Integrand size = 19, antiderivative size = 322 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=-\frac {(b c-a d)^3 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}-\frac {2 b^2 (b c-a d) e e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac {b^3 e e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac {(b c-a d)^3 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}+\frac {2 b^2 (b c-a d) e^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}-\frac {b^3 e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4} \]
-(-a*d+b*c)^3*exp(e/(d*x+c)^2)*(d*x+c)/d^4-2*b^2*(-a*d+b*c)*e*exp(e/(d*x+c )^2)*(d*x+c)/d^4+3/2*b*(-a*d+b*c)^2*exp(e/(d*x+c)^2)*(d*x+c)^2/d^4+1/4*b^3 *e*exp(e/(d*x+c)^2)*(d*x+c)^2/d^4-b^2*(-a*d+b*c)*exp(e/(d*x+c)^2)*(d*x+c)^ 3/d^4+1/4*b^3*exp(e/(d*x+c)^2)*(d*x+c)^4/d^4-3/2*b*(-a*d+b*c)^2*e*Ei(e/(d* x+c)^2)/d^4-1/4*b^3*e^2*Ei(e/(d*x+c)^2)/d^4+2*b^2*(-a*d+b*c)*e^(3/2)*erfi( e^(1/2)/(d*x+c))*Pi^(1/2)/d^4+(-a*d+b*c)^3*erfi(e^(1/2)/(d*x+c))*e^(1/2)*P i^(1/2)/d^4
Time = 0.18 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.75 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=-\frac {c \left (6 a^2 b c d^2-4 a^3 d^3-4 a b^2 d \left (c^2+2 e\right )+b^3 \left (c^3+7 c e\right )\right ) e^{\frac {e}{(c+d x)^2}}}{4 d^4}+\frac {d e^{\frac {e}{(c+d x)^2}} x \left (4 a^3 d^3+6 a^2 b d^3 x+4 a b^2 d \left (2 e+d^2 x^2\right )+b^3 \left (-6 c e+d e x+d^3 x^3\right )\right )+4 (b c-a d) \sqrt {e} \left (-2 a b c d+a^2 d^2+b^2 \left (c^2+2 e\right )\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )-b e \left (-12 a b c d+6 a^2 d^2+b^2 \left (6 c^2+e\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4} \]
-1/4*(c*(6*a^2*b*c*d^2 - 4*a^3*d^3 - 4*a*b^2*d*(c^2 + 2*e) + b^3*(c^3 + 7* c*e))*E^(e/(c + d*x)^2))/d^4 + (d*E^(e/(c + d*x)^2)*x*(4*a^3*d^3 + 6*a^2*b *d^3*x + 4*a*b^2*d*(2*e + d^2*x^2) + b^3*(-6*c*e + d*e*x + d^3*x^3)) + 4*( b*c - a*d)*Sqrt[e]*(-2*a*b*c*d + a^2*d^2 + b^2*(c^2 + 2*e))*Sqrt[Pi]*Erfi[ Sqrt[e]/(c + d*x)] - b*e*(-12*a*b*c*d + 6*a^2*d^2 + b^2*(6*c^2 + e))*ExpIn tegralEi[e/(c + d*x)^2])/(4*d^4)
Time = 0.61 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
| \(\displaystyle \int (a+b x)^3 e^{\frac {e}{(c+d x)^2}} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (-\frac {3 b^2 (c+d x)^2 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {(a d-b c)^3 e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {3 b (c+d x) (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {b^3 (c+d x)^3 e^{\frac {e}{(c+d x)^2}}}{d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {\pi } b^2 e^{3/2} (b c-a d) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^4}-\frac {2 b^2 e (c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^4}+\frac {\sqrt {\pi } \sqrt {e} (b c-a d)^3 \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b e (b c-a d)^2 \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{(c+d x)^2}}}{d^4}-\frac {b^3 e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}+\frac {b^3 (c+d x)^4 e^{\frac {e}{(c+d x)^2}}}{4 d^4}+\frac {b^3 e (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{4 d^4}\) |
-(((b*c - a*d)^3*E^(e/(c + d*x)^2)*(c + d*x))/d^4) - (2*b^2*(b*c - a*d)*e* E^(e/(c + d*x)^2)*(c + d*x))/d^4 + (3*b*(b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x)^2)/(2*d^4) + (b^3*e*E^(e/(c + d*x)^2)*(c + d*x)^2)/(4*d^4) - (b^2* (b*c - a*d)*E^(e/(c + d*x)^2)*(c + d*x)^3)/d^4 + (b^3*E^(e/(c + d*x)^2)*(c + d*x)^4)/(4*d^4) + ((b*c - a*d)^3*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x )])/d^4 + (2*b^2*(b*c - a*d)*e^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 - (3*b*(b*c - a*d)^2*e*ExpIntegralEi[e/(c + d*x)^2])/(2*d^4) - (b^3*e^2*E xpIntegralEi[e/(c + d*x)^2])/(4*d^4)
