\(\int e^{(a+b x) (c+d x)} x^3 \, dx\) [438]

Optimal result

Integrand size = 17, antiderivative size = 297 \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=-\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {3 (b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}-\frac {(b c+a d)^3 e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{16 b^{7/2} d^{7/2}} \]

[Out]

-1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)/b^2/d^2+1/8*(a*d+b*c)^2*exp(a*c+(a*d+b*c)*x+b*d*x^2)/b^3/d^3-1/4*(a*d+b*c)*e
xp(a*c+(a*d+b*c)*x+b*d*x^2)*x/b^2/d^2+1/2*exp(a*c+(a*d+b*c)*x+b*d*x^2)*x^2/b/d+3/8*(a*d+b*c)*erfi(1/2*(2*b*d*x
+a*d+b*c)/b^(1/2)/d^(1/2))*Pi^(1/2)/b^(5/2)/d^(5/2)/exp(1/4*(-a*d+b*c)^2/b/d)-1/16*(a*d+b*c)^3*erfi(1/2*(2*b*d
*x+a*d+b*c)/b^(1/2)/d^(1/2))*Pi^(1/2)/b^(7/2)/d^(7/2)/exp(1/4*(-a*d+b*c)^2/b/d)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2276, 2273, 2272, 2266, 2235} \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=-\frac {\sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} (a d+b c)^3 \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{16 b^{7/2} d^{7/2}}+\frac {3 \sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} (a d+b c) \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}+\frac {(a d+b c)^2 e^{x (a d+b c)+a c+b d x^2}}{8 b^3 d^3}-\frac {x (a d+b c) e^{x (a d+b c)+a c+b d x^2}}{4 b^2 d^2}-\frac {e^{x (a d+b c)+a c+b d x^2}}{2 b^2 d^2}+\frac {x^2 e^{x (a d+b c)+a c+b d x^2}}{2 b d} \]

[In]

Int[E^((a + b*x)*(c + d*x))*x^3,x]

[Out]

-1/2*E^(a*c + (b*c + a*d)*x + b*d*x^2)/(b^2*d^2) + ((b*c + a*d)^2*E^(a*c + (b*c + a*d)*x + b*d*x^2))/(8*b^3*d^
3) - ((b*c + a*d)*E^(a*c + (b*c + a*d)*x + b*d*x^2)*x)/(4*b^2*d^2) + (E^(a*c + (b*c + a*d)*x + b*d*x^2)*x^2)/(
2*b*d) + (3*(b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(8*b^(5/2)*d^(5/2)*E^((b*c -
a*d)^2/(4*b*d))) - ((b*c + a*d)^3*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(16*b^(7/2)*d^(7/
2)*E^((b*c - a*d)^2/(4*b*d)))

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2272

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] - Dist[(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]

Rule 2273

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[(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]

Rule 2276

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \int e^{a c+(b c+a d) x+b d x^2} x^3 \, dx \\ & = \frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}-\frac {\int e^{a c+(b c+a d) x+b d x^2} x \, dx}{b d}-\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} x^2 \, dx}{2 b d} \\ & = -\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{4 b^2 d^2}+\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{2 b^2 d^2}+\frac {(b c+a d)^2 \int e^{a c+(b c+a d) x+b d x^2} x \, dx}{4 b^2 d^2} \\ & = -\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}-\frac {(b c+a d)^3 \int e^{a c+(b c+a d) x+b d x^2} \, dx}{8 b^3 d^3}+\frac {\left ((b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{4 b^2 d^2}+\frac {\left ((b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{2 b^2 d^2} \\ & = -\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {3 (b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}-\frac {\left ((b c+a d)^3 e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{8 b^3 d^3} \\ & = -\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {3 (b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}-\frac {(b c+a d)^3 e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{16 b^{7/2} d^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.64 \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=\frac {e^{-\frac {(b c-a d)^2}{4 b d}} \left (2 \sqrt {b} \sqrt {d} e^{\frac {(a d+b (c+2 d x))^2}{4 b d}} \left (a^2 d^2-2 b d (2-a c+a d x)+b^2 \left (c^2-2 c d x+4 d^2 x^2\right )\right )-\left (b^3 c^3+3 b^2 c (-2+a c) d+3 a b (-2+a c) d^2+a^3 d^3\right ) \sqrt {\pi } \text {erfi}\left (\frac {a d+b (c+2 d x)}{2 \sqrt {b} \sqrt {d}}\right )\right )}{16 b^{7/2} d^{7/2}} \]

