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

Optimal. Leaf size=64 \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 e^{a+b x+c x^2} \left (a+b x+c x^2\right )+2 e^{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.244845, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 e^{a+b x+c x^2} \left (a+b x+c x^2\right )+2 e^{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 78.5493, size = 60, normalized size = 0.94 \[ \left (a + b x + c x^{2}\right )^{2} e^{a + b x + c x^{2}} - 2 \left (a + b x + c x^{2}\right ) e^{a + b x + c x^{2}} + 2 e^{a + b x + c x^{2}} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0166974, size = 36, normalized size = 0.56 \[ e^{a+x (b+c x)} \left ((a+x (b+c x))^2-2 (a+x (b+c x))+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 64, normalized size = 1. \[ \left ({c}^{2}{x}^{4}+2\,cb{x}^{3}+2\,ac{x}^{2}+{b}^{2}{x}^{2}+2\,abx-2\,c{x}^{2}+{a}^{2}-2\,bx-2\,a+2 \right ){{\rm e}^{c{x}^{2}+bx+a}} \]

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)^2,x)

[Out]

(c^2*x^4+2*b*c*x^3+2*a*c*x^2+b^2*x^2+2*a*b*x-2*c*x^2+a^2-2*b*x-2*a+2)*exp(c*x^2+
b*x+a)

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Maxima [A]  time = 1.07892, size = 1651, normalized size = 25.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

1/2*sqrt(pi)*a^2*b*erf(sqrt(-c)*x - 1/2*b/sqrt(-c))*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*b^2*e^(a - 1/4*b^2/c)
/sqrt(c) + 1/8*(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^3
*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^2*sqrt(c)*e^(a - 1/4*b^2/c) + 3/4*(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)))*a*b*sqrt(c)*e^(a - 1/4*b^2/c) - 1/4*(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))*b^2*sqrt(c)*e^(a - 1/4*b^2/c) - 1/4*(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))*a*c^(
3/2)*e^(a - 1/4*b^2/c) + 5/32*(sqrt(pi)*(2*c*x + b)*b^4*(erf(1/2*sqrt(-(2*c*x +
b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(9/2)) - 8*b^3*e^(1/4*(2*c*x + b)^2/c)/c
^(7/2) - 24*(2*c*x + b)^3*b^2*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/
c)^(3/2)*c^(9/2)) + 32*b*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(5/2) - 16*(2*c*x + b)
^5*gamma(5/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(5/2)*c^(9/2)))*b*c^(3/2
)*e^(a - 1/4*b^2/c) - 1/32*(sqrt(pi)*(2*c*x + b)*b^5*(erf(1/2*sqrt(-(2*c*x + b)^
2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(11/2)) - 10*b^4*e^(1/4*(2*c*x + b)^2/c)/c^
(9/2) - 40*(2*c*x + b)^3*b^3*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c
)^(3/2)*c^(11/2)) + 80*b^2*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(7/2) - 80*(2*c*x +
b)^5*b*gamma(5/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(5/2)*c^(11/2)) - 32
*gamma(3, -1/4*(2*c*x + b)^2/c)/c^(5/2))*c^(5/2)*e^(a - 1/4*b^2/c)

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Fricas [A]  time = 0.264676, size = 74, normalized size = 1.16 \[{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \,{\left (a - 1\right )} b x +{\left (b^{2} + 2 \,{\left (a - 1\right )} c\right )} x^{2} + a^{2} - 2 \, a + 2\right )} e^{\left (c x^{2} + b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

(c^2*x^4 + 2*b*c*x^3 + 2*(a - 1)*b*x + (b^2 + 2*(a - 1)*c)*x^2 + a^2 - 2*a + 2)*
e^(c*x^2 + b*x + a)

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Sympy [A]  time = 0.22048, size = 68, normalized size = 1.06 \[ \left (a^{2} + 2 a b x + 2 a c x^{2} - 2 a + b^{2} x^{2} + 2 b c x^{3} - 2 b x + c^{2} x^{4} - 2 c x^{2} + 2\right ) e^{a + b x + c x^{2}} \]

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)**2,x)

[Out]

(a**2 + 2*a*b*x + 2*a*c*x**2 - 2*a + b**2*x**2 + 2*b*c*x**3 - 2*b*x + c**2*x**4
- 2*c*x**2 + 2)*exp(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.290573, size = 161, normalized size = 2.52 \[ \frac{{\left (c^{4}{\left (2 \, x + \frac{b}{c}\right )}^{4} - 2 \, b^{2} c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} + 8 \, a c^{3}{\left (2 \, x + \frac{b}{c}\right )}^{2} - 8 \, c^{3}{\left (2 \, x + \frac{b}{c}\right )}^{2} + b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2} + 8 \, b^{2} c - 32 \, a c^{2} + 32 \, c^{2}\right )} e^{\left (c x^{2} + b x + a\right )}}{16 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*e^(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

1/16*(c^4*(2*x + b/c)^4 - 2*b^2*c^2*(2*x + b/c)^2 + 8*a*c^3*(2*x + b/c)^2 - 8*c^
3*(2*x + b/c)^2 + b^4 - 8*a*b^2*c + 16*a^2*c^2 + 8*b^2*c - 32*a*c^2 + 32*c^2)*e^
(c*x^2 + b*x + a)/c^2