\(\int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx\) [212]

Optimal result

Integrand size = 29, antiderivative size = 38 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]

[Out]

-1/3*(b*x+a)*GAMMA(1/3,-(b*x+a)^3)/b/(-(b*x+a)^3)^(1/3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2259, 2239} \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]

[In]

Int[E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3),x]

[Out]

-1/3*((a + b*x)*Gamma[1/3, -(a + b*x)^3])/(b*(-(a + b*x)^3)^(1/3))

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]

Rule 2259

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \int e^{(a+b x)^3} \, dx \\ & = -\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]

[In]

Integrate[E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3),x]

[Out]

-1/3*((a + b*x)*Gamma[1/3, -(a + b*x)^3])/(b*(-(a + b*x)^3)^(1/3))

Maple [F]

\[\int {\mathrm e}^{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x\]

[In]

int(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3),x)

[Out]

int(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3),x)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\frac {\left (-b^{3}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, b^{3}} \]

[In]

integrate(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3),x, algorithm="fricas")

[Out]

1/3*(-b^3)^(2/3)*gamma(1/3, -b^3*x^3 - 3*a*b^2*x^2 - 3*a^2*b*x - a^3)/b^3

Sympy [F]

\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=e^{a^{3}} \int e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \]

[In]

integrate(exp(b**3*x**3+3*a*b**2*x**2+3*a**2*b*x+a**3),x)

[Out]

exp(a**3)*Integral(exp(b**3*x**3)*exp(3*a*b**2*x**2)*exp(3*a**2*b*x), x)

Maxima [F]

\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\int { e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]

[In]

integrate(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3),x, algorithm="maxima")

[Out]

integrate(e^(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

Giac [F]

\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\int { e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]

[In]

integrate(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3),x, algorithm="giac")

[Out]

integrate(e^(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

Mupad [F(-1)]

Timed out. \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\int {\mathrm {e}}^{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \,d x \]

[In]

int(exp(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x),x)

[Out]

int(exp(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x), x)