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 = 0.0151144, 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 \[ \frac{(c+d x) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d} \]

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)

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Rubi in Sympy [A]  time = 1.79105, size = 36, normalized size = 0.9 \[ \frac{\sqrt [3]{- \frac{e}{\left (c + d x\right )^{3}}} \left (c + d x\right ) \Gamma{\left (- \frac{1}{3},- \frac{e}{\left (c + d x\right )^{3}} \right )}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e/(d*x+c)**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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Mathematica [A]  time = 0.00507461, size = 61, normalized size = 1.52 \[ \frac{e \text{Gamma}\left (\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{d (c+d x)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3}}+\frac{(c+d x) e^{\frac{e}{(c+d x)^3}}}{d} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(e/(c + d*x)^3),x]

[Out]

(E^(e/(c + d*x)^3)*(c + d*x))/d + (e*Gamma[2/3, -(e/(c + d*x)^3)])/(d*(-(e/(c +
d*x)^3))^(2/3)*(c + d*x)^2)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{3}}}}}\, dx \]

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. \[ 3 \, d e \int \frac{x e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} + x e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(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 [A]  time = 0.256165, size = 136, normalized size = 3.4 \[ \frac{{\left (d^{3} x + c d^{2}\right )} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} + e \Gamma \left (\frac{2}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{d^{3} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(e^(e/(d*x + c)^3), x)