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3.620 \(\int e^{a+b x+c x^2} (b+2 c x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=90 \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+6 e^{a+b x+c x^2} \left (a+b x+c x^2\right )-6 e^{a+b x+c x^2} \]

[Out]

-6*E^(a + b*x + c*x^2) + 6*E^(a + b*x + c*x^2)*(a + b*x + c*x^2) - 3*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2 +
 E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^3

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Rubi [A]  time = 0.181703, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {6707, 2176, 2194} \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+6 e^{a+b x+c x^2} \left (a+b x+c x^2\right )-6 e^{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

-6*E^(a + b*x + c*x^2) + 6*E^(a + b*x + c*x^2)*(a + b*x + c*x^2) - 3*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2 +
 E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^3

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 )^3 \, dx &=\operatorname{Subst}\left (\int e^x x^3 \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-3 \operatorname{Subst}\left (\int e^x x^2 \, dx,x,a+b x+c x^2\right )\\ &=-3 e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3+6 \operatorname{Subst}\left (\int e^x x \, dx,x,a+b x+c x^2\right )\\ &=6 e^{a+b x+c x^2} \left (a+b x+c x^2\right )-3 e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3-6 \operatorname{Subst}\left (\int e^x \, dx,x,a+b x+c x^2\right )\\ &=-6 e^{a+b x+c x^2}+6 e^{a+b x+c x^2} \left (a+b x+c x^2\right )-3 e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0374432, size = 49, normalized size = 0.54 \[ e^{a+x (b+c x)} \left ((a+x (b+c x))^3-3 (a+x (b+c x))^2+6 (a+x (b+c x))-6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x + c*x^2)*(b + 2*c*x)*(a + b*x + c*x^2)^3,x]

[Out]

E^(a + x*(b + c*x))*(-6 + 6*(a + x*(b + c*x)) - 3*(a + x*(b + c*x))^2 + (a + x*(b + c*x))^3)

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Maple [A]  time = 0.039, size = 145, normalized size = 1.6 \begin{align*} \left ({c}^{3}{x}^{6}+3\,{c}^{2}b{x}^{5}+3\,a{c}^{2}{x}^{4}+3\,{b}^{2}c{x}^{4}+6\,abc{x}^{3}+{b}^{3}{x}^{3}-3\,{c}^{2}{x}^{4}+3\,{a}^{2}c{x}^{2}+3\,a{b}^{2}{x}^{2}-6\,bc{x}^{3}+3\,{a}^{2}bx-6\,ac{x}^{2}-3\,{b}^{2}{x}^{2}+{a}^{3}-6\,abx+6\,c{x}^{2}-3\,{a}^{2}+6\,bx+6\,a-6 \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)^3,x)

[Out]

(c^3*x^6+3*b*c^2*x^5+3*a*c^2*x^4+3*b^2*c*x^4+6*a*b*c*x^3+b^3*x^3-3*c^2*x^4+3*a^2*c*x^2+3*a*b^2*x^2-6*b*c*x^3+3
*a^2*b*x-6*a*c*x^2-3*b^2*x^2+a^3-6*a*b*x+6*c*x^2-3*a^2+6*b*x+6*a-6)*exp(c*x^2+b*x+a)

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Maxima [C]  time = 2.13694, size = 3214, normalized size = 35.71 \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)^3,x, algorithm="maxima")

[Out]

1/2*sqrt(pi)*a^3*b*erf(sqrt(-c)*x - 1/2*b/sqrt(-c))*e^(a - 1/4*b^2/c)/sqrt(-c) - 3/4*(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))*a^2
*b^2*e^(a - 1/4*b^2/c)/sqrt(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)))*a*b^3*e^(a - 1/4*b^2/c)/sqrt(c) - 1/16*(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^4*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^3*sqrt(c)*e^(a - 1/4*b^2/c) + 9/
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
)))*a^2*b*sqrt(c)*e^(a - 1/4*b^2/c) - 3/4*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqr
t(-(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*b^2*sqrt(c)*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^3*sqrt(c)*e^(a - 1/4*b^2/c) - 3/8*(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))*a^2*c^(3/2)*e^(a - 1/4*b^2/c) + 15/32*(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)))*a*b*c^(3/2)
*e^(a - 1/4*b^2/c) - 9/64*(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*gamm
a(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))*
b^2*c^(3/2)*e^(a - 1/4*b^2/c) - 3/32*(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))*a*c^(5/2)*e^(a - 1/4*b^2/c) + 7/128*(sqrt(pi)*(2*c*x + b)*b^6*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)
/(sqrt(-(2*c*x + b)^2/c)*c^(13/2)) - 12*b^5*e^(1/4*(2*c*x + b)^2/c)/c^(11/2) - 60*(2*c*x + b)^3*b^4*gamma(3/2,
 -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(13/2)) + 160*b^3*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(9/2) -
240*(2*c*x + b)^5*b^2*gamma(5/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(5/2)*c^(13/2)) - 192*b*gamma(3, -1
/4*(2*c*x + b)^2/c)/c^(7/2) - 64*(2*c*x + b)^7*gamma(7/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(7/2)*c^(1
3/2)))*b*c^(5/2)*e^(a - 1/4*b^2/c) - 1/128*(sqrt(pi)*(2*c*x + b)*b^7*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sq
rt(-(2*c*x + b)^2/c)*c^(15/2)) - 14*b^6*e^(1/4*(2*c*x + b)^2/c)/c^(13/2) - 84*(2*c*x + b)^3*b^5*gamma(3/2, -1/
4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(15/2)) + 280*b^4*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(11/2) - 560
*(2*c*x + b)^5*b^3*gamma(5/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(5/2)*c^(15/2)) - 672*b^2*gamma(3, -1/
4*(2*c*x + b)^2/c)/c^(9/2) - 448*(2*c*x + b)^7*b*gamma(7/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(7/2)*c^
(15/2)) + 128*gamma(4, -1/4*(2*c*x + b)^2/c)/c^(7/2))*c^(7/2)*e^(a - 1/4*b^2/c)

