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

Optimal. Leaf size=38 \[ -\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2227, 2208} \[ -\frac {(a+b x) \text {Gamma}\left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 2227

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*} \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx &=\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]  time = 0.01, size = 38, normalized size = 1.00 \[ -\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]

Antiderivative was successfully verified.

[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))

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 44, normalized size = 1.16 \[ \frac {\left (-b^{3}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {e}}^{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a^{3}} \int e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________