Integrand size = 17, antiderivative size = 111 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=-\frac {(b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^2}+\frac {b e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^2}-\frac {b e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2} \]
-(-a*d+b*c)*exp(e/(d*x+c)^2)*(d*x+c)/d^2+1/2*b*exp(e/(d*x+c)^2)*(d*x+c)^2/ d^2-1/2*b*e*Ei(e/(d*x+c)^2)/d^2+(-a*d+b*c)*erfi(e^(1/2)/(d*x+c))*e^(1/2)*P i^(1/2)/d^2
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=-\frac {e^{\frac {e}{(c+d x)^2}} (c+d x) (b c-2 a d-b d x)+2 (-b c+a d) \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )+b e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2} \]
-1/2*(E^(e/(c + d*x)^2)*(c + d*x)*(b*c - 2*a*d - b*d*x) + 2*(-(b*c) + a*d) *Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)] + b*e*ExpIntegralEi[e/(c + d*x)^ 2])/d^2
Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2656, 2009}
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.
\(\displaystyle \int (a+b x) e^{\frac {e}{(c+d x)^2}} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {(a d-b c) e^{\frac {e}{(c+d x)^2}}}{d}+\frac {b (c+d x) e^{\frac {e}{(c+d x)^2}}}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } \sqrt {e} (b c-a d) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^2}-\frac {(c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^2}-\frac {b e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2}+\frac {b (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{2 d^2}\) |
-(((b*c - a*d)*E^(e/(c + d*x)^2)*(c + d*x))/d^2) + (b*E^(e/(c + d*x)^2)*(c + d*x)^2)/(2*d^2) + ((b*c - a*d)*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)] )/d^2 - (b*e*ExpIntegralEi[e/(c + d*x)^2])/(2*d^2)
3.5.11.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(-\frac {a \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )+\frac {b \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}-\frac {b c \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d}}{d}\) | \(140\) |
default | \(-\frac {a \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )+\frac {b \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}-\frac {b c \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d}}{d}\) | \(140\) |
risch | \(a \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {a \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {a e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}+\frac {b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{2}}{2}-\frac {b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c^{2}}{2 d^{2}}+\frac {b e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2 d^{2}}+\frac {b c e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d^{2} \sqrt {-e}}\) | \(143\) |
parts | \(b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{2}+a \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c x}{d}+\frac {a \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) b x}{d \sqrt {-e}}-\frac {a e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}+\frac {b \left (-\frac {e \sqrt {\pi }\, \left (-\frac {\operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) \left (d x +c \right )}{\sqrt {-e}}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{\sqrt {\pi }}\right )}{d}+\frac {-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}}{d}\right )}{d}\) | \(220\) |
-1/d*(a*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d *x+c)))+b/d*(-1/2*exp(e/(d*x+c)^2)*(d*x+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2))-b/d *c*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c) )))
Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.10 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=-\frac {b e {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, \sqrt {\pi } {\left (b c d - a d^{2}\right )} \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) - {\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{2 \, d^{2}} \]
-1/2*(b*e*Ei(e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*sqrt(pi)*(b*c*d - a*d^2)*sqr t(-e/d^2)*erf(d*sqrt(-e/d^2)/(d*x + c)) - (b*d^2*x^2 + 2*a*d^2*x - b*c^2 + 2*a*c*d)*e^(e/(d^2*x^2 + 2*c*d*x + c^2)))/d^2
\[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=\int \left (a + b x\right ) e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \]
\[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=\int { {\left (b x + a\right )} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]
1/2*(b*x^2 + 2*a*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2)) + integrate((b*d*e*x^2 + 2*a*d*e*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c ^2*d*x + c^3), x)
\[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=\int { {\left (b x + a\right )} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]
Timed out. \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,\left (a+b\,x\right ) \,d x \]