3.419 \(\int e^{\frac{e}{(c+d x)^3}} \, dx\)

Optimal. Leaf size=40 \[ \frac{(c+d x) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d} \]

[Out]

((-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d)

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Rubi [A]  time = 0.0055857, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2208} \[ \frac{(c+d x) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x)^3),x]

[Out]

((-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d)

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int e^{\frac{e}{(c+d x)^3}} \, dx &=\frac{\sqrt [3]{-\frac{e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0060606, size = 40, normalized size = 1. \[ \frac{(c+d x) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(e/(c + d*x)^3),x]

[Out]

((-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d)

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{3}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 3 \, d e \int \frac{x e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} + x e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*d*e*integrate(x*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c
^3*d*x + c^4), x) + x*e^(e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))

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Fricas [B]  time = 1.57395, size = 189, normalized size = 4.72 \begin{align*} -\frac{d \left (-\frac{e}{d^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{2}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (d x + c\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*(-e/d^3)^(1/3)*gamma(2/3, -e/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - (d*x + c)*e^(e/(d^3*x^3 + 3*c*d^
2*x^2 + 3*c^2*d*x + c^3)))/d

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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)**3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (\frac{e}{{\left (d x + c\right )}^{3}}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(e/(d*x + c)^3), x)