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 = 0.0150699, 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 \[ -\frac{(c+d x) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]

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))

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Rubi in Sympy [A]  time = 1.69646, size = 36, normalized size = 0.9 \[ - \frac{\left (c + d x\right ) \Gamma{\left (\frac{1}{3},- e \left (c + d x\right )^{3} \right )}}{3 d \sqrt [3]{- e \left (c + d x\right )^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e*(d*x+c)**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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Mathematica [A]  time = 0.00250611, 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 = 0.019, size = 0, normalized size = 0. \[ \int{{\rm e}^{e \left ( dx+c \right ) ^{3}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^((d*x + c)^3*e),x, algorithm="maxima")

[Out]

integrate(e^((d*x + c)^3*e), x)

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Fricas [A]  time = 0.234419, size = 62, normalized size = 1.55 \[ -\frac{\Gamma \left (\frac{1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, \left (-d^{3} e\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*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)^(1/3)

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^((d*x + c)^3*e),x, algorithm="giac")

[Out]

integrate(e^((d*x + c)^3*e), x)