3.5.9 \(\int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx\) [409]

Optimal. Leaf size=322 \[ -\frac {(b c-a d)^3 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}-\frac {2 b^2 (b c-a d) e e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac {b^3 e e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac {(b c-a d)^3 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}+\frac {2 b^2 (b c-a d) e^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4} \]

[Out]

-(-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)*Pi^(1/2)/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.24, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2258, 2237, 2242, 2235, 2245, 2241} \begin {gather*} \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 \text {Ei}\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 \text {Ei}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x)^2)*(a + b*x)^3,x]

[Out]

-(((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*ExpIntegralEi[e/(c + d*x)^2])/(4*
d^4)

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 2237

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2242

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2245

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {align*} \int e^{\frac {e}{(c+d x)^2}} (a+b x)^3 \, dx &=\int \left (\frac {(-b c+a d)^3 e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}-\frac {3 b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac {b^3 \int e^{\frac {e}{(c+d x)^2}} (c+d x)^3 \, dx}{d^3}-\frac {\left (3 b^2 (b c-a d)\right ) \int e^{\frac {e}{(c+d x)^2}} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2\right ) \int e^{\frac {e}{(c+d x)^2}} (c+d x) \, dx}{d^3}-\frac {(b c-a d)^3 \int e^{\frac {e}{(c+d x)^2}} \, dx}{d^3}\\ &=-\frac {(b c-a d)^3 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^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 {\left (b^3 e\right ) \int e^{\frac {e}{(c+d x)^2}} (c+d x) \, dx}{2 d^3}-\frac {\left (2 b^2 (b c-a d) e\right ) \int e^{\frac {e}{(c+d x)^2}} \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2 e\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{c+d x} \, dx}{d^3}-\frac {\left (2 (b c-a d)^3 e\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^3}\\ &=-\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 {3 b (b c-a d)^2 e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}+\frac {\left (2 (b c-a d)^3 e\right ) \text {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^4}+\frac {\left (b^3 e^2\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{c+d x} \, dx}{2 d^3}-\frac {\left (4 b^2 (b c-a d) e^2\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^3}\\ &=-\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 {3 b (b c-a d)^2 e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}+\frac {\left (4 b^2 (b c-a d) e^2\right ) \text {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^4}\\ &=-\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 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 243, normalized size = 0.75 \begin {gather*} -\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 ) \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(e/(c + d*x)^2)*(a + b*x)^3,x]

[Out]

-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))*ExpIntegralEi[e/(c + d*x)^2])/(4*d^4)

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 560, normalized size = 1.74 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(d*x+c)^2)*(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/d*(a^3*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c)))+b^3/d^3*(-1/4*(d*x+c)^4*ex
p(e/(d*x+c)^2)+1/2*e*(-1/2*exp(e/(d*x+c)^2)*(d*x+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2)))+3*b^2/d^2*a*(-1/3*(d*x+c)^3*e
xp(e/(d*x+c)^2)+2/3*e*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c))))-3*b^3/d^3*c*(
-1/3*(d*x+c)^3*exp(e/(d*x+c)^2)+2/3*e*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c))
))+3*b/d*a^2*(-1/2*exp(e/(d*x+c)^2)*(d*x+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2))+3*b^3/d^3*c^2*(-1/2*exp(e/(d*x+c)^2)*(
d*x+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2))-b^3/d^3*c^3*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)
/(d*x+c)))-6*b^2/d^2*c*a*(-1/2*exp(e/(d*x+c)^2)*(d*x+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2))-3*b/d*c*a^2*(-(d*x+c)*exp(
e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c)))+3*b^2/d^2*c^2*a*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1
/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)^2)*(b*x+a)^3,x, algorithm="maxima")

[Out]

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)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 309, normalized size = 0.96 \begin {gather*} \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 {1}{d^{2}}} \operatorname {erfi}\left (\frac {d \sqrt {\frac {1}{d^{2}}} e^{\frac {1}{2}}}{d x + c}\right ) e^{\frac {1}{2}} - {\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} + 6 \, a^{2} b d^{4} x^{2} + 4 \, a^{3} d^{4} x - 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 (b^{3} d^{2} x^{2} - 7 \, b^{3} c^{2} + 8 \, a b^{2} c d - 2 \, {\left (3 \, b^{3} c d - 4 \, a b^{2} d^{2}\right )} x\right )} e\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)^2)*(b*x+a)^3,x, algorithm="fricas")

[Out]

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(d^(-2
))*erfi(d*sqrt(d^(-2))*e^(1/2)/(d*x + c))*e^(1/2) - (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 + 6*a^2*b*d^4*x^2 + 4*a^3*d^4*x - 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 + (b^3*d^2*x^2 - 7*b^3*c^2 + 8*a*b^2*c*d - 2*(3*b^3*c*d - 4*a*b^2*d^2)*
x)*e)*e^(e/(d^2*x^2 + 2*c*d*x + c^2)))/d^4

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right )^{3} e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)**2)*(b*x+a)**3,x)

[Out]

Integral((a + b*x)**3*exp(e/(c**2 + 2*c*d*x + d**2*x**2)), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c)^2)*(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((b*x + a)^3*e^(e/(d*x + c)^2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,{\left (a+b\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x)^2)*(a + b*x)^3,x)

[Out]

int(exp(e/(c + d*x)^2)*(a + b*x)^3, x)

________________________________________________________________________________________