[In]

Integrate[E^((a + b*x)*(c + d*x))*x^3,x]

[Out]

(2*Sqrt[b]*Sqrt[d]*E^((a*d + b*(c + 2*d*x))^2/(4*b*d))*(a^2*d^2 - 2*b*d*(2 - a*c + a*d*x) + b^2*(c^2 - 2*c*d*x
+ 4*d^2*x^2)) - (b^3*c^3 + 3*b^2*c*(-2 + a*c)*d + 3*a*b*(-2 + a*c)*d^2 + a^3*d^3)*Sqrt[Pi]*Erfi[(a*d + b*(c +
2*d*x))/(2*Sqrt[b]*Sqrt[d])])/(16*b^(7/2)*d^(7/2)*E^((b*c - a*d)^2/(4*b*d)))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {{\mathrm e}^{c a +\left (a d +c b \right ) x +b d \,x^{2}} x^{2}}{2 b d}-\frac {\left (a d +c b \right ) \left (\frac {{\mathrm e}^{c a +\left (a d +c b \right ) x +b d \,x^{2}} x}{2 b d}-\frac {\left (a d +c b \right ) \left (\frac {{\mathrm e}^{c a +\left (a d +c b \right ) x +b d \,x^{2}}}{2 b d}+\frac {\left (a d +c b \right ) \sqrt {\pi }\, {\mathrm e}^{c a -\frac {\left (a d +c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right )}{4 b d \sqrt {-b d}}\right )}{2 b d}+\frac {\sqrt {\pi }\, {\mathrm e}^{c a -\frac {\left (a d +c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right )}{4 b d \sqrt {-b d}}\right )}{2 b d}-\frac {\frac {{\mathrm e}^{c a +\left (a d +c b \right ) x +b d \,x^{2}}}{2 b d}+\frac {\left (a d +c b \right ) \sqrt {\pi }\, {\mathrm e}^{c a -\frac {\left (a d +c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right )}{4 b d \sqrt {-b d}}}{b d}\)	\(368\)
risch	\(\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )} x^{2}}{2 b d}-\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )} x a}{4 b^{2} d}-\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )} x c}{4 b \,d^{2}}+\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )} a^{2}}{8 b^{3} d}+\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )} a c}{4 b^{2} d^{2}}+\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )} c^{2}}{8 b \,d^{3}}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) a^{3}}{16 b^{3} \sqrt {-b d}}+\frac {3 \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) a^{2} c}{16 b^{2} d \sqrt {-b d}}+\frac {3 \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) a \,c^{2}}{16 b \,d^{2} \sqrt {-b d}}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) c^{3}}{16 d^{3} \sqrt {-b d}}-\frac {3 \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) a}{8 b^{2} d \sqrt {-b d}}-\frac {3 \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) c}{8 b \,d^{2} \sqrt {-b d}}-\frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )}}{2 b^{2} d^{2}}\)	\(539\)
parts	\(-\frac {\sqrt {\pi }\, {\mathrm e}^{c a -\frac {\left (a d +c b \right )^{2}}{4 b d}} \operatorname {erf}\left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right ) x^{3}}{2 \sqrt {-b d}}-\frac {{\mathrm e}^{c a -\frac {\left (a d +c b \right )^{2}}{4 b d}} \left (8 \,\operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) x^{3} d^{3} b^{3} \sqrt {\pi }\, \sqrt {-b d}+\sqrt {\pi }\, \sqrt {-b d}\, \operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) a^{3} d^{3}+3 \sqrt {\pi }\, \sqrt {-b d}\, \operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) a^{2} b c \,d^{2}+3 \sqrt {\pi }\, \sqrt {-b d}\, \operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) a \,b^{2} c^{2} d +\sqrt {\pi }\, \sqrt {-b d}\, \operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) b^{3} c^{3}-8 b^{3} d^{3} {\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} x^{2}+4 b^{2} d^{3} x \,{\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} a +4 b^{3} d^{2} x \,{\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} c -6 \sqrt {\pi }\, \sqrt {-b d}\, \operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) a b \,d^{2}-6 \sqrt {\pi }\, \sqrt {-b d}\, \operatorname {erf}\left (\frac {2 b d x +a d +c b}{2 \sqrt {-b d}}\right ) b^{2} c d -2 \,{\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} a^{2} b \,d^{3}-4 \,{\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} a \,b^{2} c \,d^{2}-2 \,{\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} b^{3} c^{2} d +8 \,{\mathrm e}^{\frac {\left (2 b d x +a d +c b \right )^{2}}{4 b d}} b^{2} d^{2}\right )}{16 b^{4} d^{4}}\)	\(592\)

