \sqrt{\log (f)}}-\frac{b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{x f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

-(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Log[f]^(3/2)) - (b*f^(a
+ b*x + c*x^2))/(4*c^2*Log[f]) + (f^(a + b*x + c*x^2)*x)/(2*c*Log[f]) + (b^2*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(
(b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Sqrt[Log[f]])

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Rubi [A]  time = 0.0925418, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2241, 2240, 2234, 2204} \[ -\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{\sqrt{\pi } b^2 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \sqrt{\log (f)}}-\frac{b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{x f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*x^2,x]

[Out]

-(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Log[f]^(3/2)) - (b*f^(a
+ b*x + c*x^2))/(4*c^2*Log[f]) + (f^(a + b*x + c*x^2)*x)/(2*c*Log[f]) + (b^2*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(
(b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Sqrt[Log[f]])

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

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 f^{a+b x+c x^2} x^2 \, dx &=\frac{f^{a+b x+c x^2} x}{2 c \log (f)}-\frac{b \int f^{a+b x+c x^2} x \, dx}{2 c}-\frac{\int f^{a+b x+c x^2} \, dx}{2 c \log (f)}\\ &=-\frac{b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x}{2 c \log (f)}+\frac{b^2 \int f^{a+b x+c x^2} \, dx}{4 c^2}-\frac{f^{a-\frac{b^2}{4 c}} \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c \log (f)}\\ &=-\frac{f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}-\frac{b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x}{2 c \log (f)}+\frac{\left (b^2 f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c^2}\\ &=-\frac{f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 c^{3/2} \log ^{\frac{3}{2}}(f)}-\frac{b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x}{2 c \log (f)}+\frac{b^2 f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{8 c^{5/2} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.115661, size = 104, normalized size = 0.63 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (\sqrt{\pi } \left (b^2 \log (f)-2 c\right ) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )-2 \sqrt{c} \sqrt{\log (f)} (b-2 c x) f^{\frac{(b+2 c x)^2}{4 c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x + c*x^2)*x^2,x]

[Out]

(f^(a - b^2/(4*c))*(-2*Sqrt[c]*f^((b + 2*c*x)^2/(4*c))*(b - 2*c*x)*Sqrt[Log[f]] + Sqrt[Pi]*Erfi[((b + 2*c*x)*S
qrt[Log[f]])/(2*Sqrt[c])]*(-2*c + b^2*Log[f])))/(8*c^(5/2)*Log[f]^(3/2))

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Maple [A]  time = 0.031, size = 163, normalized size = 1. \begin{align*}{\frac{x{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{2\,c\ln \left ( f \right ) }}-{\frac{b{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{4\,{c}^{2}\ln \left ( f \right ) }}-{\frac{{b}^{2}\sqrt{\pi }{f}^{a}}{8\,{c}^{2}}{f}^{-{\frac{{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 ) }}}}+{\frac{\sqrt{\pi }{f}^{a}}{4\,c\ln \left ( f \right ) }{f}^{-{\frac{{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 ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c/ln(f)*x*f^(c*x^2)*f^(b*x)*f^a-1/4/c^2*b/ln(f)*f^(c*x^2)*f^(b*x)*f^a-1/8/c^2*b^2*Pi^(1/2)*f^a*f^(-1/4*b^2
/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+1/4/c/ln(f)*Pi^(1/2)*f^a*f^(-1/4*b^
2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))

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Maxima [A]  time = 1.1764, size = 224, normalized size = 1.37 \begin{align*} \frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b^{2}{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{4 \,{\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac{3}{2}} \left (c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{4 \, b c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac{5}{2}}}\right )} f^{a - \frac{b^{2}}{4 \, c}}}{8 \, \sqrt{c \log \left (f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^3/(sqrt(-(2*c*x + b)^2*log(f
)/c)*(c*log(f))^(5/2)) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^3/((-(2*c*x + b)^2*log
(f)/c)^(3/2)*(c*log(f))^(5/2)) - 4*b*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^2/(c*log(f))^(5/2))*f^(a - 1/4*b^2/c)/sq
rt(c*log(f))

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Fricas [A]  time = 1.55679, size = 238, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (2 \, c^{2} x - b c\right )} f^{c x^{2} + b x + a} \log \left (f\right ) - \frac{\sqrt{\pi }{\left (b^{2} \log \left (f\right ) - 2 \, c\right )} \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c^{3} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(2*c^2*x - b*c)*f^(c*x^2 + b*x + a)*log(f) - sqrt(pi)*(b^2*log(f) - 2*c)*sqrt(-c*log(f))*erf(1/2*(2*c*x
+ b)*sqrt(-c*log(f))/c)/f^(1/4*(b^2 - 4*a*c)/c))/(c^3*log(f)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22308, size = 146, normalized size = 0.89 \begin{align*} -\frac{\frac{\sqrt{\pi }{\left (b^{2} \log \left (f\right ) - 2 \, c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )} \log \left (f\right )} - \frac{2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right )\right )}}{\log \left (f\right )}}{8 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(sqrt(pi)*(b^2*log(f) - 2*c)*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f))/c)
/(sqrt(-c*log(f))*log(f)) - 2*(c*(2*x + b/c) - 2*b)*e^(c*x^2*log(f) + b*x*log(f) + a*log(f))/log(f))/c^2