3.88.68 \(\int (125000 x^4 \log (x)+312500 x^4 \log ^2(x)) \, dx\)

Optimal. Leaf size=9 \[ 62500 x^5 \log ^2(x) \]

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Rubi [A]  time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2304, 2305} \begin {gather*} 62500 x^5 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[125000*x^4*Log[x] + 312500*x^4*Log[x]^2,x]

[Out]

62500*x^5*Log[x]^2

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} &=125000 \int x^4 \log (x) \, dx+312500 \int x^4 \log ^2(x) \, dx\\ &=-5000 x^5+25000 x^5 \log (x)+62500 x^5 \log ^2(x)-125000 \int x^4 \log (x) \, dx\\ &=62500 x^5 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} 62500 x^5 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[125000*x^4*Log[x] + 312500*x^4*Log[x]^2,x]

[Out]

62500*x^5*Log[x]^2

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fricas [A]  time = 0.58, size = 9, normalized size = 1.00 \begin {gather*} 62500 \, x^{5} \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(312500*x^4*log(x)^2+125000*x^4*log(x),x, algorithm="fricas")

[Out]

62500*x^5*log(x)^2

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giac [A]  time = 0.15, size = 9, normalized size = 1.00 \begin {gather*} 62500 \, x^{5} \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(312500*x^4*log(x)^2+125000*x^4*log(x),x, algorithm="giac")

[Out]

62500*x^5*log(x)^2

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maple [A]  time = 0.02, size = 10, normalized size = 1.11

method	result	size
default	\(62500 x^{5} \ln \relax (x )^{2}\)	\(10\)
norman	\(62500 x^{5} \ln \relax (x )^{2}\)	\(10\)
risch	\(62500 x^{5} \ln \relax (x )^{2}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(312500*x^4*ln(x)^2+125000*x^4*ln(x),x,method=_RETURNVERBOSE)

[Out]

62500*x^5*ln(x)^2

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maxima [B]  time = 0.34, size = 30, normalized size = 3.33 \begin {gather*} 2500 \, {\left (25 \, \log \relax (x)^{2} - 10 \, \log \relax (x) + 2\right )} x^{5} + 25000 \, x^{5} \log \relax (x) - 5000 \, x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(312500*x^4*log(x)^2+125000*x^4*log(x),x, algorithm="maxima")

[Out]

2500*(25*log(x)^2 - 10*log(x) + 2)*x^5 + 25000*x^5*log(x) - 5000*x^5

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mupad [B]  time = 5.50, size = 9, normalized size = 1.00 \begin {gather*} 62500\,x^5\,{\ln \relax (x)}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(125000*x^4*log(x) + 312500*x^4*log(x)^2,x)

[Out]

62500*x^5*log(x)^2

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sympy [A]  time = 0.09, size = 8, normalized size = 0.89 \begin {gather*} 62500 x^{5} \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(312500*x**4*ln(x)**2+125000*x**4*ln(x),x)

[Out]

62500*x**5*log(x)**2

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