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Fricas [A]  time = 0.831962, size = 261, normalized size = 2.9 \begin{align*}{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c +{\left (a - 1\right )} c^{2}\right )} x^{4} +{\left (b^{3} + 6 \,{\left (a - 1\right )} b c\right )} x^{3} + a^{3} + 3 \,{\left (a^{2} - 2 \, a + 2\right )} b x + 3 \,{\left ({\left (a - 1\right )} b^{2} +{\left (a^{2} - 2 \, a + 2\right )} c\right )} x^{2} - 3 \, a^{2} + 6 \, a - 6\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)^3,x, algorithm="fricas")

[Out]

(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + (a - 1)*c^2)*x^4 + (b^3 + 6*(a - 1)*b*c)*x^3 + a^3 + 3*(a^2 - 2*a + 2)*b*x
 + 3*((a - 1)*b^2 + (a^2 - 2*a + 2)*c)*x^2 - 3*a^2 + 6*a - 6)*e^(c*x^2 + b*x + a)

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Sympy [A]  time = 0.235616, size = 160, normalized size = 1.78 \begin{align*} \left (a^{3} + 3 a^{2} b x + 3 a^{2} c x^{2} - 3 a^{2} + 3 a b^{2} x^{2} + 6 a b c x^{3} - 6 a b x + 3 a c^{2} x^{4} - 6 a c x^{2} + 6 a + b^{3} x^{3} + 3 b^{2} c x^{4} - 3 b^{2} x^{2} + 3 b c^{2} x^{5} - 6 b c x^{3} + 6 b x + c^{3} x^{6} - 3 c^{2} x^{4} + 6 c x^{2} - 6\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)**3,x)

[Out]

(a**3 + 3*a**2*b*x + 3*a**2*c*x**2 - 3*a**2 + 3*a*b**2*x**2 + 6*a*b*c*x**3 - 6*a*b*x + 3*a*c**2*x**4 - 6*a*c*x
**2 + 6*a + b**3*x**3 + 3*b**2*c*x**4 - 3*b**2*x**2 + 3*b*c**2*x**5 - 6*b*c*x**3 + 6*b*x + c**3*x**6 - 3*c**2*
x**4 + 6*c*x**2 - 6)*exp(a + b*x + c*x**2)

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Giac [B]  time = 1.26402, size = 360, normalized size = 4. \begin{align*} \frac{{\left (c^{6}{\left (2 \, x + \frac{b}{c}\right )}^{6} - 3 \, b^{2} c^{4}{\left (2 \, x + \frac{b}{c}\right )}^{4} + 12 \, a c^{5}{\left (2 \, x + \frac{b}{c}\right )}^{4} - 12 \, c^{5}{\left (2 \, x + \frac{b}{c}\right )}^{4} + 3 \, b^{4} c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} - 24 \, a b^{2} c^{3}{\left (2 \, x + \frac{b}{c}\right )}^{2} + 48 \, a^{2} c^{4}{\left (2 \, x + \frac{b}{c}\right )}^{2} + 24 \, b^{2} c^{3}{\left (2 \, x + \frac{b}{c}\right )}^{2} - 96 \, a c^{4}{\left (2 \, x + \frac{b}{c}\right )}^{2} - b^{6} + 12 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \, c^{4}{\left (2 \, x + \frac{b}{c}\right )}^{2} - 12 \, b^{4} c + 96 \, a b^{2} c^{2} - 192 \, a^{2} c^{3} - 96 \, b^{2} c^{2} + 384 \, a c^{3} - 384 \, c^{3}\right )} e^{\left (c x^{2} + b x + a\right )}}{64 \, c^{3}} \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)^3,x, algorithm="giac")

[Out]

1/64*(c^6*(2*x + b/c)^6 - 3*b^2*c^4*(2*x + b/c)^4 + 12*a*c^5*(2*x + b/c)^4 - 12*c^5*(2*x + b/c)^4 + 3*b^4*c^2*
(2*x + b/c)^2 - 24*a*b^2*c^3*(2*x + b/c)^2 + 48*a^2*c^4*(2*x + b/c)^2 + 24*b^2*c^3*(2*x + b/c)^2 - 96*a*c^4*(2
*x + b/c)^2 - b^6 + 12*a*b^4*c - 48*a^2*b^2*c^2 + 64*a^3*c^3 + 96*c^4*(2*x + b/c)^2 - 12*b^4*c + 96*a*b^2*c^2
- 192*a^2*c^3 - 96*b^2*c^2 + 384*a*c^3 - 384*c^3)*e^(c*x^2 + b*x + a)/c^3

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