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

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 = 0.147627, 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 \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2} \]

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)

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Rubi in Sympy [A]  time = 32.2764, size = 32, normalized size = 0.84 \[ \left (a + b x + c x^{2}\right ) e^{a + b x + c x^{2}} - e^{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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)*exp(a + b*x + c*x**2) - exp(a + b*x + c*x**2)

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Mathematica [A]  time = 0.0208447, 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 = 0.005, size = 24, normalized size = 0.6 \[ \left ( c{x}^{2}+bx+a-1 \right ){{\rm e}^{c{x}^{2}+bx+a}} \]

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 [A]  time = 0.928268, size = 676, normalized size = 17.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(2*c*x + b)*e^(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*(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))*b^2*e^(a - 1/4*b^2/c)/sqr
t(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 = 0.245006, size = 31, normalized size = 0.82 \[{\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(2*c*x + b)*e^(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 = 0.146431, 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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GIAC/XCAS [A]  time = 0.295464, size = 59, normalized size = 1.55 \[ \frac{{\left (c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} - b^{2} + 4 \, a c - 4 \, c\right )} e^{\left (c x^{2} + b x + a\right )}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(2*c*x + b)*e^(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