3.69.6 $\int e^{1+x}(1+2x+x^2+x^3)dx$

Optimal. Leaf size=21 $$1+e^{1+x}(1-x)(-5+x-x^2)$$

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Rubi [A]  time = 0.09, antiderivative size = 35, normalized size of antiderivative = 1.67, number of steps used = 12, number of rules used = 3, integrand size = 17, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.176, Rules used = {2196, 2194, 2176} $$\begin{array}{c}{e}^{x+1}{x}^3-2e^{x+1}{x}^2+6e^{x+1}x-5e^{x+1}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[E^(1 + x)*(1 + 2*x + x^2 + x^3),x]

[Out]

-5*E^(1 + x) + 6*E^(1 + x)*x - 2*E^(1 + x)*x^2 + E^(1 + x)*x^3

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

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 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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (e^{1+x}+2e^{1+x}x+e^{1+x}{x}^2+e^{1+x}{x}^3)dx\\ & =2\int e^{1+x}xdx+\int e^{1+x}dx+\int e^{1+x}{x}^{2}dx+\int e^{1+x}{x}^{3}dx\\ & =e^{1+x}+2e^{1+x}x+e^{1+x}{x}^2+e^{1+x}{x}^3-2\int e^{1+x}dx-2\int e^{1+x}xdx-3\int e^{1+x}{x}^{2}dx\\ & =-e^{1+x}-2e^{1+x}{x}^2+e^{1+x}{x}^3+2\int e^{1+x}dx+6\int e^{1+x}xdx\\ & =e^{1+x}+6e^{1+x}x-2e^{1+x}{x}^2+e^{1+x}{x}^3-6\int e^{1+x}dx\\ & =-5e^{1+x}+6e^{1+x}x-2e^{1+x}{x}^2+e^{1+x}{x}^3\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 19, normalized size = 0.90 $$\begin{array}{c}{e}^{1+x}(-5+6x-2x^2+x^3)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[E^(1 + x)*(1 + 2*x + x^2 + x^3),x]

[Out]

E^(1 + x)*(-5 + 6*x - 2*x^2 + x^3)

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fricas [A]  time = 0.51, size = 18, normalized size = 0.86 $$\begin{array}{c}(x^3-2x^2+6x-5)e^{(x+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2+2*x+1)*exp(x+1),x, algorithm="fricas")

[Out]

(x^3 - 2*x^2 + 6*x - 5)*e^(x + 1)

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giac [A]  time = 0.21, size = 18, normalized size = 0.86 $$\begin{array}{c}(x^3-2x^2+6x-5)e^{(x+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2+2*x+1)*exp(x+1),x, algorithm="giac")

[Out]

(x^3 - 2*x^2 + 6*x - 5)*e^(x + 1)

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maple [A]  time = 0.04, size = 19, normalized size = 0.90

method	result	size
gosper	$(x^3-2x^2+6x-5){\mathrm{e}}^{x+1}$	$19$
risch	$(x^3-2x^2+6x-5){\mathrm{e}}^{x+1}$	$19$
norman	$x^3{\mathrm{e}}^{x+1}+6x{\mathrm{e}}^{x+1}-2x^2{\mathrm{e}}^{x+1}-5{\mathrm{e}}^{x+1}$	$32$
derivativedivides	${\mathrm{e}}^{x+1}{(x+1)}^3-5{\mathrm{e}}^{x+1}{(x+1)}^2+13{\mathrm{e}}^{x+1}(x+1)-14{\mathrm{e}}^{x+1}$	$38$
default	${\mathrm{e}}^{x+1}{(x+1)}^3-5{\mathrm{e}}^{x+1}{(x+1)}^2+13{\mathrm{e}}^{x+1}(x+1)-14{\mathrm{e}}^{x+1}$	$38$
meijerg	$\mathrm{e}(6-\frac{(-4x^3+12x^2-24x+24){\mathrm{e}}^x}{4})-\mathrm{e}(2-\frac{(3x^2-6x+6){\mathrm{e}}^x}{3})+2\mathrm{e}(1-\frac{(-2x+2){\mathrm{e}}^x}{2})-(1-{\mathrm{e}}^x)\mathrm{e}$	$71$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+x^2+2*x+1)*exp(x+1),x,method=_RETURNVERBOSE)

[Out]

(x^3-2*x^2+6*x-5)*exp(x+1)

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maxima [B]  time = 0.37, size = 63, normalized size = 3.00 $$\begin{array}{c}(x^{3}e-3x^{2}e+6xe-6e)e^x+(x^{2}e-2xe+2e)e^x+2(xe-e)e^x+e^{(x+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2+2*x+1)*exp(x+1),x, algorithm="maxima")

[Out]

(x^3*e - 3*x^2*e + 6*x*e - 6*e)*e^x + (x^2*e - 2*x*e + 2*e)*e^x + 2*(x*e - e)*e^x + e^(x + 1)

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mupad [B]  time = 0.04, size = 18, normalized size = 0.86 $$\begin{array}{c}{\mathrm{e}}^{x+1}(x^3-2x^2+6x-5)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + 1)*(2*x + x^2 + x^3 + 1),x)

[Out]

exp(x + 1)*(6*x - 2*x^2 + x^3 - 5)

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sympy [A]  time = 0.09, size = 17, normalized size = 0.81 $$\begin{array}{c}(x^3-2x^2+6x-5)e^{x+1}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+x**2+2*x+1)*exp(x+1),x)

[Out]

(x**3 - 2*x**2 + 6*x - 5)*exp(x + 1)

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