3.81.17 \(\int (-2-2 x+10 x^2+4 x^3+(-2-4 x+30 x^2+16 x^3) \log (x)) \, dx\)

Optimal. Leaf size=18 \[ x (-1+2 x) (2+x (6+2 x)) \log (x) \]

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Rubi [A]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.50, number of steps used = 7, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2356, 2295, 2304} \begin {gather*} 4 x^4 \log (x)+10 x^3 \log (x)-2 x^2 \log (x)-2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2 - 2*x + 10*x^2 + 4*x^3 + (-2 - 4*x + 30*x^2 + 16*x^3)*Log[x],x]

[Out]

-2*x*Log[x] - 2*x^2*Log[x] + 10*x^3*Log[x] + 4*x^4*Log[x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, 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
]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-2 x-x^2+\frac {10 x^3}{3}+x^4+\int \left (-2-4 x+30 x^2+16 x^3\right ) \log (x) \, dx\\ &=-2 x-x^2+\frac {10 x^3}{3}+x^4+\int \left (-2 \log (x)-4 x \log (x)+30 x^2 \log (x)+16 x^3 \log (x)\right ) \, dx\\ &=-2 x-x^2+\frac {10 x^3}{3}+x^4-2 \int \log (x) \, dx-4 \int x \log (x) \, dx+16 \int x^3 \log (x) \, dx+30 \int x^2 \log (x) \, dx\\ &=-2 x \log (x)-2 x^2 \log (x)+10 x^3 \log (x)+4 x^4 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.50 \begin {gather*} -2 x \log (x)-2 x^2 \log (x)+10 x^3 \log (x)+4 x^4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2 - 2*x + 10*x^2 + 4*x^3 + (-2 - 4*x + 30*x^2 + 16*x^3)*Log[x],x]

[Out]

-2*x*Log[x] - 2*x^2*Log[x] + 10*x^3*Log[x] + 4*x^4*Log[x]

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fricas [A]  time = 0.92, size = 23, normalized size = 1.28 \begin {gather*} 2 \, {\left (2 \, x^{4} + 5 \, x^{3} - x^{2} - x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^3+30*x^2-4*x-2)*log(x)+4*x^3+10*x^2-2*x-2,x, algorithm="fricas")

[Out]

2*(2*x^4 + 5*x^3 - x^2 - x)*log(x)

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giac [A]  time = 0.17, size = 27, normalized size = 1.50 \begin {gather*} 4 \, x^{4} \log \relax (x) + 10 \, x^{3} \log \relax (x) - 2 \, x^{2} \log \relax (x) - 2 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^3+30*x^2-4*x-2)*log(x)+4*x^3+10*x^2-2*x-2,x, algorithm="giac")

[Out]

4*x^4*log(x) + 10*x^3*log(x) - 2*x^2*log(x) - 2*x*log(x)

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maple [A]  time = 0.02, size = 23, normalized size = 1.28

method	result	size
risch	\(\left (4 x^{4}+10 x^{3}-2 x^{2}-2 x \right ) \ln \relax (x )\)	\(23\)
default	\(4 x^{4} \ln \relax (x )+10 x^{3} \ln \relax (x )-2 x^{2} \ln \relax (x )-2 x \ln \relax (x )\)	\(28\)
norman	\(4 x^{4} \ln \relax (x )+10 x^{3} \ln \relax (x )-2 x^{2} \ln \relax (x )-2 x \ln \relax (x )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^3+30*x^2-4*x-2)*ln(x)+4*x^3+10*x^2-2*x-2,x,method=_RETURNVERBOSE)

[Out]

(4*x^4+10*x^3-2*x^2-2*x)*ln(x)

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maxima [A]  time = 0.37, size = 23, normalized size = 1.28 \begin {gather*} 2 \, {\left (2 \, x^{4} + 5 \, x^{3} - x^{2} - x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^3+30*x^2-4*x-2)*log(x)+4*x^3+10*x^2-2*x-2,x, algorithm="maxima")

[Out]

2*(2*x^4 + 5*x^3 - x^2 - x)*log(x)

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mupad [B]  time = 5.02, size = 18, normalized size = 1.00 \begin {gather*} 2\,x\,\ln \relax (x)\,\left (2\,x-1\right )\,\left (x^2+3\,x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*x^2 - 2*x + 4*x^3 - log(x)*(4*x - 30*x^2 - 16*x^3 + 2) - 2,x)

[Out]

2*x*log(x)*(2*x - 1)*(3*x + x^2 + 1)

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sympy [A]  time = 0.11, size = 20, normalized size = 1.11 \begin {gather*} \left (4 x^{4} + 10 x^{3} - 2 x^{2} - 2 x\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**3+30*x**2-4*x-2)*ln(x)+4*x**3+10*x**2-2*x-2,x)

[Out]

(4*x**4 + 10*x**3 - 2*x**2 - 2*x)*log(x)

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