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

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

[Out]

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

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Rubi [A]  time = 0.0051903, 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} \[ -\frac{(c+d x) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]

Antiderivative was successfully verified.

[In]

Int[E^(e*(c + d*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0061005, size = 40, normalized size = 1. \[ -\frac{(c+d x) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(e*(c + d*x)^3),x]

[Out]

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

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Maple [F]  time = 0.012, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{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*} \int e^{\left ({\left (d x + c\right )}^{3} e\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="maxima")

[Out]

integrate(e^((d*x + c)^3*e), x)

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Fricas [A]  time = 1.53867, size = 120, normalized size = 3. \begin{align*} \frac{\left (-d^{3} e\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{3} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c^{3} e} \int e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e*(d*x+c)**3),x)

[Out]

exp(c**3*e)*Integral(exp(d**3*e*x**3)*exp(3*c*d**2*e*x**2)*exp(3*c**2*d*e*x), x)

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