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]

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

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Rubi [A]  time = 0.10, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.40, size = 23, normalized size = 0.61 \[ {\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \]

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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giac [A]  time = 0.23, size = 44, normalized size = 1.16 \[ \frac {{\left (c^{2} {\left (2 \, x + \frac {b}{c}\right )}^{2} - b^{2} + 4 \, a c - 4 \, c\right )} e^{\left (c x^{2} + b x + a\right )}}{4 \, c} \]

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

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maple [A]  time = 0.02, size = 24, normalized size = 0.63 \[ \left (c \,x^{2}+b x +a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a} \]

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 = 2.96, size = 501, normalized size = 13.18 \[ \frac {\sqrt {\pi } a b \operatorname {erf}\left (\sqrt {-c} x - \frac {b}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{2 \, \sqrt {-c}} - \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}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} b^{2} e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {1}{2} \, {\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}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} a \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} + \frac {3}{8} \, {\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}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {5}{2}}} - \frac {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} b \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} - \frac {1}{8} \, {\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}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {7}{2}}} - \frac {6 \, b^{2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {5}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {7}{2}}} + \frac {8 \, \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{c^{\frac {3}{2}}}\right )} c^{\frac {3}{2}} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} \]

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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mupad [B]  time = 0.10, size = 23, normalized size = 0.61 \[ {\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (c\,x^2+b\,x+a-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.15, size = 22, normalized size = 0.58 \[ \left (a + b x + c x^{2} - 1\right ) e^{a + b x + c x^{2}} \]

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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