3.1 \(\int e^x (1-x^3+x^4-x^5+x^6) \, dx\)

Optimal. Leaf size=51 \[ e^x x^6-7 e^x x^5+36 e^x x^4-145 e^x x^3+435 e^x x^2-870 e^x x+871 e^x \]

[Out]

871*E^x - 870*E^x*x + 435*E^x*x^2 - 145*E^x*x^3 + 36*E^x*x^4 - 7*E^x*x^5 + E^x*x^6

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Rubi [A]  time = 0.171166, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2196, 2194, 2176} \[ e^x x^6-7 e^x x^5+36 e^x x^4-145 e^x x^3+435 e^x x^2-870 e^x x+871 e^x \]

Antiderivative was successfully verified.

[In]

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

[Out]

871*E^x - 870*E^x*x + 435*E^x*x^2 - 145*E^x*x^3 + 36*E^x*x^4 - 7*E^x*x^5 + E^x*x^6

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^x \left (1-x^3+x^4-x^5+x^6\right ) \, dx &=\int \left (e^x-e^x x^3+e^x x^4-e^x x^5+e^x x^6\right ) \, dx\\ &=\int e^x \, dx-\int e^x x^3 \, dx+\int e^x x^4 \, dx-\int e^x x^5 \, dx+\int e^x x^6 \, dx\\ &=e^x-e^x x^3+e^x x^4-e^x x^5+e^x x^6+3 \int e^x x^2 \, dx-4 \int e^x x^3 \, dx+5 \int e^x x^4 \, dx-6 \int e^x x^5 \, dx\\ &=e^x+3 e^x x^2-5 e^x x^3+6 e^x x^4-7 e^x x^5+e^x x^6-6 \int e^x x \, dx+12 \int e^x x^2 \, dx-20 \int e^x x^3 \, dx+30 \int e^x x^4 \, dx\\ &=e^x-6 e^x x+15 e^x x^2-25 e^x x^3+36 e^x x^4-7 e^x x^5+e^x x^6+6 \int e^x \, dx-24 \int e^x x \, dx+60 \int e^x x^2 \, dx-120 \int e^x x^3 \, dx\\ &=7 e^x-30 e^x x+75 e^x x^2-145 e^x x^3+36 e^x x^4-7 e^x x^5+e^x x^6+24 \int e^x \, dx-120 \int e^x x \, dx+360 \int e^x x^2 \, dx\\ &=31 e^x-150 e^x x+435 e^x x^2-145 e^x x^3+36 e^x x^4-7 e^x x^5+e^x x^6+120 \int e^x \, dx-720 \int e^x x \, dx\\ &=151 e^x-870 e^x x+435 e^x x^2-145 e^x x^3+36 e^x x^4-7 e^x x^5+e^x x^6+720 \int e^x \, dx\\ &=871 e^x-870 e^x x+435 e^x x^2-145 e^x x^3+36 e^x x^4-7 e^x x^5+e^x x^6\\ \end{align*}

Mathematica [A]  time = 0.0248788, size = 32, normalized size = 0.63 \[ e^x \left (x^6-7 x^5+36 x^4-145 x^3+435 x^2-870 x+871\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^x*(871 - 870*x + 435*x^2 - 145*x^3 + 36*x^4 - 7*x^5 + x^6)

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Maple [A]  time = 0.003, size = 32, normalized size = 0.6 \begin{align*} \left ({x}^{6}-7\,{x}^{5}+36\,{x}^{4}-145\,{x}^{3}+435\,{x}^{2}-870\,x+871 \right ){{\rm e}^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-x^5+x^4-x^3+1)*exp(x),x)

[Out]

(x^6-7*x^5+36*x^4-145*x^3+435*x^2-870*x+871)*exp(x)

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Maxima [B]  time = 0.939958, size = 134, normalized size = 2.63 \begin{align*}{\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} -{\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} +{\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} -{\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-x^5+x^4-x^3+1)*exp(x),x, algorithm="maxima")

[Out]

(x^6 - 6*x^5 + 30*x^4 - 120*x^3 + 360*x^2 - 720*x + 720)*e^x - (x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e
^x + (x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x - (x^3 - 3*x^2 + 6*x - 6)*e^x + e^x

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Fricas [A]  time = 1.5627, size = 84, normalized size = 1.65 \begin{align*}{\left (x^{6} - 7 \, x^{5} + 36 \, x^{4} - 145 \, x^{3} + 435 \, x^{2} - 870 \, x + 871\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-x^5+x^4-x^3+1)*exp(x),x, algorithm="fricas")

[Out]

(x^6 - 7*x^5 + 36*x^4 - 145*x^3 + 435*x^2 - 870*x + 871)*e^x

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Sympy [A]  time = 0.094271, size = 31, normalized size = 0.61 \begin{align*} \left (x^{6} - 7 x^{5} + 36 x^{4} - 145 x^{3} + 435 x^{2} - 870 x + 871\right ) e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-x**5+x**4-x**3+1)*exp(x),x)

[Out]

(x**6 - 7*x**5 + 36*x**4 - 145*x**3 + 435*x**2 - 870*x + 871)*exp(x)

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Giac [A]  time = 1.07739, size = 42, normalized size = 0.82 \begin{align*}{\left (x^{6} - 7 \, x^{5} + 36 \, x^{4} - 145 \, x^{3} + 435 \, x^{2} - 870 \, x + 871\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-x^5+x^4-x^3+1)*exp(x),x, algorithm="giac")

[Out]

(x^6 - 7*x^5 + 36*x^4 - 145*x^3 + 435*x^2 - 870*x + 871)*e^x