Integrand size = 17, antiderivative size = 181 \[ \int e^{a+b x-c x^2} x^3 \, dx=-\frac {b^2 e^{a+b x-c x^2}}{8 c^3}-\frac {e^{a+b x-c x^2}}{2 c^2}-\frac {b e^{a+b x-c x^2} x}{4 c^2}-\frac {e^{a+b x-c x^2} x^2}{2 c}-\frac {b^3 e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{16 c^{7/2}}-\frac {3 b e^{a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{8 c^{5/2}} \] Output:
-1/8*b^2*exp(-c*x^2+b*x+a)/c^3-1/2*exp(-c*x^2+b*x+a)/c^2-1/4*b*exp(-c*x^2+ b*x+a)*x/c^2-1/2*exp(-c*x^2+b*x+a)*x^2/c-1/16*b^3*exp(a+1/4*b^2/c)*Pi^(1/2 )*erf(1/2*(-2*c*x+b)/c^(1/2))/c^(7/2)-3/8*b*exp(a+1/4*b^2/c)*Pi^(1/2)*erf( 1/2*(-2*c*x+b)/c^(1/2))/c^(5/2)
Time = 0.46 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.50 \[ \int e^{a+b x-c x^2} x^3 \, dx=-\frac {e^a \left (2 \sqrt {c} e^{x (b-c x)} \left (b^2+2 b c x+4 c \left (1+c x^2\right )\right )+b \left (b^2+6 c\right ) e^{\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )\right )}{16 c^{7/2}} \] Input:
Integrate[E^(a + b*x - c*x^2)*x^3,x]
Output:
-1/16*(E^a*(2*Sqrt[c]*E^(x*(b - c*x))*(b^2 + 2*b*c*x + 4*c*(1 + c*x^2)) + b*(b^2 + 6*c)*E^(b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])]))/c^(7/2 )
Time = 1.33 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {2671, 2670, 2664, 2634, 2671, 2664, 2634, 2670, 2664, 2634}
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.
| \(\displaystyle \int x^3 e^{a+b x-c x^2} \, dx\) |
| \(\Big \downarrow \) 2671 |
| \(\displaystyle \frac {\int e^{-c x^2+b x+a} xdx}{c}+\frac {b \int e^{-c x^2+b x+a} x^2dx}{2 c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2670 |
| \(\displaystyle \frac {\frac {b \int e^{-c x^2+b x+a}dx}{2 c}-\frac {e^{a+b x-c x^2}}{2 c}}{c}+\frac {b \int e^{-c x^2+b x+a} x^2dx}{2 c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {\frac {b e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx}{2 c}-\frac {e^{a+b x-c x^2}}{2 c}}{c}+\frac {b \int e^{-c x^2+b x+a} x^2dx}{2 c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {b \int e^{-c x^2+b x+a} x^2dx}{2 c}+\frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2671 |
| \(\displaystyle \frac {b \left (\frac {\int e^{-c x^2+b x+a}dx}{2 c}+\frac {b \int e^{-c x^2+b x+a} xdx}{2 c}-\frac {x e^{a+b x-c x^2}}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {b \left (\frac {e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx}{2 c}+\frac {b \int e^{-c x^2+b x+a} xdx}{2 c}-\frac {x e^{a+b x-c x^2}}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {b \left (\frac {b \int e^{-c x^2+b x+a} xdx}{2 c}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {x e^{a+b x-c x^2}}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2670 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \int e^{-c x^2+b x+a}dx}{2 c}-\frac {e^{a+b x-c x^2}}{2 c}\right )}{2 c}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {x e^{a+b x-c x^2}}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b e^{a+\frac {b^2}{4 c}} \int e^{-\frac {(b-2 c x)^2}{4 c}}dx}{2 c}-\frac {e^{a+b x-c x^2}}{2 c}\right )}{2 c}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {x e^{a+b x-c x^2}}{2 c}\right )}{2 c}+\frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}}{c}+\frac {b \left (\frac {b \left (-\frac {\sqrt {\pi } b e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {e^{a+b x-c x^2}}{2 c}\right )}{2 c}-\frac {\sqrt {\pi } e^{a+\frac {b^2}{4 c}} \text {erf}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {x e^{a+b x-c x^2}}{2 c}\right )}{2 c}-\frac {x^2 e^{a+b x-c x^2}}{2 c}\) |
