Integrand size = 29, antiderivative size = 38 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx=-e^{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right ) \]
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx=e^{a+x (b+c x)} \left (-1+a+b x+c x^2\right ) \]
Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {7258, 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.
\(\displaystyle \int (b+2 c x) e^{a+b x+c x^2} \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 7258 |
\(\displaystyle \int e^{a+b x+c x^2} \left (a+b x+c x^2\right )d\left (a+b x+c x^2\right )\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )-\int e^{c x^2+b x+a}d\left (c x^2+b x+a\right )\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2}\) |
3.7.22.3.1 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 = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\left (c \,x^{2}+b x +a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(24\) |
risch | \(\left (c \,x^{2}+b x +a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(24\) |
derivativedivides | \(-{\mathrm e}^{c \,x^{2}+b x +a}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )\) | \(37\) |
default | \(-{\mathrm e}^{c \,x^{2}+b x +a}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )\) | \(37\) |
norman | \(\left (a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a}+x \,{\mathrm e}^{c \,x^{2}+b x +a} b +x^{2} {\mathrm e}^{c \,x^{2}+b x +a} c\) | \(47\) |
parallelrisch | \(x^{2} {\mathrm e}^{c \,x^{2}+b x +a} c +x \,{\mathrm e}^{c \,x^{2}+b x +a} b +a \,{\mathrm e}^{c \,x^{2}+b x +a}-{\mathrm e}^{c \,x^{2}+b x +a}\) | \(58\) |
parts | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) c^{2} x^{3}}{\sqrt {-c}}-\frac {3 \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) c b \,x^{2}}{2 \sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) c a x}{\sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) b^{2} x}{2 \sqrt {-c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) a b}{2 \sqrt {-c}}-\frac {{\mathrm e}^{a -\frac {b^{2}}{4 c}} \left (2 c^{2} \operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) x^{3} \sqrt {\pi }\, \sqrt {-c}+3 c b \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) x^{2} \sqrt {\pi }\, \sqrt {-c}+2 \sqrt {\pi }\, \sqrt {-c}\, x \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) a c +\sqrt {\pi }\, \sqrt {-c}\, x \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) b^{2}-2 c^{2} x^{2} {\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}}+a b \,\operatorname {erf}\left (\frac {2 x c +b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, \sqrt {-c}-2 \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} a c -2 b c \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} x +2 \,{\mathrm e}^{\frac {\left (2 x c +b \right )^{2}}{4 c}} c \right )}{2 c}\) | \(448\) |
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx={\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \]
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.58 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx=\left (a + b x + c x^{2} - 1\right ) e^{a + b x + c x^{2}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 501, normalized size of antiderivative = 13.18 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } a b \operatorname {erf}\left (\sqrt {-c} x - \frac {b}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{2 \, \sqrt {-c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} b^{2} e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {1}{2} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} a \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} + \frac {3}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {5}{2}}} - \frac {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} b \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} - \frac {1}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {7}{2}}} - \frac {6 \, b^{2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {5}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {7}{2}}} + \frac {8 \, \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{c^{\frac {3}{2}}}\right )} c^{\frac {3}{2}} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} \]
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)/(sq rt(-(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)/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*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)/(sq rt(-(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)
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx={\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \]
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx={\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (c\,x^2+b\,x+a-1\right ) \]