[In]

int(exp((b*x+a)*(d*x+c))*x^3,x,method=_RETURNVERBOSE)

[Out]

1/2*exp(c*a+(a*d+b*c)*x+b*d*x^2)*x^2/b/d-1/2*(a*d+b*c)/b/d*(1/2*exp(c*a+(a*d+b*c)*x+b*d*x^2)*x/b/d-1/2*(a*d+b*
c)/b/d*(1/2*exp(c*a+(a*d+b*c)*x+b*d*x^2)/b/d+1/4*(a*d+b*c)/b/d*Pi^(1/2)*exp(c*a-1/4*(a*d+b*c)^2/b/d)/(-b*d)^(1
/2)*erf(-(-b*d)^(1/2)*x+1/2*(a*d+b*c)/(-b*d)^(1/2)))+1/4/b/d*Pi^(1/2)*exp(c*a-1/4*(a*d+b*c)^2/b/d)/(-b*d)^(1/2
)*erf(-(-b*d)^(1/2)*x+1/2*(a*d+b*c)/(-b*d)^(1/2)))-1/b/d*(1/2*exp(c*a+(a*d+b*c)*x+b*d*x^2)/b/d+1/4*(a*d+b*c)/b
/d*Pi^(1/2)*exp(c*a-1/4*(a*d+b*c)^2/b/d)/(-b*d)^(1/2)*erf(-(-b*d)^(1/2)*x+1/2*(a*d+b*c)/(-b*d)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.71 \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=\frac {\sqrt {\pi } {\left (b^{3} c^{3} + a^{3} d^{3} + 3 \, {\left (a^{2} b c - 2 \, a b\right )} d^{2} + 3 \, {\left (a b^{2} c^{2} - 2 \, b^{2} c\right )} d\right )} \sqrt {-b d} \operatorname {erf}\left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d}}{2 \, b d}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )} + 2 \, {\left (4 \, b^{3} d^{3} x^{2} + b^{3} c^{2} d + a^{2} b d^{3} + 2 \, {\left (a b^{2} c - 2 \, b^{2}\right )} d^{2} - 2 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} e^{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )}}{16 \, b^{4} d^{4}} \]

[In]

integrate(exp((b*x+a)*(d*x+c))*x^3,x, algorithm="fricas")

[Out]

1/16*(sqrt(pi)*(b^3*c^3 + a^3*d^3 + 3*(a^2*b*c - 2*a*b)*d^2 + 3*(a*b^2*c^2 - 2*b^2*c)*d)*sqrt(-b*d)*erf(1/2*(2
*b*d*x + b*c + a*d)*sqrt(-b*d)/(b*d))*e^(-1/4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(b*d)) + 2*(4*b^3*d^3*x^2 + b^3*
c^2*d + a^2*b*d^3 + 2*(a*b^2*c - 2*b^2)*d^2 - 2*(b^3*c*d^2 + a*b^2*d^3)*x)*e^(b*d*x^2 + a*c + (b*c + a*d)*x))/
(b^4*d^4)

Sympy [F(-1)]

Timed out. \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=\text {Timed out} \]

[In]

integrate(exp((b*x+a)*(d*x+c))*x**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.90 \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=-\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, b d x + b c + a d\right )} {\left (b c + a d\right )}^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}\right ) - 1\right )}}{\left (b d\right )^{\frac {7}{2}} \sqrt {-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}} - \frac {6 \, {\left (b c + a d\right )}^{2} b d e^{\left (\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}}{\left (b d\right )^{\frac {7}{2}}} + \frac {8 \, b^{2} d^{2} \Gamma \left (2, -\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}{\left (b d\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, b d x + b c + a d\right )}^{3} {\left (b c + a d\right )} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}{\left (b d\right )^{\frac {7}{2}} \left (-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}\right )^{\frac {3}{2}}}\right )} e^{\left (a c - \frac {{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{16 \, \sqrt {b d}} \]

[In]

integrate(exp((b*x+a)*(d*x+c))*x^3,x, algorithm="maxima")

[Out]

