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

Optimal. Leaf size=50 \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]

[Out]

(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d

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Rubi [A]  time = 0.0629938, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]  Int[E^(e/(c + d*x)^2),x]

[Out]

(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d

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Rubi in Sympy [A]  time = 7.34943, size = 39, normalized size = 0.78 \[ - \frac{\sqrt{\pi } \sqrt{e} \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d} + \frac{\left (c + d x\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e/(d*x+c)**2),x)

[Out]

-sqrt(pi)*sqrt(e)*erfi(sqrt(e)/(c + d*x))/d + (c + d*x)*exp(e/(c + d*x)**2)/d

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Mathematica [A]  time = 0.0168698, size = 50, normalized size = 1. \[ \frac{(c+d x) e^{\frac{e}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{e} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(e/(c + d*x)^2),x]

[Out]

(E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d

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Maple [A]  time = 0.007, size = 48, normalized size = 1. \[ -{\frac{1}{d} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c)^2),x)

[Out]

-1/d*(-(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. \[ 2 \, d e \int \frac{x e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + x e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(e/(d*x + c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.256635, size = 104, normalized size = 2.08 \[ -\frac{\sqrt{\pi } e \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) -{\left (d^{2} x + c d\right )} \sqrt{-\frac{e}{d^{2}}} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{2} \sqrt{-\frac{e}{d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(e/(d*x + c)^2),x, algorithm="fricas")

[Out]

-(sqrt(pi)*e*erf(d*sqrt(-e/d^2)/(d*x + c)) - (d^2*x + c*d)*sqrt(-e/d^2)*e^(e/(d^
2*x^2 + 2*c*d*x + c^2)))/(d^2*sqrt(-e/d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(e/(d*x+c)**2),x)

[Out]

Integral(exp(e/(c + d*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(e/(d*x + c)^2),x, algorithm="giac")

[Out]

integrate(e^(e/(d*x + c)^2), x)