Input:
Int[E^(a + b*x - c*x^2)*x^3,x]
Output:
-1/2*(E^(a + b*x - c*x^2)*x^2)/c + (-1/2*E^(a + b*x - c*x^2)/c - (b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2)))/c + (b*(-1 /2*(E^(a + b*x - c*x^2)*x)/c - (E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x) /(2*Sqrt[c])])/(4*c^(3/2)) + (b*(-1/2*E^(a + b*x - c*x^2)/c - (b*E^(a + b^ 2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2))))/(2*c)))/(2*c )
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - Simp[(b*e - 2*c*d)/(2* c) Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ [b*e - 2*c*d, 0]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + (-Simp[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] , x] - Simp[(m - 1)*(e^2/(2*c*Log[F])) Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{-c \,x^{2}+b x +a} x^{2}}{2 c}-\frac {b \,{\mathrm e}^{-c \,x^{2}+b x +a} x}{4 c^{2}}-\frac {b^{2} {\mathrm e}^{-c \,x^{2}+b x +a}}{8 c^{3}}-\frac {b^{3} \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{16 c^{\frac {7}{2}}}-\frac {3 b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {5}{2}}}-\frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{2 c^{2}}\) | \(154\) |
| default | \(-\frac {{\mathrm e}^{-c \,x^{2}+b x +a} x^{2}}{2 c}+\frac {b \left (-\frac {{\mathrm e}^{-c \,x^{2}+b x +a} x}{2 c}+\frac {b \left (-\frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{2 c}-\frac {b \sqrt {\pi }\, {\mathrm e}^{a +\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}\right )}{2 c}-\frac {\sqrt {\pi }\, {\mathrm e}^{a +\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}\right )}{2 c}+\frac {-\frac {{\mathrm e}^{-c \,x^{2}+b x +a}}{2 c}-\frac {b \sqrt {\pi }\, {\mathrm e}^{a +\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}}{c}\) | \(194\) |
| parts | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{a +\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) x^{3}}{2 \sqrt {c}}+\frac {{\mathrm e}^{a +\frac {b^{2}}{4 c}} \left (8 \,\operatorname {erf}\left (\frac {-2 c x +b}{2 \sqrt {c}}\right ) x^{3} c^{\frac {11}{2}} \sqrt {\pi }-\sqrt {\pi }\, c^{\frac {5}{2}} \operatorname {erf}\left (\frac {-2 c x +b}{2 \sqrt {c}}\right ) b^{3}-8 \,{\mathrm e}^{-\frac {\left (-2 c x +b \right )^{2}}{4 c}} x^{2} c^{5}-6 \sqrt {\pi }\, c^{\frac {7}{2}} \operatorname {erf}\left (\frac {-2 c x +b}{2 \sqrt {c}}\right ) b -4 b \,{\mathrm e}^{-\frac {\left (-2 c x +b \right )^{2}}{4 c}} x \,c^{4}-2 \,{\mathrm e}^{-\frac {\left (-2 c x +b \right )^{2}}{4 c}} b^{2} c^{3}-8 \,{\mathrm e}^{-\frac {\left (-2 c x +b \right )^{2}}{4 c}} c^{4}\right )}{16 c^{6}}\) | \(206\) |
Input:
int(exp(-c*x^2+b*x+a)*x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*exp(-c*x^2+b*x+a)*x^2/c-1/4*b*exp(-c*x^2+b*x+a)*x/c^2-1/8*b^2*exp(-c* x^2+b*x+a)/c^3-1/16*b^3/c^(7/2)*Pi^(1/2)*exp(1/4*(4*a*c+b^2)/c)*erf(-c^(1/ 2)*x+1/2*b/c^(1/2))-3/8*b/c^(5/2)*Pi^(1/2)*exp(1/4*(4*a*c+b^2)/c)*erf(-c^( 1/2)*x+1/2*b/c^(1/2))-1/2*exp(-c*x^2+b*x+a)/c^2
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.49 \[ \int e^{a+b x-c x^2} x^3 \, dx=\frac {\sqrt {\pi } {\left (b^{3} + 6 \, b c\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x - b}{2 \, \sqrt {c}}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )} - 2 \, {\left (4 \, c^{3} x^{2} + 2 \, b c^{2} x + b^{2} c + 4 \, c^{2}\right )} e^{\left (-c x^{2} + b x + a\right )}}{16 \, c^{4}} \] Input:
