3.42.97 \(\int e^{2 x} (14 x+11 x^2-2 x^3) \, dx\)

Optimal. Leaf size=16 \[ e^{2 x} x \left (7 x-x^2\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 21, normalized size of antiderivative = 1.31, number of steps used = 12, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1594, 2196, 2176, 2194} \begin {gather*} 7 e^{2 x} x^2-e^{2 x} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*x)*(14*x + 11*x^2 - 2*x^3),x]

[Out]

7*E^(2*x)*x^2 - E^(2*x)*x^3

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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} &=\int e^{2 x} x \left (14+11 x-2 x^2\right ) \, dx\\ &=\int \left (14 e^{2 x} x+11 e^{2 x} x^2-2 e^{2 x} x^3\right ) \, dx\\ &=-\left (2 \int e^{2 x} x^3 \, dx\right )+11 \int e^{2 x} x^2 \, dx+14 \int e^{2 x} x \, dx\\ &=7 e^{2 x} x+\frac {11}{2} e^{2 x} x^2-e^{2 x} x^3+3 \int e^{2 x} x^2 \, dx-7 \int e^{2 x} \, dx-11 \int e^{2 x} x \, dx\\ &=-\frac {7 e^{2 x}}{2}+\frac {3}{2} e^{2 x} x+7 e^{2 x} x^2-e^{2 x} x^3-3 \int e^{2 x} x \, dx+\frac {11}{2} \int e^{2 x} \, dx\\ &=-\frac {3 e^{2 x}}{4}+7 e^{2 x} x^2-e^{2 x} x^3+\frac {3}{2} \int e^{2 x} \, dx\\ &=7 e^{2 x} x^2-e^{2 x} x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 13, normalized size = 0.81 \begin {gather*} -e^{2 x} (-7+x) x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)*(14*x + 11*x^2 - 2*x^3),x]

[Out]

-(E^(2*x)*(-7 + x)*x^2)

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fricas [A]  time = 0.55, size = 15, normalized size = 0.94 \begin {gather*} -{\left (x^{3} - 7 \, x^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+11*x^2+14*x)*exp(x)^2,x, algorithm="fricas")

[Out]

-(x^3 - 7*x^2)*e^(2*x)

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giac [A]  time = 0.18, size = 15, normalized size = 0.94 \begin {gather*} -{\left (x^{3} - 7 \, x^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+11*x^2+14*x)*exp(x)^2,x, algorithm="giac")

[Out]

-(x^3 - 7*x^2)*e^(2*x)

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maple [A]  time = 0.02, size = 13, normalized size = 0.81

method	result	size
gosper	\(-{\mathrm e}^{2 x} \left (x -7\right ) x^{2}\)	\(13\)
risch	\(\left (-x^{3}+7 x^{2}\right ) {\mathrm e}^{2 x}\)	\(17\)
default	\(7 \,{\mathrm e}^{2 x} x^{2}-{\mathrm e}^{2 x} x^{3}\)	\(20\)
norman	\(7 \,{\mathrm e}^{2 x} x^{2}-{\mathrm e}^{2 x} x^{3}\)	\(20\)
meijerg	\(\frac {\left (-32 x^{3}+48 x^{2}-48 x +24\right ) {\mathrm e}^{2 x}}{32}+\frac {11 \left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{24}-\frac {7 \left (-4 x +2\right ) {\mathrm e}^{2 x}}{4}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3+11*x^2+14*x)*exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

-exp(x)^2*(x-7)*x^2

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maxima [B]  time = 0.35, size = 49, normalized size = 3.06 \begin {gather*} -\frac {1}{4} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {11}{4} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {7}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+11*x^2+14*x)*exp(x)^2,x, algorithm="maxima")

[Out]

-1/4*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 11/4*(2*x^2 - 2*x + 1)*e^(2*x) + 7/2*(2*x - 1)*e^(2*x)

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mupad [B]  time = 0.04, size = 12, normalized size = 0.75 \begin {gather*} -x^2\,{\mathrm {e}}^{2\,x}\,\left (x-7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*(14*x + 11*x^2 - 2*x^3),x)

[Out]

-x^2*exp(2*x)*(x - 7)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} \left (- x^{3} + 7 x^{2}\right ) e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3+11*x**2+14*x)*exp(x)**2,x)

[Out]

(-x**3 + 7*x**2)*exp(2*x)

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