3.92.69 \(\int e^{5+6 x^3} (-1-18 x^3) \, dx\)

Optimal. Leaf size=12 \[ -e^{5+6 x^3} x \]

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Rubi [C]  time = 0.05, antiderivative size = 62, normalized size of antiderivative = 5.17, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2226, 2208, 2218} \begin {gather*} \frac {e^5 x \Gamma \left (\frac {1}{3},-6 x^3\right )}{3 \sqrt [3]{6} \sqrt [3]{-x^3}}+\frac {e^5 x^4 \Gamma \left (\frac {4}{3},-6 x^3\right )}{\sqrt [3]{6} \left (-x^3\right )^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(5 + 6*x^3)*(-1 - 18*x^3),x]

[Out]

(E^5*x*Gamma[1/3, -6*x^3])/(3*6^(1/3)*(-x^3)^(1/3)) + (E^5*x^4*Gamma[4/3, -6*x^3])/(6^(1/3)*(-x^3)^(4/3))

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 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{5+6 x^3}-18 e^{5+6 x^3} x^3\right ) \, dx\\ &=-\left (18 \int e^{5+6 x^3} x^3 \, dx\right )-\int e^{5+6 x^3} \, dx\\ &=\frac {e^5 x \Gamma \left (\frac {1}{3},-6 x^3\right )}{3 \sqrt [3]{6} \sqrt [3]{-x^3}}+\frac {e^5 x^4 \Gamma \left (\frac {4}{3},-6 x^3\right )}{\sqrt [3]{6} \left (-x^3\right )^{4/3}}\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.02, size = 43, normalized size = 3.58 \begin {gather*} \frac {e^5 x \left (\Gamma \left (\frac {1}{3},-6 x^3\right )-3 \Gamma \left (\frac {4}{3},-6 x^3\right )\right )}{3 \sqrt [3]{6} \sqrt [3]{-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(5 + 6*x^3)*(-1 - 18*x^3),x]

[Out]

(E^5*x*(Gamma[1/3, -6*x^3] - 3*Gamma[4/3, -6*x^3]))/(3*6^(1/3)*(-x^3)^(1/3))

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fricas [A]  time = 0.71, size = 11, normalized size = 0.92 \begin {gather*} -x e^{\left (6 \, x^{3} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x^3-1)*exp(6*x^3+5),x, algorithm="fricas")

[Out]

-x*e^(6*x^3 + 5)

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giac [A]  time = 0.13, size = 11, normalized size = 0.92 \begin {gather*} -x e^{\left (6 \, x^{3} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x^3-1)*exp(6*x^3+5),x, algorithm="giac")

[Out]

-x*e^(6*x^3 + 5)

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maple [A]  time = 0.06, size = 12, normalized size = 1.00

method	result	size
gosper	\(-x \,{\mathrm e}^{6 x^{3}+5}\)	\(12\)
norman	\(-x \,{\mathrm e}^{6 x^{3}+5}\)	\(12\)
risch	\(-x \,{\mathrm e}^{6 x^{3}+5}\)	\(12\)
meijerg	\(-\frac {{\mathrm e}^{5} 6^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} \left (\frac {2 x \left (-1\right )^{\frac {1}{3}} \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) \left (-x^{3}\right )^{\frac {1}{3}}}-x 6^{\frac {1}{3}} \left (-1\right )^{\frac {1}{3}} {\mathrm e}^{6 x^{3}}-\frac {x \left (-1\right )^{\frac {1}{3}} \Gamma \left (\frac {1}{3}, -6 x^{3}\right )}{3 \left (-x^{3}\right )^{\frac {1}{3}}}\right )}{6}+\frac {{\mathrm e}^{5} 6^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} \left (\frac {2 x \left (-1\right )^{\frac {1}{3}} \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) \left (-x^{3}\right )^{\frac {1}{3}}}-\frac {x \left (-1\right )^{\frac {1}{3}} \Gamma \left (\frac {1}{3}, -6 x^{3}\right )}{\left (-x^{3}\right )^{\frac {1}{3}}}\right )}{18}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*x^3-1)*exp(6*x^3+5),x,method=_RETURNVERBOSE)

[Out]

-x*exp(6*x^3+5)

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maxima [C]  time = 0.37, size = 47, normalized size = 3.92 \begin {gather*} \frac {6^{\frac {2}{3}} x^{4} e^{5} \Gamma \left (\frac {4}{3}, -6 \, x^{3}\right )}{6 \, \left (-x^{3}\right )^{\frac {4}{3}}} + \frac {6^{\frac {2}{3}} x e^{5} \Gamma \left (\frac {1}{3}, -6 \, x^{3}\right )}{18 \, \left (-x^{3}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x^3-1)*exp(6*x^3+5),x, algorithm="maxima")

[Out]

1/6*6^(2/3)*x^4*e^5*gamma(4/3, -6*x^3)/(-x^3)^(4/3) + 1/18*6^(2/3)*x*e^5*gamma(1/3, -6*x^3)/(-x^3)^(1/3)

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mupad [B]  time = 0.04, size = 11, normalized size = 0.92 \begin {gather*} -x\,{\mathrm {e}}^5\,{\mathrm {e}}^{6\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(6*x^3 + 5)*(18*x^3 + 1),x)

[Out]

-x*exp(5)*exp(6*x^3)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.83 \begin {gather*} - x e^{6 x^{3} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x**3-1)*exp(6*x**3+5),x)

[Out]

-x*exp(6*x**3 + 5)

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