\(\int e^{\frac {e}{(c+d x)^3}} \, dx\) [419]

Optimal result

Integrand size = 11, antiderivative size = 40 \[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\frac {\sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2239} \[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\frac {(c+d x) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d} \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\frac {\sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d} \]

[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)

Maple [F]

\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (34) = 68\).

Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.22 \[ \int e^{\frac {e}{(c+d x)^3}} \, dx=-\frac {d \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (d x + c\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d} \]

[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

Sympy [F]

\[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\int e^{\frac {e}{\left (c + d x\right )^{3}}}\, dx \]

[In]

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

[Out]

Integral(exp(e/(c + d*x)**3), x)

Maxima [F]

\[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]

[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))

Giac [F]

\[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\int { e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]

[In]

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

[Out]

integrate(e^(e/(d*x + c)^3), x)

Mupad [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.52 \[ \int e^{\frac {e}{(c+d x)^3}} \, dx=\frac {\left (c+d\,x\right )\,\left ({\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {e}{{\left (c+d\,x\right )}^3}\right )}^{1/3}-{\left (-\frac {e}{{\left (c+d\,x\right )}^3}\right )}^{1/3}\,\Gamma \left (\frac {2}{3},-\frac {e}{{\left (c+d\,x\right )}^3}\right )\right )}{d} \]

[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