3.5.9.3.1 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 = 0.42 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} b^{3} x^{4}}{4}+{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a \,b^{2} x^{3}+\frac {3 \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a^{2} b \,x^{2}}{2}+{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a^{3} x +\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} b^{3} e \,x^{2}}{4 d^{2}}+\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a^{3} c}{d}-\frac {3 \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a^{2} b \,c^{2}}{2 d^{2}}+\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a \,b^{2} c^{3}}{d^{3}}+\frac {2 \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a \,b^{2} e x}{d^{2}}-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} b^{3} c^{4}}{4 d^{4}}-\frac {3 \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} b^{3} c e x}{2 d^{3}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) a^{3} e}{d \sqrt {-e}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) a^{2} b c e}{d^{2} \sqrt {-e}}-\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) a \,b^{2} c^{2} e}{d^{3} \sqrt {-e}}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) b^{3} c^{3} e}{d^{4} \sqrt {-e}}+\frac {2 \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} a \,b^{2} c e}{d^{3}}-\frac {7 \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} b^{3} c^{2} e}{4 d^{4}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right ) a^{2} b e}{2 d^{2}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right ) a \,b^{2} c e}{d^{3}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right ) b^{3} c^{2} e}{2 d^{4}}-\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) a \,b^{2} e^{2}}{d^{3} \sqrt {-e}}+\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) b^{3} c \,e^{2}}{d^{4} \sqrt {-e}}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right ) b^{3} e^{2}}{4 d^{4}}\) | \(553\) |
| derivativedivides | \(-\frac {a^{3} \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^{3} \left (-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{4}+\frac {e \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 )}{2}\right )}{d^{3}}+\frac {3 b^{2} a \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \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 )}{3}\right )}{d^{2}}-\frac {3 b^{3} c \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \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 )}{3}\right )}{d^{3}}+\frac {3 b \,a^{2} \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 {3 b^{3} c^{2} \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^{3}}-\frac {b^{3} c^{3} \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^{3}}-\frac {6 b^{2} c a \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^{2}}-\frac {3 b c \,a^{2} \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}+\frac {3 b^{2} c^{2} 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 )}{d^{2}}}{d}\) | \(560\) |