-1/16*(sqrt(pi)*(2*b*d*x + b*c + a*d)*(b*c + a*d)^3*(erf(1/2*sqrt(-(2*b*d*x + b*c + a*d)^2/(b*d))) - 1)/((b*d)
^(7/2)*sqrt(-(2*b*d*x + b*c + a*d)^2/(b*d))) - 6*(b*c + a*d)^2*b*d*e^(1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/(b*d)
^(7/2) + 8*b^2*d^2*gamma(2, -1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/(b*d)^(7/2) - 12*(2*b*d*x + b*c + a*d)^3*(b*c
+ a*d)*gamma(3/2, -1/4*(2*b*d*x + b*c + a*d)^2/(b*d))/((b*d)^(7/2)*(-(2*b*d*x + b*c + a*d)^2/(b*d))^(3/2)))*e^
(a*c - 1/4*(b*c + a*d)^2/(b*d))/sqrt(b*d)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.84 \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=\frac {\frac {\sqrt {\pi } {\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} - 6 \, b^{2} c d - 6 \, a b d^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-b d} {\left (2 \, x + \frac {b c + a d}{b d}\right )}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{\sqrt {-b d}} + 2 \, {\left (b^{2} d^{2} {\left (2 \, x + \frac {b c + a d}{b d}\right )}^{2} - 3 \, b^{2} c d {\left (2 \, x + \frac {b c + a d}{b d}\right )} - 3 \, a b d^{2} {\left (2 \, x + \frac {b c + a d}{b d}\right )} + 3 \, b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} - 4 \, b d\right )} e^{\left (b d x^{2} + b c x + a d x + a c\right )}}{16 \, b^{3} d^{3}} \]

[In]

integrate(exp((b*x+a)*(d*x+c))*x^3,x, algorithm="giac")

[Out]

1/16*(sqrt(pi)*(b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 - 6*b^2*c*d - 6*a*b*d^2)*erf(-1/2*sqrt(-b*d)
*(2*x + (b*c + a*d)/(b*d)))*e^(-1/4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(b*d))/sqrt(-b*d) + 2*(b^2*d^2*(2*x + (b*c
+ a*d)/(b*d))^2 - 3*b^2*c*d*(2*x + (b*c + a*d)/(b*d)) - 3*a*b*d^2*(2*x + (b*c + a*d)/(b*d)) + 3*b^2*c^2 + 6*a
*b*c*d + 3*a^2*d^2 - 4*b*d)*e^(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^3*d^3)

Mupad [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.77 \[ \int e^{(a+b x) (c+d x)} x^3 \, dx=\frac {{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}\,\left (\frac {a^2\,d^2}{8}-b\,\left (\frac {d}{2}-\frac {a\,c\,d}{4}\right )+\frac {b^2\,c^2}{8}\right )}{b^3\,d^3}+\frac {x^2\,{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}}{2\,b\,d}-\frac {x\,{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}\,\left (a\,d+b\,c\right )}{4\,b^2\,d^2}-\frac {\sqrt {\pi }\,{\mathrm {e}}^{\frac {a\,c}{2}-\frac {a^2\,d}{4\,b}-\frac {b\,c^2}{4\,d}}\,\mathrm {erfi}\left (\frac {\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x}{\sqrt {b\,d}}\right )\,\left (a^3\,d^3+3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-6\,a\,b\,d^2+b^3\,c^3-6\,b^2\,c\,d\right )}{16\,b^3\,d^3\,\sqrt {b\,d}} \]

[In]

int(x^3*exp((a + b*x)*(c + d*x)),x)

[Out]

(exp(a*c + a*d*x + b*c*x + b*d*x^2)*((a^2*d^2)/8 - b*(d/2 - (a*c*d)/4) + (b^2*c^2)/8))/(b^3*d^3) + (x^2*exp(a*
c + a*d*x + b*c*x + b*d*x^2))/(2*b*d) - (x*exp(a*c + a*d*x + b*c*x + b*d*x^2)*(a*d + b*c))/(4*b^2*d^2) - (pi^(
1/2)*exp((a*c)/2 - (a^2*d)/(4*b) - (b*c^2)/(4*d))*erfi(((a*d)/2 + (b*c)/2 + b*d*x)/(b*d)^(1/2))*(a^3*d^3 + b^3
*c^3 - 6*a*b*d^2 - 6*b^2*c*d + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2))/(16*b^3*d^3*(b*d)^(1/2))