3.673 \(\int e^{3 x} (-8+2 x^3+x^5) \, dx\)

Optimal. Leaf size=68 \[ \frac{1}{3} e^{3 x} x^5-\frac{5}{9} e^{3 x} x^4+\frac{38}{27} e^{3 x} x^3-\frac{38}{27} e^{3 x} x^2+\frac{76}{81} e^{3 x} x-\frac{724 e^{3 x}}{243} \]

[Out]

(-724*E^(3*x))/243 + (76*E^(3*x)*x)/81 - (38*E^(3*x)*x^2)/27 + (38*E^(3*x)*x^3)/27 - (5*E^(3*x)*x^4)/9 + (E^(3
*x)*x^5)/3

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Rubi [A]  time = 0.107394, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2196, 2194, 2176} \[ \frac{1}{3} e^{3 x} x^5-\frac{5}{9} e^{3 x} x^4+\frac{38}{27} e^{3 x} x^3-\frac{38}{27} e^{3 x} x^2+\frac{76}{81} e^{3 x} x-\frac{724 e^{3 x}}{243} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*x)*(-8 + 2*x^3 + x^5),x]

[Out]

(-724*E^(3*x))/243 + (76*E^(3*x)*x)/81 - (38*E^(3*x)*x^2)/27 + (38*E^(3*x)*x^3)/27 - (5*E^(3*x)*x^4)/9 + (E^(3
*x)*x^5)/3

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx &=\int \left (-8 e^{3 x}+2 e^{3 x} x^3+e^{3 x} x^5\right ) \, dx\\ &=2 \int e^{3 x} x^3 \, dx-8 \int e^{3 x} \, dx+\int e^{3 x} x^5 \, dx\\ &=-\frac{8 e^{3 x}}{3}+\frac{2}{3} e^{3 x} x^3+\frac{1}{3} e^{3 x} x^5-\frac{5}{3} \int e^{3 x} x^4 \, dx-2 \int e^{3 x} x^2 \, dx\\ &=-\frac{8 e^{3 x}}{3}-\frac{2}{3} e^{3 x} x^2+\frac{2}{3} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5+\frac{4}{3} \int e^{3 x} x \, dx+\frac{20}{9} \int e^{3 x} x^3 \, dx\\ &=-\frac{8 e^{3 x}}{3}+\frac{4}{9} e^{3 x} x-\frac{2}{3} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5-\frac{4}{9} \int e^{3 x} \, dx-\frac{20}{9} \int e^{3 x} x^2 \, dx\\ &=-\frac{76 e^{3 x}}{27}+\frac{4}{9} e^{3 x} x-\frac{38}{27} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5+\frac{40}{27} \int e^{3 x} x \, dx\\ &=-\frac{76 e^{3 x}}{27}+\frac{76}{81} e^{3 x} x-\frac{38}{27} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5-\frac{40}{81} \int e^{3 x} \, dx\\ &=-\frac{724 e^{3 x}}{243}+\frac{76}{81} e^{3 x} x-\frac{38}{27} e^{3 x} x^2+\frac{38}{27} e^{3 x} x^3-\frac{5}{9} e^{3 x} x^4+\frac{1}{3} e^{3 x} x^5\\ \end{align*}

Mathematica [A]  time = 0.0212306, size = 34, normalized size = 0.5 \[ \frac{1}{243} e^{3 x} \left (81 x^5-135 x^4+342 x^3-342 x^2+228 x-724\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*x)*(-8 + 2*x^3 + x^5),x]

[Out]

(E^(3*x)*(-724 + 228*x - 342*x^2 + 342*x^3 - 135*x^4 + 81*x^5))/243

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Maple [A]  time = 0.024, size = 32, normalized size = 0.5 \begin{align*}{\frac{{{\rm e}^{3\,x}} \left ( 81\,{x}^{5}-135\,{x}^{4}+342\,{x}^{3}-342\,{x}^{2}+228\,x-724 \right ) }{243}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(3*x)*(x^5+2*x^3-8),x)

[Out]

1/243*exp(3*x)*(81*x^5-135*x^4+342*x^3-342*x^2+228*x-724)

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Maxima [A]  time = 0.983796, size = 80, normalized size = 1.18 \begin{align*} \frac{1}{243} \,{\left (81 \, x^{5} - 135 \, x^{4} + 180 \, x^{3} - 180 \, x^{2} + 120 \, x - 40\right )} e^{\left (3 \, x\right )} + \frac{2}{27} \,{\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} - \frac{8}{3} \, e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3*x)*(x^5+2*x^3-8),x, algorithm="maxima")

[Out]

1/243*(81*x^5 - 135*x^4 + 180*x^3 - 180*x^2 + 120*x - 40)*e^(3*x) + 2/27*(9*x^3 - 9*x^2 + 6*x - 2)*e^(3*x) - 8
/3*e^(3*x)

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Fricas [A]  time = 0.853035, size = 92, normalized size = 1.35 \begin{align*} \frac{1}{243} \,{\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3*x)*(x^5+2*x^3-8),x, algorithm="fricas")

[Out]

1/243*(81*x^5 - 135*x^4 + 342*x^3 - 342*x^2 + 228*x - 724)*e^(3*x)

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Sympy [A]  time = 0.101229, size = 31, normalized size = 0.46 \begin{align*} \frac{\left (81 x^{5} - 135 x^{4} + 342 x^{3} - 342 x^{2} + 228 x - 724\right ) e^{3 x}}{243} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3*x)*(x**5+2*x**3-8),x)

[Out]

(81*x**5 - 135*x**4 + 342*x**3 - 342*x**2 + 228*x - 724)*exp(3*x)/243

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Giac [A]  time = 1.2537, size = 42, normalized size = 0.62 \begin{align*} \frac{1}{243} \,{\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3*x)*(x^5+2*x^3-8),x, algorithm="giac")

[Out]

1/243*(81*x^5 - 135*x^4 + 342*x^3 - 342*x^2 + 228*x - 724)*e^(3*x)