3.621 \(\int e^{a+b x+c x^2} (b+2 c x) (a+b x+c x^2)^2 \, dx\)
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 = 0.160683, 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, Rules used =
{6707, 2176, 2194} \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 e^{a+b x+c x^2} \left (a+b x+c x^2\right )+2 e^{a+b x+c x^2} \]
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
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 )^2 \, dx &=\operatorname{Subst}\left (\int e^x x^2 \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \operatorname{Subst}\left (\int e^x x \, dx,x,a+b x+c x^2\right )\\ &=-2 e^{a+b x+c x^2} \left (a+b x+c x^2\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+2 \operatorname{Subst}\left (\int e^x \, dx,x,a+b x+c x^2\right )\\ &=2 e^{a+b x+c x^2}-2 e^{a+b x+c x^2} \left (a+b x+c x^2\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2\\ \end{align*}
Mathematica [A] time = 0.0312684, 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 = 0.039, size = 64, normalized size = 1. \begin{align*} \left ({c}^{2}{x}^{4}+2\,bc{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}} \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)^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 [C] time = 1.68291, size = 1651, normalized size = 25.8 \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)^2,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*gam
ma(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^(1
1/2)) - 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 = 0.80876, size = 136, normalized size = 2.12 \begin{align*}{\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 )} \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)^2,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 = 0.180534, size = 68, normalized size = 1.06 \begin{align*} \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}} \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)**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 [A] time = 1.31223, size = 161, normalized size = 2.52 \begin{align*} \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}} \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)^2,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