\(\int e^{3 x} (-8+2 x^3+x^5) \, dx\) [673]

Optimal result

Integrand size = 16, antiderivative size = 68 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=-\frac {724 e^{3 x}}{243}+\frac {76}{81} e^{3 x} x-\frac {38}{27} e^{3 x} x^2+\frac {38}{27} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5 \]

[Out]

-724/243*exp(3*x)+76/81*exp(3*x)*x-38/27*exp(3*x)*x^2+38/27*exp(3*x)*x^3-5/9*exp(3*x)*x^4+1/3*exp(3*x)*x^5

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2227, 2225, 2207} \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{3} e^{3 x} x^5-\frac {5}{9} e^{3 x} x^4+\frac {38}{27} e^{3 x} x^3-\frac {38}{27} e^{3 x} x^2+\frac {76}{81} e^{3 x} x-\frac {724 e^{3 x}}{243} \]

[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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

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] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.50 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} e^{3 x} \left (-724+228 x-342 x^2+342 x^3-135 x^4+81 x^5\right ) \]

[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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46

method	result	size
risch	\(\left (\frac {1}{3} x^{5}-\frac {5}{9} x^{4}+\frac {38}{27} x^{3}-\frac {38}{27} x^{2}+\frac {76}{81} x -\frac {724}{243}\right ) {\mathrm e}^{3 x}\)	\(31\)
gosper	\(\frac {{\mathrm e}^{3 x} \left (81 x^{5}-135 x^{4}+342 x^{3}-342 x^{2}+228 x -724\right )}{243}\)	\(32\)
derivativedivides	\(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\)	\(51\)
default	\(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\)	\(51\)
norman	\(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\)	\(51\)
parallelrisch	\(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\)	\(51\)
parts	\(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\)	\(51\)
meijerg	\(\frac {724}{243}-\frac {\left (-1458 x^{5}+2430 x^{4}-3240 x^{3}+3240 x^{2}-2160 x +720\right ) {\mathrm e}^{3 x}}{4374}-\frac {\left (-108 x^{3}+108 x^{2}-72 x +24\right ) {\mathrm e}^{3 x}}{162}-\frac {8 \,{\mathrm e}^{3 x}}{3}\)	\(61\)

[In]

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

[Out]

(1/3*x^5-5/9*x^4+38/27*x^3-38/27*x^2+76/81*x-724/243)*exp(3*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {\left (81 x^{5} - 135 x^{4} + 342 x^{3} - 342 x^{2} + 228 x - 724\right ) e^{3 x}}{243} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 180 \, x^{3} - 180 \, x^{2} + 120 \, x - 40\right )} e^{\left (3 \, x\right )} + \frac {2}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} - \frac {8}{3} \, e^{\left (3 \, x\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {{\mathrm {e}}^{3\,x}\,\left (81\,x^5-135\,x^4+342\,x^3-342\,x^2+228\,x-724\right )}{243} \]

[In]

int(exp(3*x)*(2*x^3 + x^5 - 8),x)

[Out]

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