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

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 = 0.01, 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 = {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*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.00 \[ -\frac {(c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(e*(c + d*x)^3),x]

[Out]

-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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fricas [A]  time = 0.43, size = 52, normalized size = 1.30 \[ \frac {\left (-d^{3} e\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{3} e} \]

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

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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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\left (d x +c \right )^{3} e}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c^{3} e} \int e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx \]

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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