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

Optimal. Leaf size=90 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}}+\frac{e f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

(e*f^(a + b*x + c*x^2))/(2*c*Log[f]) + ((2*c*d - b*e)*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 = 0.0411925, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2240, 2234, 2204} \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}}+\frac{e f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*(d + e*x),x]

[Out]

(e*f^(a + b*x + c*x^2))/(2*c*Log[f]) + ((2*c*d - b*e)*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]])

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} (d+e x) \, dx &=\frac{e f^{a+b x+c x^2}}{2 c \log (f)}-\frac{(-2 c d+b e) \int f^{a+b x+c x^2} \, dx}{2 c}\\ &=\frac{e f^{a+b x+c x^2}}{2 c \log (f)}+\frac{\left ((2 c d-b e) f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c}\\ &=\frac{e f^{a+b x+c x^2}}{2 c \log (f)}+\frac{(2 c d-b e) 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} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.1029, size = 96, normalized size = 1.07 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (\sqrt{\pi } \sqrt{\log (f)} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )+2 \sqrt{c} e f^{\frac{(b+2 c x)^2}{4 c}}\right )}{4 c^{3/2} \log (f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 131, normalized size = 1.5 \begin{align*} -{\frac{d\sqrt{\pi }{f}^{a}}{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{e{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{2\,c\ln \left ( f \right ) }}+{\frac{be\sqrt{\pi }{f}^{a}}{4\,c}{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)*(e*x+d),x)

[Out]

-1/2*d*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*
e/c/ln(f)*f^(c*x^2)*f^(b*x)*f^a+1/4*e*b/c*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 [B]  time = 1.14236, size = 216, normalized size = 2.4 \begin{align*} -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\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 )^{2}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{3}{2}}} - \frac{2 \, c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{3}{2}}}\right )} e f^{a - \frac{b^{2}}{4 \, c}}}{4 \, \sqrt{c \log \left (f\right )}} + \frac{\sqrt{\pi } d 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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^2/(sqrt(-(2*c*x + b)^2*log(f)
/c)*(c*log(f))^(3/2)) - 2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)/(c*log(f))^(3/2))*e*f^(a - 1/4*b^2/c)/sqrt(c*log(f)
) + 1/2*sqrt(pi)*d*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 = 1.56755, size = 203, normalized size = 2.26 \begin{align*} \frac{2 \, c e f^{c x^{2} + b x + a} - \frac{\sqrt{\pi }{\left (2 \, c d - b e\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}}}}{4 \, c^{2} \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)*(e*x+d),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25363, size = 184, normalized size = 2.04 \begin{align*} -\frac{\sqrt{\pi } d \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 )}}{2 \, \sqrt{-c \log \left (f\right )}} + \frac{\frac{\sqrt{\pi } b \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}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )}} + \frac{2 \, e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right ) + 1\right )}}{\log \left (f\right )}}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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