Integrand size = 31, antiderivative size = 64 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx=2 e^{a+b x+c x^2}-2 e^{a+b x+c x^2} \left (a+b x+c x^2\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2 \]
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.56 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx=e^{a+x (b+c x)} \left (2-2 (a+x (b+c x))+(a+x (b+c x))^2\right ) \]
Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {7258, 2607, 2607, 2624}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (b+2 c x) e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 7258 |
\(\displaystyle \int e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2d\left (a+b x+c x^2\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \int e^{c x^2+b x+a} \left (c x^2+b x+a\right )d\left (c x^2+b x+a\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )-\int e^{c x^2+b x+a}d\left (c x^2+b x+a\right )\right )\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )^2-2 \left (e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2}\right )\) |
E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2 - 2*(-E^(a + b*x + c*x^2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2))
3.7.21.3.1 Defintions of rubi rules used
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] - Simp[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] && !TrueQ[$UseGamma]
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q Subst[Int[x^m*F^x, x], x, v], x] /; !FalseQ[q]] /; FreeQ[ {F, m}, x] && EqQ[w, v]
Time = 0.62 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(2 \,{\mathrm e}^{c \,x^{2}+b x +a}-2 \,{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{2}\) | \(62\) |
default | \(2 \,{\mathrm e}^{c \,x^{2}+b x +a}-2 \,{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )^{2}\) | \(62\) |
gosper | \(\left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x -2 c \,x^{2}+a^{2}-2 b x -2 a +2\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(64\) |
risch | \(\left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x -2 c \,x^{2}+a^{2}-2 b x -2 a +2\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(64\) |
norman | \(\left (a^{2}-2 a +2\right ) {\mathrm e}^{c \,x^{2}+b x +a}+x^{4} {\mathrm e}^{c \,x^{2}+b x +a} c^{2}+\left (2 a b -2 b \right ) x \,{\mathrm e}^{c \,x^{2}+b x +a}+\left (2 c a +b^{2}-2 c \right ) x^{2} {\mathrm e}^{c \,x^{2}+b x +a}+2 x^{3} {\mathrm e}^{c \,x^{2}+b x +a} b c\) | \(105\) |
parallelrisch | \(x^{4} {\mathrm e}^{c \,x^{2}+b x +a} c^{2}+2 x^{3} {\mathrm e}^{c \,x^{2}+b x +a} b c +2 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} a c +x^{2} {\mathrm e}^{c \,x^{2}+b x +a} b^{2}-2 x^{2} {\mathrm e}^{c \,x^{2}+b x +a} c +2 x \,{\mathrm e}^{c \,x^{2}+b x +a} a b -2 x \,{\mathrm e}^{c \,x^{2}+b x +a} b +{\mathrm e}^{c \,x^{2}+b x +a} a^{2}-2 a \,{\mathrm e}^{c \,x^{2}+b x +a}+2 \,{\mathrm e}^{c \,x^{2}+b x +a}\) | \(164\) |
parts | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) c^{3} x^{5}}{\sqrt {-c}}-\frac {5 \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) b \,c^{2} x^{4}}{2 \sqrt {-c}}-\frac {2 \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) a \,c^{2} x^{3}}{\sqrt {-c}}-\frac {2 \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) b^{2} c \,x^{3}}{\sqrt {-c}}-\frac {3 \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) a b c \,x^{2}}{\sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) b^{3} x^{2}}{2 \sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) a^{2} c x}{\sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) b^{2} x a}{\sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) a^{2} b}{2 \sqrt {-c}}-\frac {{\mathrm e}^{a -\frac {b^{2}}{4 c}} \left (2 c^{3} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) x^{5} \sqrt {\pi }\, \sqrt {-c}+5 b \,c^{2} x^{4} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, \sqrt {-c}+4 \sqrt {\pi }\, \sqrt {-c}\, x^{3} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) a \,c^{2}+4 \sqrt {\pi }\, \sqrt {-c}\, x^{3} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) b^{2} c -2 c^{3} {\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} x^{4}+6 \sqrt {\pi }\, \sqrt {-c}\, x^{2} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) a b c +\sqrt {\pi }\, \sqrt {-c}\, x^{2} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) b^{3}+2 \sqrt {\pi }\, \sqrt {-c}\, x \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) a^{2} c +2 \sqrt {\pi }\, \sqrt {-c}\, x \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) a \,b^{2}-4 c^{2} x^{2} {\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} a -2 c \,x^{2} {\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} b^{2}-4 c^{2} b \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} x^{3}+a^{2} b \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, \sqrt {-c}+4 c^{2} x^{2} {\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}}-4 b c \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} x a +4 b c \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} x -2 \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} a^{2} c +4 \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} a c -4 \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} c \right )}{2 c}\) | \(897\) |
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx={\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, {\left (a - 1\right )} b x + {\left (b^{2} + 2 \, {\left (a - 1\right )} c\right )} x^{2} + a^{2} - 2 \, a + 2\right )} e^{\left (c x^{2} + b x + a\right )} \]
(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)
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.06 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx=\left (a^{2} + 2 a b x + 2 a c x^{2} - 2 a + b^{2} x^{2} + 2 b c x^{3} - 2 b x + c^{2} x^{4} - 2 c x^{2} + 2\right ) e^{a + b x + c x^{2}} \]
(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)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.59 (sec) , antiderivative size = 1223, normalized size of antiderivative = 19.11 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
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*gam ma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(7/2)) + 8*ga...
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.66 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx=-{\left (2 \, c x^{2} - {\left (c x^{2} + b x + a\right )}^{2} + 2 \, b x + 2 \, a - 2\right )} e^{\left (c x^{2} + b x + a\right )} \]
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx={\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,\left (a^2+2\,a\,b\,x+2\,a\,c\,x^2-2\,a+b^2\,x^2+2\,b\,c\,x^3-2\,b\,x+c^2\,x^4-2\,c\,x^2+2\right ) \]