3.410 \(\int e^{\frac{e}{(c+d x)^2}} (a+b x)^2 \, dx\)

Optimal. Leaf size=215 \[ -\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^2 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^3}+\frac{b e (b c-a d) \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{d^3}-\frac{b (c+d x)^2 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^3}+\frac{(c+d x) (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{d^3}-\frac{2 \sqrt{\pi } b^2 e^{3/2} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{3 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{(c+d x)^2}}}{3 d^3}+\frac{2 b^2 e (c+d x) e^{\frac{e}{(c+d x)^2}}}{3 d^3} \]

[Out]

((b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x))/d^3 + (2*b^2*e*E^(e/(c + d*x)^2)*(c + d*x))/(3*d^3) - (b*(b*c - a*
d)*E^(e/(c + d*x)^2)*(c + d*x)^2)/d^3 + (b^2*E^(e/(c + d*x)^2)*(c + d*x)^3)/(3*d^3) - ((b*c - a*d)^2*Sqrt[e]*S
qrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^3 - (2*b^2*e^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/(3*d^3) + (b*(b*c - a*
d)*e*ExpIntegralEi[e/(c + d*x)^2])/d^3

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Rubi [A]  time = 0.233229, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ -\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^2 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^3}+\frac{b e (b c-a d) \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{d^3}-\frac{b (c+d x)^2 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^3}+\frac{(c+d x) (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{d^3}-\frac{2 \sqrt{\pi } b^2 e^{3/2} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{3 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{(c+d x)^2}}}{3 d^3}+\frac{2 b^2 e (c+d x) e^{\frac{e}{(c+d x)^2}}}{3 d^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x)^2)*(a + b*x)^2,x]

[Out]

((b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x))/d^3 + (2*b^2*e*E^(e/(c + d*x)^2)*(c + d*x))/(3*d^3) - (b*(b*c - a*
d)*E^(e/(c + d*x)^2)*(c + d*x)^2)/d^3 + (b^2*E^(e/(c + d*x)^2)*(c + d*x)^3)/(3*d^3) - ((b*c - a*d)^2*Sqrt[e]*S
qrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^3 - (2*b^2*e^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/(3*d^3) + (b*(b*c - a*
d)*e*ExpIntegralEi[e/(c + d*x)^2])/d^3

Rule 2226

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]

Rule 2206

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 2211

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 2204

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 2214

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 2210

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]

Rubi steps

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

Mathematica [A]  time = 0.188064, size = 176, normalized size = 0.82 \[ \frac{-\sqrt{\pi } \sqrt{e} \left (3 a^2 d^2-6 a b c d+b^2 \left (3 c^2+2 e\right )\right ) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )+d x e^{\frac{e}{(c+d x)^2}} \left (3 a^2 d^2+3 a b d^2 x+b^2 \left (d^2 x^2+2 e\right )\right )+3 b e (b c-a d) \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{3 d^3}+\frac{c e^{\frac{e}{(c+d x)^2}} \left (3 a^2 d^2-3 a b c d+b^2 \left (c^2+2 e\right )\right )}{3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-3*a*b*c*d + 3*a^2*d^2 + b^2*(c^2 + 2*e))*E^(e/(c + d*x)^2))/(3*d^3) + (d*E^(e/(c + d*x)^2)*x*(3*a^2*d^2 +
3*a*b*d^2*x + b^2*(2*e + d^2*x^2)) - Sqrt[e]*(-6*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 2*e))*Sqrt[Pi]*Erfi[Sqrt[
e]/(c + d*x)] + 3*b*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^2])/(3*d^3)

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Maple [A]  time = 0.009, size = 313, normalized size = 1.5 \begin{align*} -{\frac{1}{d} \left ({a}^{2} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) +{\frac{{b}^{2}}{{d}^{2}} \left ( -{\frac{ \left ( dx+c \right ) ^{3}}{3}{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}}+{\frac{2\,e}{3} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) } \right ) }+{\frac{{b}^{2}{c}^{2}}{{d}^{2}} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) }+2\,{\frac{ab}{d} \left ( -1/2\,{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}} \left ( dx+c \right ) ^{2}-1/2\,e{\it Ei} \left ( 1,-{\frac{e}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) }-2\,{\frac{{b}^{2}c}{{d}^{2}} \left ( -1/2\,{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}} \left ( dx+c \right ) ^{2}-1/2\,e{\it Ei} \left ( 1,-{\frac{e}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) }-2\,{\frac{abc}{d} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{e\sqrt{\pi }}{\sqrt{-e}}{\it Erf} \left ({\frac{\sqrt{-e}}{dx+c}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{2} d^{2} x^{3} + 3 \, a b d^{2} x^{2} +{\left (3 \, a^{2} d^{2} + 2 \, b^{2} e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{2}} + \int -\frac{2 \,{\left (b^{2} c^{3} e + 3 \,{\left (b^{2} c d^{2} e - a b d^{3} e\right )} x^{2} -{\left (3 \, a^{2} d^{3} e -{\left (3 \, c^{2} d e - 2 \, d e^{2}\right )} b^{2}\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \,{\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*d^2*x^3 + 3*a*b*d^2*x^2 + (3*a^2*d^2 + 2*b^2*e)*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2))/d^2 + integrate(-2
/3*(b^2*c^3*e + 3*(b^2*c*d^2*e - a*b*d^3*e)*x^2 - (3*a^2*d^3*e - (3*c^2*d*e - 2*d*e^2)*b^2)*x)*e^(e/(d^2*x^2 +
2*c*d*x + c^2))/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2), x)

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Fricas [A]  time = 1.57501, size = 420, normalized size = 1.95 \begin{align*} \frac{3 \,{\left (b^{2} c - a b d\right )} e{\rm Ei}\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \sqrt{\pi }{\left (3 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 3 \, a^{2} d^{3} + 2 \, b^{2} d e\right )} \sqrt{-\frac{e}{d^{2}}} \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) +{\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2} + 2 \, b^{2} c e +{\left (3 \, a^{2} d^{3} + 2 \, b^{2} d e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{2} e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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