3.4.94 \(\int e^{e (c+d x)^3} \, dx\) [394]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {2239} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(e*(c + d*x)^3),x]

[Out]

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

Rule 2239

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*}

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Mathematica [A]
time = 0.05, size = 40, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(e*(c + d*x)^3),x]

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{e \left (d x +c \right )^{3}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.13, size = 49, normalized size = 1.22 \begin {gather*} \frac {\left (-d^{3} e\right )^{\frac {2}{3}} e^{\left (-1\right )} \Gamma \left (\frac {1}{3}, -{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} e\right )}{3 \, d^{3}} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e*(c + d*x)^3),x)

[Out]

int(exp(e*(c + d*x)^3), x)

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