| default | \(-\frac {a^{3} \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^{3} \left (-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{4}+\frac {e \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 )}{2}\right )}{d^{3}}+\frac {3 b^{2} a \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \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 )}{3}\right )}{d^{2}}-\frac {3 b^{3} c \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \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 )}{3}\right )}{d^{3}}+\frac {3 b \,a^{2} \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 {3 b^{3} c^{2} \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^{3}}-\frac {b^{3} c^{3} \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^{3}}-\frac {6 b^{2} c a \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^{2}}-\frac {3 b c \,a^{2} \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}+\frac {3 b^{2} c^{2} 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 )}{d^{2}}}{d}\) | \(560\) |
| parts | \(\text {Expression too large to display}\) | \(966\) |
1/4*exp(e/(d*x+c)^2)*b^3*x^4+exp(e/(d*x+c)^2)*a*b^2*x^3+3/2*exp(e/(d*x+c)^ 2)*a^2*b*x^2+exp(e/(d*x+c)^2)*a^3*x+1/4/d^2*exp(e/(d*x+c)^2)*b^3*e*x^2+1/d *exp(e/(d*x+c)^2)*a^3*c-3/2/d^2*exp(e/(d*x+c)^2)*a^2*b*c^2+1/d^3*exp(e/(d* x+c)^2)*a*b^2*c^3+2/d^2*exp(e/(d*x+c)^2)*a*b^2*e*x-1/4/d^4*exp(e/(d*x+c)^2 )*b^3*c^4-3/2/d^3*exp(e/(d*x+c)^2)*b^3*c*e*x-1/d/(-e)^(1/2)*Pi^(1/2)*erf(( -e)^(1/2)/(d*x+c))*a^3*e+3/d^2/(-e)^(1/2)*Pi^(1/2)*erf((-e)^(1/2)/(d*x+c)) *a^2*b*c*e-3/d^3/(-e)^(1/2)*Pi^(1/2)*erf((-e)^(1/2)/(d*x+c))*a*b^2*c^2*e+1 /d^4/(-e)^(1/2)*Pi^(1/2)*erf((-e)^(1/2)/(d*x+c))*b^3*c^3*e+2/d^3*exp(e/(d* x+c)^2)*a*b^2*c*e-7/4/d^4*exp(e/(d*x+c)^2)*b^3*c^2*e+3/2/d^2*Ei(1,-e/(d*x+ c)^2)*a^2*b*e-3/d^3*Ei(1,-e/(d*x+c)^2)*a*b^2*c*e+3/2/d^4*Ei(1,-e/(d*x+c)^2 )*b^3*c^2*e-2/d^3/(-e)^(1/2)*Pi^(1/2)*erf((-e)^(1/2)/(d*x+c))*a*b^2*e^2+2/ d^4/(-e)^(1/2)*Pi^(1/2)*erf((-e)^(1/2)/(d*x+c))*b^3*c*e^2+1/4/d^4*Ei(1,-e/ (d*x+c)^2)*b^3*e^2
Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.97 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=-\frac {4 \, \sqrt {\pi } {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4} + 2 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} e\right )} \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (b^{3} e^{2} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} e\right )} {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (b^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + {\left (6 \, a^{2} b d^{4} + b^{3} d^{2} e\right )} x^{2} - {\left (7 \, b^{3} c^{2} - 8 \, a b^{2} c d\right )} e + 2 \, {\left (2 \, a^{3} d^{4} - {\left (3 \, b^{3} c d - 4 \, a b^{2} d^{2}\right )} e\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{4}} \]
-1/4*(4*sqrt(pi)*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4 + 2*(b^3*c*d - a*b^2*d^2)*e)*sqrt(-e/d^2)*erf(d*sqrt(-e/d^2)/(d*x + c)) + (b ^3*e^2 + 6*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*e)*Ei(e/(d^2*x^2 + 2*c*d*x + c^2)) - (b^3*d^4*x^4 + 4*a*b^2*d^4*x^3 - b^3*c^4 + 4*a*b^2*c^3*d - 6*a^2 *b*c^2*d^2 + 4*a^3*c*d^3 + (6*a^2*b*d^4 + b^3*d^2*e)*x^2 - (7*b^3*c^2 - 8* a*b^2*c*d)*e + 2*(2*a^3*d^4 - (3*b^3*c*d - 4*a*b^2*d^2)*e)*x)*e^(e/(d^2*x^ 2 + 2*c*d*x + c^2)))/d^4
\[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=\int \left (a + b x\right )^{3} e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \]
\[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]
1/4*(b^3*d^3*x^4 + 4*a*b^2*d^3*x^3 + (6*a^2*b*d^3 + b^3*d*e)*x^2 + 2*(2*a^ 3*d^3 - 3*b^3*c*e + 4*a*b^2*d*e)*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2))/d^3 + integrate(1/2*(3*b^3*c^4*e - 4*a*b^2*c^3*d*e - (12*a*b^2*c*d^3*e - 6*a^2*b *d^4*e - (6*c^2*d^2*e + d^2*e^2)*b^3)*x^2 + 2*(2*a^3*d^4*e - 2*(3*c^2*d^2* e - 2*d^2*e^2)*a*b^2 + (4*c^3*d*e - 3*c*d*e^2)*b^3)*x)*e^(e/(d^2*x^2 + 2*c *d*x + c^2))/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3), x)
\[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )} \,d x } \]
Timed out. \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,{\left (a+b\,x\right )}^3 \,d x \]