3.63.36 \(\int \frac {1}{2} (3+e (-2-4 x)-18 x-9 x^2) \, dx\)

Optimal. Leaf size=19 \[ \left (3+e+\frac {3 x}{2}\right ) \left (3-x-x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.68, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12} \begin {gather*} -\frac {3 x^3}{2}-\frac {9 x^2}{2}+\frac {3 x}{2}-\frac {1}{4} e (2 x+1)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + E*(-2 - 4*x) - 18*x - 9*x^2)/2,x]

[Out]

(3*x)/2 - (9*x^2)/2 - (3*x^3)/2 - (E*(1 + 2*x)^2)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (3+e (-2-4 x)-18 x-9 x^2\right ) \, dx\\ &=\frac {3 x}{2}-\frac {9 x^2}{2}-\frac {3 x^3}{2}-\frac {1}{4} e (1+2 x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.58 \begin {gather*} \frac {3 x}{2}-e x-\frac {9 x^2}{2}-e x^2-\frac {3 x^3}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E*(-2 - 4*x) - 18*x - 9*x^2)/2,x]

[Out]

(3*x)/2 - E*x - (9*x^2)/2 - E*x^2 - (3*x^3)/2

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fricas [A]  time = 0.53, size = 23, normalized size = 1.21 \begin {gather*} -\frac {3}{2} \, x^{3} - \frac {9}{2} \, x^{2} - {\left (x^{2} + x\right )} e + \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*x-2)*exp(1)-9/2*x^2-9*x+3/2,x, algorithm="fricas")

[Out]

-3/2*x^3 - 9/2*x^2 - (x^2 + x)*e + 3/2*x

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giac [A]  time = 0.16, size = 23, normalized size = 1.21 \begin {gather*} -\frac {3}{2} \, x^{3} - \frac {9}{2} \, x^{2} - {\left (x^{2} + x\right )} e + \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*x-2)*exp(1)-9/2*x^2-9*x+3/2,x, algorithm="giac")

[Out]

-3/2*x^3 - 9/2*x^2 - (x^2 + x)*e + 3/2*x

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maple [A]  time = 0.03, size = 23, normalized size = 1.21

method	result	size
gosper	\(-\frac {x \left (2 x \,{\mathrm e}+3 x^{2}+2 \,{\mathrm e}+9 x -3\right )}{2}\)	\(23\)
norman	\(\left (-{\mathrm e}-\frac {9}{2}\right ) x^{2}+\left (-{\mathrm e}+\frac {3}{2}\right ) x -\frac {3 x^{3}}{2}\)	\(25\)
risch	\(-x^{2} {\mathrm e}-x \,{\mathrm e}-\frac {3 x^{3}}{2}-\frac {9 x^{2}}{2}+\frac {3 x}{2}\)	\(27\)
default	\(\frac {{\mathrm e} \left (-2 x^{2}-2 x \right )}{2}-\frac {3 x^{3}}{2}-\frac {9 x^{2}}{2}+\frac {3 x}{2}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-4*x-2)*exp(1)-9/2*x^2-9*x+3/2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x*(2*x*exp(1)+3*x^2+2*exp(1)+9*x-3)

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maxima [A]  time = 0.37, size = 23, normalized size = 1.21 \begin {gather*} -\frac {3}{2} \, x^{3} - \frac {9}{2} \, x^{2} - {\left (x^{2} + x\right )} e + \frac {3}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*x-2)*exp(1)-9/2*x^2-9*x+3/2,x, algorithm="maxima")

[Out]

-3/2*x^3 - 9/2*x^2 - (x^2 + x)*e + 3/2*x

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mupad [B]  time = 0.04, size = 22, normalized size = 1.16 \begin {gather*} -\frac {3\,x^3}{2}+\left (-\mathrm {e}-\frac {9}{2}\right )\,x^2+\left (\frac {3}{2}-\mathrm {e}\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3/2 - (9*x^2)/2 - (exp(1)*(4*x + 2))/2 - 9*x,x)

[Out]

- x*(exp(1) - 3/2) - (3*x^3)/2 - x^2*(exp(1) + 9/2)

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sympy [A]  time = 0.05, size = 26, normalized size = 1.37 \begin {gather*} - \frac {3 x^{3}}{2} + x^{2} \left (- \frac {9}{2} - e\right ) + x \left (\frac {3}{2} - e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*x-2)*exp(1)-9/2*x**2-9*x+3/2,x)

[Out]

-3*x**3/2 + x**2*(-9/2 - E) + x*(3/2 - E)

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