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

Optimal. Leaf size=38 \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2} \]

[Out]

-E^(a + b*x + c*x^2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)

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Rubi [A]  time = 0.0956658, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6707, 2176, 2194} \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2),x]

[Out]

-E^(a + b*x + c*x^2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)

Rule 6707

Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Dist[q, Subst[Int[x^m*F^x,
x], x, v], x] /;  !FalseQ[q]] /; FreeQ[{F, m}, x] && EqQ[w, v]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx &=\operatorname{Subst}\left (\int e^x x \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )-\operatorname{Subst}\left (\int e^x \, dx,x,a+b x+c x^2\right )\\ &=-e^{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0288192, size = 23, normalized size = 0.61 \[ e^{a+x (b+c x)} \left (a+b x+c x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(a + x*(b + c*x))*(-1 + a + b*x + c*x^2)

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Maple [A]  time = 0.037, size = 24, normalized size = 0.6 \begin{align*} \left ( c{x}^{2}+bx+a-1 \right ){{\rm e}^{c{x}^{2}+bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2+b*x+a-1)*exp(c*x^2+b*x+a)

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Maxima [C]  time = 1.37013, size = 676, normalized size = 17.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi)*a*b*erf(sqrt(-c)*x - 1/2*b/sqrt(-c))*e^(a - 1/4*b^2/c)/sqrt(-c) - 1/4*(sqrt(pi)*(2*c*x + b)*b*(er
f(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(3/2)) - 2*e^(1/4*(2*c*x + b)^2/c)/sqrt(c))*b^2*e
^(a - 1/4*b^2/c)/sqrt(c) - 1/2*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b
)^2/c)*c^(3/2)) - 2*e^(1/4*(2*c*x + b)^2/c)/sqrt(c))*a*sqrt(c)*e^(a - 1/4*b^2/c) + 3/8*(sqrt(pi)*(2*c*x + b)*b
^2*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 4*b*e^(1/4*(2*c*x + b)^2/c)/c^(3/2
) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(5/2)))*b*sqrt(c)*e^(a - 1/4*
b^2/c) - 1/8*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(7/2))
- 6*b^2*e^(1/4*(2*c*x + b)^2/c)/c^(5/2) - 12*(2*c*x + b)^3*b*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2
/c)^(3/2)*c^(7/2)) + 8*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(3/2))*c^(3/2)*e^(a - 1/4*b^2/c)

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Fricas [A]  time = 0.825417, size = 58, normalized size = 1.53 \begin{align*}{\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 + b*x + a - 1)*e^(c*x^2 + b*x + a)

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Sympy [A]  time = 0.135125, size = 22, normalized size = 0.58 \begin{align*} \left (a + b x + c x^{2} - 1\right ) e^{a + b x + c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a + b*x + c*x**2 - 1)*exp(a + b*x + c*x**2)

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Giac [A]  time = 1.19335, size = 59, normalized size = 1.55 \begin{align*} \frac{{\left (c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} - b^{2} + 4 \, a c - 4 \, c\right )} e^{\left (c x^{2} + b x + a\right )}}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^2*(2*x + b/c)^2 - b^2 + 4*a*c - 4*c)*e^(c*x^2 + b*x + a)/c