3.73.41 \(\int (50+e^{3-x} (-1+x)+292 x+381 x^2+140 x^3+15 x^4) \, dx\)

Optimal. Leaf size=30 \[ x \left (-e^{3-x}+x+\left (3+\frac {2}{x}\right ) x (1+x) (5+x)^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.53, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2176, 2194} \begin {gather*} 3 x^5+35 x^4+127 x^3+146 x^2+50 x-e^{3-x}+e^{3-x} (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[50 + E^(3 - x)*(-1 + x) + 292*x + 381*x^2 + 140*x^3 + 15*x^4,x]

[Out]

-E^(3 - x) + E^(3 - x)*(1 - x) + 50*x + 146*x^2 + 127*x^3 + 35*x^4 + 3*x^5

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=50 x+146 x^2+127 x^3+35 x^4+3 x^5+\int e^{3-x} (-1+x) \, dx\\ &=e^{3-x} (1-x)+50 x+146 x^2+127 x^3+35 x^4+3 x^5+\int e^{3-x} \, dx\\ &=-e^{3-x}+e^{3-x} (1-x)+50 x+146 x^2+127 x^3+35 x^4+3 x^5\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 34, normalized size = 1.13 \begin {gather*} 50 x-e^{3-x} x+146 x^2+127 x^3+35 x^4+3 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[50 + E^(3 - x)*(-1 + x) + 292*x + 381*x^2 + 140*x^3 + 15*x^4,x]

[Out]

50*x - E^(3 - x)*x + 146*x^2 + 127*x^3 + 35*x^4 + 3*x^5

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fricas [A]  time = 0.73, size = 33, normalized size = 1.10 \begin {gather*} 3 \, x^{5} + 35 \, x^{4} + 127 \, x^{3} + 146 \, x^{2} - x e^{\left (-x + 3\right )} + 50 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-1)*exp(3-x)+15*x^4+140*x^3+381*x^2+292*x+50,x, algorithm="fricas")

[Out]

3*x^5 + 35*x^4 + 127*x^3 + 146*x^2 - x*e^(-x + 3) + 50*x

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giac [A]  time = 0.15, size = 33, normalized size = 1.10 \begin {gather*} 3 \, x^{5} + 35 \, x^{4} + 127 \, x^{3} + 146 \, x^{2} - x e^{\left (-x + 3\right )} + 50 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-1)*exp(3-x)+15*x^4+140*x^3+381*x^2+292*x+50,x, algorithm="giac")

[Out]

3*x^5 + 35*x^4 + 127*x^3 + 146*x^2 - x*e^(-x + 3) + 50*x

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maple [A]  time = 0.02, size = 34, normalized size = 1.13

method	result	size
norman	\(50 x +146 x^{2}+127 x^{3}+35 x^{4}+3 x^{5}-x \,{\mathrm e}^{3-x}\)	\(34\)
risch	\(50 x +146 x^{2}+127 x^{3}+35 x^{4}+3 x^{5}-x \,{\mathrm e}^{3-x}\)	\(34\)
default	\(50 x +{\mathrm e}^{3-x} \left (3-x \right )-3 \,{\mathrm e}^{3-x}+146 x^{2}+127 x^{3}+35 x^{4}+3 x^{5}\)	\(45\)
derivativedivides	\(146 \left (3-x \right )^{2}-2778+926 x +127 x^{3}+35 x^{4}+3 x^{5}+{\mathrm e}^{3-x} \left (3-x \right )-3 \,{\mathrm e}^{3-x}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-1)*exp(3-x)+15*x^4+140*x^3+381*x^2+292*x+50,x,method=_RETURNVERBOSE)

[Out]

50*x+146*x^2+127*x^3+35*x^4+3*x^5-x*exp(3-x)

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maxima [A]  time = 0.36, size = 33, normalized size = 1.10 \begin {gather*} 3 \, x^{5} + 35 \, x^{4} + 127 \, x^{3} + 146 \, x^{2} - x e^{\left (-x + 3\right )} + 50 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-1)*exp(3-x)+15*x^4+140*x^3+381*x^2+292*x+50,x, algorithm="maxima")

[Out]

3*x^5 + 35*x^4 + 127*x^3 + 146*x^2 - x*e^(-x + 3) + 50*x

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mupad [B]  time = 0.06, size = 30, normalized size = 1.00 \begin {gather*} x\,\left (146\,x-{\mathrm {e}}^{3-x}+127\,x^2+35\,x^3+3\,x^4+50\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(292*x + exp(3 - x)*(x - 1) + 381*x^2 + 140*x^3 + 15*x^4 + 50,x)

[Out]

x*(146*x - exp(3 - x) + 127*x^2 + 35*x^3 + 3*x^4 + 50)

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sympy [A]  time = 0.11, size = 29, normalized size = 0.97 \begin {gather*} 3 x^{5} + 35 x^{4} + 127 x^{3} + 146 x^{2} - x e^{3 - x} + 50 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-1)*exp(3-x)+15*x**4+140*x**3+381*x**2+292*x+50,x)

[Out]

3*x**5 + 35*x**4 + 127*x**3 + 146*x**2 - x*exp(3 - x) + 50*x

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