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

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

[Out]

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

________________________________________________________________________________________

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

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)

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[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)

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 89, normalized size = 2.22 \[ -\frac {d \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (d x + c\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(e/(d*x + c)^3), x)

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/(d*x+c)^3*e),x)

[Out]

int(exp(1/(d*x+c)^3*e),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 3 \, d e \int \frac {x e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} + x e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(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))

________________________________________________________________________________________

mupad [B]  time = 3.95, size = 61, normalized size = 1.52 \[ \frac {\left (c+d\,x\right )\,\left ({\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {e}{{\left (c+d\,x\right )}^3}\right )}^{1/3}-{\left (-\frac {e}{{\left (c+d\,x\right )}^3}\right )}^{1/3}\,\Gamma \left (\frac {2}{3},-\frac {e}{{\left (c+d\,x\right )}^3}\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x)^3),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(e/(c + d*x)**3), x)

________________________________________________________________________________________