3.82.42 \(\int \frac {1}{324} (100 x^4 \log (4 x^2)+125 x^4 \log ^2(4 x^2)) \, dx\)

Optimal. Leaf size=15 \[ \frac {25}{324} x^5 \log ^2\left (4 x^2\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2304, 2305} \begin {gather*} \frac {25}{324} x^5 \log ^2\left (4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(100*x^4*Log[4*x^2] + 125*x^4*Log[4*x^2]^2)/324,x]

[Out]

(25*x^5*Log[4*x^2]^2)/324

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{324} \int \left (100 x^4 \log \left (4 x^2\right )+125 x^4 \log ^2\left (4 x^2\right )\right ) \, dx\\ &=\frac {25}{81} \int x^4 \log \left (4 x^2\right ) \, dx+\frac {125}{324} \int x^4 \log ^2\left (4 x^2\right ) \, dx\\ &=-\frac {2 x^5}{81}+\frac {5}{81} x^5 \log \left (4 x^2\right )+\frac {25}{324} x^5 \log ^2\left (4 x^2\right )-\frac {25}{81} \int x^4 \log \left (4 x^2\right ) \, dx\\ &=\frac {25}{324} x^5 \log ^2\left (4 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {25}{324} x^5 \log ^2\left (4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100*x^4*Log[4*x^2] + 125*x^4*Log[4*x^2]^2)/324,x]

[Out]

(25*x^5*Log[4*x^2]^2)/324

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fricas [A]  time = 0.57, size = 13, normalized size = 0.87 \begin {gather*} \frac {25}{324} \, x^{5} \log \left (4 \, x^{2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(125/324*x^4*log(4*x^2)^2+25/81*x^4*log(4*x^2),x, algorithm="fricas")

[Out]

25/324*x^5*log(4*x^2)^2

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giac [A]  time = 0.14, size = 13, normalized size = 0.87 \begin {gather*} \frac {25}{324} \, x^{5} \log \left (4 \, x^{2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(125/324*x^4*log(4*x^2)^2+25/81*x^4*log(4*x^2),x, algorithm="giac")

[Out]

25/324*x^5*log(4*x^2)^2

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maple [A]  time = 0.03, size = 14, normalized size = 0.93

method	result	size
norman	\(\frac {25 x^{5} \ln \left (4 x^{2}\right )^{2}}{324}\)	\(14\)
risch	\(\frac {25 x^{5} \ln \left (4 x^{2}\right )^{2}}{324}\)	\(14\)
default	\(\frac {25 x^{5} \ln \left (x^{2}\right )^{2}}{324}+\frac {25 x^{5} \ln \relax (2)^{2}}{81}+\frac {25 \ln \relax (2) x^{5} \ln \left (x^{2}\right )}{81}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(125/324*x^4*ln(4*x^2)^2+25/81*x^4*ln(4*x^2),x,method=_RETURNVERBOSE)

[Out]

25/324*x^5*ln(4*x^2)^2

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maxima [A]  time = 0.37, size = 13, normalized size = 0.87 \begin {gather*} \frac {25}{324} \, x^{5} \log \left (4 \, x^{2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(125/324*x^4*log(4*x^2)^2+25/81*x^4*log(4*x^2),x, algorithm="maxima")

[Out]

25/324*x^5*log(4*x^2)^2

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mupad [B]  time = 5.41, size = 13, normalized size = 0.87 \begin {gather*} \frac {25\,x^5\,{\ln \left (4\,x^2\right )}^2}{324} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((125*x^4*log(4*x^2)^2)/324 + (25*x^4*log(4*x^2))/81,x)

[Out]

(25*x^5*log(4*x^2)^2)/324

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sympy [A]  time = 0.11, size = 14, normalized size = 0.93 \begin {gather*} \frac {25 x^{5} \log {\left (4 x^{2} \right )}^{2}}{324} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(125/324*x**4*ln(4*x**2)**2+25/81*x**4*ln(4*x**2),x)

[Out]

25*x**5*log(4*x**2)**2/324

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