3.93.20 \(\int (8-4 x-3 x^2-3 e x^2+2 x \log (4)) \, dx\)

Optimal. Leaf size=27 \[ x^2 \left (\frac {8}{x}-x+x \left (-e+\frac {-2+\log (4)}{x}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6} \begin {gather*} -\left ((1+e) x^3\right )-x^2 (2-\log (4))+8 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8 - 4*x - 3*x^2 - 3*E*x^2 + 2*x*Log[4],x]

[Out]

8*x - (1 + E)*x^3 - x^2*(2 - Log[4])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8-3 x^2-3 e x^2+x (-4+2 \log (4))\right ) \, dx\\ &=\int \left (8+(-3-3 e) x^2+x (-4+2 \log (4))\right ) \, dx\\ &=8 x-(1+e) x^3-x^2 (2-\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 0.96 \begin {gather*} 8 x-2 x^2-x^3-e x^3+x^2 \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8 - 4*x - 3*x^2 - 3*E*x^2 + 2*x*Log[4],x]

[Out]

8*x - 2*x^2 - x^3 - E*x^3 + x^2*Log[4]

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fricas [A]  time = 0.90, size = 28, normalized size = 1.04 \begin {gather*} -x^{3} e - x^{3} + 2 \, x^{2} \log \relax (2) - 2 \, x^{2} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*log(2)-3*x^2*exp(1)-3*x^2-4*x+8,x, algorithm="fricas")

[Out]

-x^3*e - x^3 + 2*x^2*log(2) - 2*x^2 + 8*x

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giac [A]  time = 0.15, size = 28, normalized size = 1.04 \begin {gather*} -x^{3} e - x^{3} + 2 \, x^{2} \log \relax (2) - 2 \, x^{2} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*log(2)-3*x^2*exp(1)-3*x^2-4*x+8,x, algorithm="giac")

[Out]

-x^3*e - x^3 + 2*x^2*log(2) - 2*x^2 + 8*x

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maple [A]  time = 0.03, size = 23, normalized size = 0.85

method	result	size
gosper	\(-x \left (x^{2} {\mathrm e}-2 x \ln \relax (2)+x^{2}+2 x -8\right )\)	\(23\)
norman	\(\left (-{\mathrm e}-1\right ) x^{3}+\left (2 \ln \relax (2)-2\right ) x^{2}+8 x\)	\(25\)
default	\(2 x^{2} \ln \relax (2)-x^{3} {\mathrm e}-x^{3}-2 x^{2}+8 x\)	\(29\)
risch	\(2 x^{2} \ln \relax (2)-x^{3} {\mathrm e}-x^{3}-2 x^{2}+8 x\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x*ln(2)-3*x^2*exp(1)-3*x^2-4*x+8,x,method=_RETURNVERBOSE)

[Out]

-x*(x^2*exp(1)-2*x*ln(2)+x^2+2*x-8)

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maxima [A]  time = 0.35, size = 28, normalized size = 1.04 \begin {gather*} -x^{3} e - x^{3} + 2 \, x^{2} \log \relax (2) - 2 \, x^{2} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*log(2)-3*x^2*exp(1)-3*x^2-4*x+8,x, algorithm="maxima")

[Out]

-x^3*e - x^3 + 2*x^2*log(2) - 2*x^2 + 8*x

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mupad [B]  time = 8.36, size = 21, normalized size = 0.78 \begin {gather*} \left (-\mathrm {e}-1\right )\,x^3+\left (\ln \relax (4)-2\right )\,x^2+8\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x*log(2) - 4*x - 3*x^2*exp(1) - 3*x^2 + 8,x)

[Out]

8*x + x^2*(log(4) - 2) - x^3*(exp(1) + 1)

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sympy [A]  time = 0.06, size = 22, normalized size = 0.81 \begin {gather*} x^{3} \left (- e - 1\right ) + x^{2} \left (-2 + 2 \log {\relax (2 )}\right ) + 8 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*ln(2)-3*x**2*exp(1)-3*x**2-4*x+8,x)

[Out]

x**3*(-E - 1) + x**2*(-2 + 2*log(2)) + 8*x

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