3.426 \(\int f^{a+b x+c x^2} x^3 \, dx\)

Optimal. Leaf size=217 \[ \frac{3 \sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}-\frac{\sqrt{\pi } b^3 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{b x f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{x^2 f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

-f^(a + b*x + c*x^2)/(2*c^2*Log[f]^2) + (3*b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqr
t[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (b^2*f^(a + b*x + c*x^2))/(8*c^3*Log[f]) - (b*f^(a + b*x + c*x^2)*x)/(4*c^2
*Log[f]) + (f^(a + b*x + c*x^2)*x^2)/(2*c*Log[f]) - (b^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log
[f]])/(2*Sqrt[c])])/(16*c^(7/2)*Sqrt[Log[f]])

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Rubi [A]  time = 0.230831, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2241, 2240, 2234, 2204} \[ \frac{3 \sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}-\frac{\sqrt{\pi } b^3 f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{b x f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac{x^2 f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*x^3,x]

[Out]

-f^(a + b*x + c*x^2)/(2*c^2*Log[f]^2) + (3*b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqr
t[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (b^2*f^(a + b*x + c*x^2))/(8*c^3*Log[f]) - (b*f^(a + b*x + c*x^2)*x)/(4*c^2
*Log[f]) + (f^(a + b*x + c*x^2)*x^2)/(2*c*Log[f]) - (b^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log
[f]])/(2*Sqrt[c])])/(16*c^(7/2)*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^3 \, dx &=\frac{f^{a+b x+c x^2} x^2}{2 c \log (f)}-\frac{b \int f^{a+b x+c x^2} x^2 \, dx}{2 c}-\frac{\int f^{a+b x+c x^2} x \, dx}{c \log (f)}\\ &=-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{b f^{a+b x+c x^2} x}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x^2}{2 c \log (f)}+\frac{b^2 \int f^{a+b x+c x^2} x \, dx}{4 c^2}+\frac{b \int f^{a+b x+c x^2} \, dx}{4 c^2 \log (f)}+\frac{b \int f^{a+b x+c x^2} \, dx}{2 c^2 \log (f)}\\ &=-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{b f^{a+b x+c x^2} x}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x^2}{2 c \log (f)}-\frac{b^3 \int f^{a+b x+c x^2} \, dx}{8 c^3}+\frac{\left (b f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c^2 \log (f)}+\frac{\left (b f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c^2 \log (f)}\\ &=-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{3 b 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} \log ^{\frac{3}{2}}(f)}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{b f^{a+b x+c x^2} x}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x^2}{2 c \log (f)}-\frac{\left (b^3 f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{8 c^3}\\ &=-\frac{f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{3 b 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} \log ^{\frac{3}{2}}(f)}+\frac{b^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}-\frac{b f^{a+b x+c x^2} x}{4 c^2 \log (f)}+\frac{f^{a+b x+c x^2} x^2}{2 c \log (f)}-\frac{b^3 f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.176507, size = 122, normalized size = 0.56 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (2 \sqrt{c} f^{\frac{(b+2 c x)^2}{4 c}} \left (\log (f) \left (b^2-2 b c x+4 c^2 x^2\right )-4 c\right )+\sqrt{\pi } b \sqrt{\log (f)} \left (6 c-b^2 \log (f)\right ) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{16 c^{7/2} \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x + c*x^2)*x^3,x]

[Out]

(f^(a - b^2/(4*c))*(b*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]]*(6*c - b^2*Log[f]) +
2*Sqrt[c]*f^((b + 2*c*x)^2/(4*c))*(-4*c + (b^2 - 2*b*c*x + 4*c^2*x^2)*Log[f])))/(16*c^(7/2)*Log[f]^2)

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Maple [A]  time = 0.086, size = 218, normalized size = 1. \begin{align*}{\frac{{x}^{2}{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{2\,c\ln \left ( f \right ) }}-{\frac{bx{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{4\,\ln \left ( f \right ){c}^{2}}}+{\frac{{b}^{2}{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{8\,{c}^{3}\ln \left ( f \right ) }}+{\frac{{b}^{3}\sqrt{\pi }{f}^{a}}{16\,{c}^{3}}{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{3\,b\sqrt{\pi }{f}^{a}}{8\,\ln \left ( f \right ){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{{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x+a)*x^3,x)

[Out]

1/2/c/ln(f)*x^2*f^(c*x^2)*f^(b*x)*f^a-1/4/c^2*b/ln(f)*x*f^(c*x^2)*f^(b*x)*f^a+1/8/c^3*b^2/ln(f)*f^(c*x^2)*f^(b
*x)*f^a+1/16/c^3*b^3*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))-3/8/c^2*b/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))-1/2/c^2/ln(f)^2*f^(c*x^2)*f^(b*x)*f^a

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Maxima [A]  time = 1.16182, size = 271, normalized size = 1.25 \begin{align*} -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b^{3}{\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 )^{4}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{7}{2}}} - \frac{12 \,{\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{4}}{\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{7}{2}}} - \frac{6 \, b^{2} c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac{7}{2}}} + \frac{8 \, c^{2} \Gamma \left (2, -\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac{7}{2}}}\right )} f^{a - \frac{b^{2}}{4 \, c}}}{16 \, \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^3,x, algorithm="maxima")

[Out]

-1/16*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^4/(sqrt(-(2*c*x + b)^2*log
(f)/c)*(c*log(f))^(7/2)) - 12*(2*c*x + b)^3*b*gamma(3/2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^4/((-(2*c*x + b)^
2*log(f)/c)^(3/2)*(c*log(f))^(7/2)) - 6*b^2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^3/(c*log(f))^(7/2) + 8*c^2*gamma(
2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^2/(c*log(f))^(7/2))*f^(a - 1/4*b^2/c)/sqrt(c*log(f))

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Fricas [A]  time = 1.57129, size = 278, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (4 \, c^{2} -{\left (4 \, c^{3} x^{2} - 2 \, b c^{2} x + b^{2} c\right )} \log \left (f\right )\right )} f^{c x^{2} + b x + a} - \frac{\sqrt{\pi }{\left (b^{3} \log \left (f\right ) - 6 \, b 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}}}}{16 \, c^{4} \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^3,x, algorithm="fricas")

[Out]

-1/16*(2*(4*c^2 - (4*c^3*x^2 - 2*b*c^2*x + b^2*c)*log(f))*f^(c*x^2 + b*x + a) - sqrt(pi)*(b^3*log(f) - 6*b*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^4*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^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+b*x+a)*x**3,x)

[Out]

Integral(f**(a + b*x + c*x**2)*x**3, x)

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Giac [A]  time = 1.25069, size = 185, normalized size = 0.85 \begin{align*} \frac{\frac{\sqrt{\pi }{\left (b^{3} \log \left (f\right ) - 6 \, b 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^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} \log \left (f\right ) - 3 \, b c{\left (2 \, x + \frac{b}{c}\right )} \log \left (f\right ) + 3 \, b^{2} \log \left (f\right ) - 4 \, c\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 )^{2}}}{16 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*x^3,x, algorithm="giac")

[Out]

1/16*(sqrt(pi)*(b^3*log(f) - 6*b*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*(2*x + b/c)^2*log(f) - 3*b*c*(2*x + b/c)*log(f) + 3*b^2*log(f) - 4*c)*e^(
c*x^2*log(f) + b*x*log(f) + a*log(f))/log(f)^2)/c^3