integrate(exp(-c*x^2+b*x+a)*x^3,x, algorithm="fricas")
Output:
1/16*(sqrt(pi)*(b^3 + 6*b*c)*sqrt(c)*erf(1/2*(2*c*x - b)/sqrt(c))*e^(1/4*( b^2 + 4*a*c)/c) - 2*(4*c^3*x^2 + 2*b*c^2*x + b^2*c + 4*c^2)*e^(-c*x^2 + b* x + a))/c^4
\[ \int e^{a+b x-c x^2} x^3 \, dx=e^{a} \int x^{3} e^{b x} e^{- c x^{2}}\, dx \] Input:
integrate(exp(-c*x**2+b*x+a)*x**3,x)
Output:
exp(a)*Integral(x**3*exp(b*x)*exp(-c*x**2), x)
Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00 \[ \int e^{a+b x-c x^2} x^3 \, dx=\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {7}{2}}} - \frac {6 \, b^{2} c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, c x - b\right )}^{3} b \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {7}{2}}} - \frac {8 \, c^{2} \Gamma \left (2, \frac {{\left (2 \, c x - b\right )}^{2}}{4 \, c}\right )}{\left (-c\right )^{\frac {7}{2}}}\right )} e^{\left (a + \frac {b^{2}}{4 \, c}\right )}}{16 \, \sqrt {-c}} \] Input:
integrate(exp(-c*x^2+b*x+a)*x^3,x, algorithm="maxima")
Output:
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*c*e^(-1/4*(2*c*x - b)^2/c)/(-c)^(7/2) - 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*c^2*gamma(2, 1/4*(2*c*x - b)^2/c)/(-c)^(7/2))*e^(a + 1/4*b^2/c)/sqrt(-c)
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.57 \[ \int e^{a+b x-c x^2} x^3 \, dx=-\frac {\frac {\sqrt {\pi } {\left (b^{3} + 6 \, b c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c^{2} {\left (2 \, x - \frac {b}{c}\right )}^{2} + 3 \, b c {\left (2 \, x - \frac {b}{c}\right )} + 3 \, b^{2} + 4 \, c\right )} e^{\left (-c x^{2} + b x + a\right )}}{16 \, c^{3}} \] Input:
integrate(exp(-c*x^2+b*x+a)*x^3,x, algorithm="giac")
Output:
-1/16*(sqrt(pi)*(b^3 + 6*b*c)*erf(-1/2*sqrt(c)*(2*x - b/c))*e^(1/4*(b^2 + 4*a*c)/c)/sqrt(c) + 2*(c^2*(2*x - b/c)^2 + 3*b*c*(2*x - b/c) + 3*b^2 + 4*c )*e^(-c*x^2 + b*x + a))/c^3
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.62 \[ \int e^{a+b x-c x^2} x^3 \, dx=-{\mathrm {e}}^{-c\,x^2+b\,x+a}\,\left (\frac {1}{2\,c^2}+\frac {b^2}{8\,c^3}\right )-\frac {x^2\,{\mathrm {e}}^{-c\,x^2+b\,x+a}}{2\,c}-\frac {b\,x\,{\mathrm {e}}^{-c\,x^2+b\,x+a}}{4\,c^2}-\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b}{2}-c\,x}{\sqrt {-c}}\right )\,{\mathrm {e}}^{a+\frac {b^2}{4\,c}}\,\left (b^3+6\,c\,b\right )}{16\,{\left (-c\right )}^{7/2}} \] Input:
int(x^3*exp(a + b*x - c*x^2),x)
Output:
- exp(a + b*x - c*x^2)*(1/(2*c^2) + b^2/(8*c^3)) - (x^2*exp(a + b*x - c*x^ 2))/(2*c) - (b*x*exp(a + b*x - c*x^2))/(4*c^2) - (pi^(1/2)*erfi((b/2 - c*x )/(-c)^(1/2))*exp(a + b^2/(4*c))*(6*b*c + b^3))/(16*(-c)^(7/2))
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.83 \[ \int e^{a+b x-c x^2} x^3 \, dx=\frac {e^{a} \left (\sqrt {\pi }\, e^{\frac {4 c^{2} x^{2}+b^{2}}{4 c}} \mathrm {erf}\left (\frac {2 c x -b}{2 \sqrt {c}}\right ) b^{3}+6 \sqrt {\pi }\, e^{\frac {4 c^{2} x^{2}+b^{2}}{4 c}} \mathrm {erf}\left (\frac {2 c x -b}{2 \sqrt {c}}\right ) b c -2 e^{b x} \sqrt {c}\, b^{2}-4 e^{b x} \sqrt {c}\, b c x -8 e^{b x} \sqrt {c}\, c^{2} x^{2}-8 e^{b x} \sqrt {c}\, c \right )}{16 e^{c \,x^{2}} \sqrt {c}\, c^{3}} \] Input:
int(exp(-c*x^2+b*x+a)*x^3,x)
Output:
(e**a*(sqrt(pi)*e**((b**2 + 4*c**2*x**2)/(4*c))*erf(( - b + 2*c*x)/(2*sqrt (c)))*b**3 + 6*sqrt(pi)*e**((b**2 + 4*c**2*x**2)/(4*c))*erf(( - b + 2*c*x) /(2*sqrt(c)))*b*c - 2*e**(b*x)*sqrt(c)*b**2 - 4*e**(b*x)*sqrt(c)*b*c*x - 8 *e**(b*x)*sqrt(c)*c**2*x**2 - 8*e**(b*x)*sqrt(c)*c))/(16*e**(c*x**2)*sqrt( c)*c**3)