3.75.52 \(\int \frac {1}{3} (6+2 x-432 x^3+2 x \log (x^2)) \, dx\)

Optimal. Leaf size=21 \[ 6+2 x-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 2304} \begin {gather*} -36 x^4+\frac {1}{3} x^2 \log \left (x^2\right )+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + 2*x - 432*x^3 + 2*x*Log[x^2])/3,x]

[Out]

2*x - 36*x^4 + (x^2*Log[x^2])/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{3} \int \left (6+2 x-432 x^3+2 x \log \left (x^2\right )\right ) \, dx\\ &=2 x+\frac {x^2}{3}-36 x^4+\frac {2}{3} \int x \log \left (x^2\right ) \, dx\\ &=2 x-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.95 \begin {gather*} 2 x-36 x^4+\frac {1}{3} x^2 \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 2*x - 432*x^3 + 2*x*Log[x^2])/3,x]

[Out]

2*x - 36*x^4 + (x^2*Log[x^2])/3

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fricas [A]  time = 0.57, size = 18, normalized size = 0.86 \begin {gather*} -36 \, x^{4} + \frac {1}{3} \, x^{2} \log \left (x^{2}\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*log(x^2)-144*x^3+2/3*x+2,x, algorithm="fricas")

[Out]

-36*x^4 + 1/3*x^2*log(x^2) + 2*x

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giac [A]  time = 0.38, size = 18, normalized size = 0.86 \begin {gather*} -36 \, x^{4} + \frac {1}{3} \, x^{2} \log \left (x^{2}\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*log(x^2)-144*x^3+2/3*x+2,x, algorithm="giac")

[Out]

-36*x^4 + 1/3*x^2*log(x^2) + 2*x

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maple [A]  time = 0.02, size = 19, normalized size = 0.90

method	result	size
default	\(2 x -36 x^{4}+\frac {x^{2} \ln \left (x^{2}\right )}{3}\)	\(19\)
norman	\(2 x -36 x^{4}+\frac {x^{2} \ln \left (x^{2}\right )}{3}\)	\(19\)
risch	\(2 x -36 x^{4}+\frac {x^{2} \ln \left (x^{2}\right )}{3}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/3*x*ln(x^2)-144*x^3+2/3*x+2,x,method=_RETURNVERBOSE)

[Out]

2*x-36*x^4+1/3*x^2*ln(x^2)

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maxima [A]  time = 0.37, size = 18, normalized size = 0.86 \begin {gather*} -36 \, x^{4} + \frac {1}{3} \, x^{2} \log \left (x^{2}\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*log(x^2)-144*x^3+2/3*x+2,x, algorithm="maxima")

[Out]

-36*x^4 + 1/3*x^2*log(x^2) + 2*x

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mupad [B]  time = 5.61, size = 18, normalized size = 0.86 \begin {gather*} 2\,x+\frac {x^2\,\ln \left (x^2\right )}{3}-36\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x)/3 + (2*x*log(x^2))/3 - 144*x^3 + 2,x)

[Out]

2*x + (x^2*log(x^2))/3 - 36*x^4

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sympy [A]  time = 0.11, size = 17, normalized size = 0.81 \begin {gather*} - 36 x^{4} + \frac {x^{2} \log {\left (x^{2} \right )}}{3} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*ln(x**2)-144*x**3+2/3*x+2,x)

[Out]

-36*x**4 + x**2*log(x**2)/3 + 2*x

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