3.1.35 \(\int (2 x+e^x (-91-95 x-2 x^2)) \, dx\)

Optimal. Leaf size=18 \[ e+x \left (x-e^x (81+2 (5+x))\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2196, 2194, 2176} \begin {gather*} -2 e^x x^2+x^2-91 e^x x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*x + E^x*(-91 - 95*x - 2*x^2),x]

[Out]

-91*E^x*x + x^2 - 2*E^x*x^2

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 {gather*} \begin {aligned} \text {integral} &=x^2+\int e^x \left (-91-95 x-2 x^2\right ) \, dx\\ &=x^2+\int \left (-91 e^x-95 e^x x-2 e^x x^2\right ) \, dx\\ &=x^2-2 \int e^x x^2 \, dx-91 \int e^x \, dx-95 \int e^x x \, dx\\ &=-91 e^x-95 e^x x+x^2-2 e^x x^2+4 \int e^x x \, dx+95 \int e^x \, dx\\ &=4 e^x-91 e^x x+x^2-2 e^x x^2-4 \int e^x \, dx\\ &=-91 e^x x+x^2-2 e^x x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} x^2-e^x \left (91 x+2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*x + E^x*(-91 - 95*x - 2*x^2),x]

[Out]

x^2 - E^x*(91*x + 2*x^2)

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fricas [A]  time = 0.65, size = 17, normalized size = 0.94 \begin {gather*} x^{2} - {\left (2 \, x^{2} + 91 \, x\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2-95*x-91)*exp(x)+2*x,x, algorithm="fricas")

[Out]

x^2 - (2*x^2 + 91*x)*e^x

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giac [A]  time = 0.26, size = 17, normalized size = 0.94 \begin {gather*} x^{2} - {\left (2 \, x^{2} + 91 \, x\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2-95*x-91)*exp(x)+2*x,x, algorithm="giac")

[Out]

x^2 - (2*x^2 + 91*x)*e^x

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maple [A]  time = 0.02, size = 17, normalized size = 0.94

method	result	size
default	\(-91 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x} x^{2}+x^{2}\)	\(17\)
norman	\(-91 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x} x^{2}+x^{2}\)	\(17\)
risch	\(\left (-2 x^{2}-91 x \right ) {\mathrm e}^{x}+x^{2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2-95*x-91)*exp(x)+2*x,x,method=_RETURNVERBOSE)

[Out]

-91*exp(x)*x-2*exp(x)*x^2+x^2

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maxima [A]  time = 0.38, size = 27, normalized size = 1.50 \begin {gather*} x^{2} - 2 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 95 \, {\left (x - 1\right )} e^{x} - 91 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2-95*x-91)*exp(x)+2*x,x, algorithm="maxima")

[Out]

x^2 - 2*(x^2 - 2*x + 2)*e^x - 95*(x - 1)*e^x - 91*e^x

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mupad [B]  time = 0.04, size = 16, normalized size = 0.89 \begin {gather*} x^2-91\,x\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - exp(x)*(95*x + 2*x^2 + 91),x)

[Out]

x^2 - 91*x*exp(x) - 2*x^2*exp(x)

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sympy [A]  time = 0.08, size = 15, normalized size = 0.83 \begin {gather*} x^{2} + \left (- 2 x^{2} - 91 x\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2-95*x-91)*exp(x)+2*x,x)

[Out]

x**2 + (-2*x**2 - 91*x)*exp(x)

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