3.21.88 \(\int (-3-3 x+9 x^2+2 x \log (x)) \, dx\)

Optimal. Leaf size=16 \[ x \left (-3-2 x+3 x^2+x \log (x)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2304} \begin {gather*} 3 x^3-2 x^2+x^2 \log (x)-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-3 - 3*x + 9*x^2 + 2*x*Log[x],x]

[Out]

-3*x - 2*x^2 + 3*x^3 + x^2*Log[x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-3 x-\frac {3 x^2}{2}+3 x^3+2 \int x \log (x) \, dx\\ &=-3 x-2 x^2+3 x^3+x^2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.25 \begin {gather*} -3 x-2 x^2+3 x^3+x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-3 - 3*x + 9*x^2 + 2*x*Log[x],x]

[Out]

-3*x - 2*x^2 + 3*x^3 + x^2*Log[x]

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fricas [A]  time = 0.57, size = 20, normalized size = 1.25 \begin {gather*} 3 \, x^{3} + x^{2} \log \relax (x) - 2 \, x^{2} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)+9*x^2-3*x-3,x, algorithm="fricas")

[Out]

3*x^3 + x^2*log(x) - 2*x^2 - 3*x

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giac [A]  time = 0.14, size = 20, normalized size = 1.25 \begin {gather*} 3 \, x^{3} + x^{2} \log \relax (x) - 2 \, x^{2} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)+9*x^2-3*x-3,x, algorithm="giac")

[Out]

3*x^3 + x^2*log(x) - 2*x^2 - 3*x

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maple [A]  time = 0.02, size = 21, normalized size = 1.31

method	result	size
default	\(-3 x -2 x^{2}+3 x^{3}+x^{2} \ln \relax (x )\)	\(21\)
norman	\(-3 x -2 x^{2}+3 x^{3}+x^{2} \ln \relax (x )\)	\(21\)
risch	\(-3 x -2 x^{2}+3 x^{3}+x^{2} \ln \relax (x )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*ln(x)+9*x^2-3*x-3,x,method=_RETURNVERBOSE)

[Out]

-3*x-2*x^2+3*x^3+x^2*ln(x)

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maxima [A]  time = 0.34, size = 20, normalized size = 1.25 \begin {gather*} 3 \, x^{3} + x^{2} \log \relax (x) - 2 \, x^{2} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)+9*x^2-3*x-3,x, algorithm="maxima")

[Out]

3*x^3 + x^2*log(x) - 2*x^2 - 3*x

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mupad [B]  time = 1.16, size = 18, normalized size = 1.12 \begin {gather*} -x\,\left (2\,x-x\,\ln \relax (x)-3\,x^2+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*log(x) - 3*x + 9*x^2 - 3,x)

[Out]

-x*(2*x - x*log(x) - 3*x^2 + 3)

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sympy [A]  time = 0.08, size = 19, normalized size = 1.19 \begin {gather*} 3 x^{3} + x^{2} \log {\relax (x )} - 2 x^{2} - 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*ln(x)+9*x**2-3*x-3,x)

[Out]

3*x**3 + x**2*log(x) - 2*x**2 - 3*x

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