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

Optimal. Leaf size=111 \[ \frac{\sqrt{\pi } \sqrt{e} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^2}-\frac{(c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^2}-\frac{b e \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^2}+\frac{b (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^2} \]

[Out]

-(((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)*Sq
rt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^2 - (b*e*ExpIntegralEi[e/(c + d*x)^2])/(2*d^2)

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Rubi [A]  time = 0.129492, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ \frac{\sqrt{\pi } \sqrt{e} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^2}-\frac{(c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^2}-\frac{b e \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^2}+\frac{b (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x)^2)*(a + b*x),x]

[Out]

-(((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)*Sq
rt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^2 - (b*e*ExpIntegralEi[e/(c + d*x)^2])/(2*d^2)

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

Mathematica [A]  time = 0.116447, size = 85, normalized size = 0.77 \[ -\frac{2 \sqrt{\pi } \sqrt{e} (a d-b c) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )+(c+d x) e^{\frac{e}{(c+d x)^2}} (-2 a d+b c-b d x)+b e \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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])/(2*d^2)

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Maple [A]  time = 0.007, size = 140, normalized size = 1.3 \begin{align*} -{\frac{1}{d} \left ( a \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}{d} \left ( -{\frac{ \left ( dx+c \right ) ^{2}}{2}{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}}-{\frac{e}{2}{\it Ei} \left ( 1,-{\frac{e}{ \left ( dx+c \right ) ^{2}}} \right ) } \right ) }-{\frac{bc}{d} \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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )} + \int \frac{{\left (b d e x^{2} + 2 \, a d e x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{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),x, algorithm="maxima")

[Out]

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)

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Fricas [A]  time = 1.47604, size = 265, normalized size = 2.39 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*sqrt(-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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )} 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),x, algorithm="giac")

[Out]

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