3.33.30 \(\int \frac {3590-5760 x+3264 x^2-768 x^3+64 x^4+(-3600+3264 x^2-1536 x^3+192 x^4) \log (x)}{x^2} \, dx\)

Optimal. Leaf size=25 \[ 2+\frac {10+16 \left (-3+2 (3-x)^2\right )^2 \log (x)}{x} \]

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Rubi [A]  time = 0.10, antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14, 2357, 2295, 2304} \begin {gather*} 64 x^3 \log (x)-768 x^2 \log (x)+\frac {10}{x}+3264 x \log (x)-5760 \log (x)+\frac {3600 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3590 - 5760*x + 3264*x^2 - 768*x^3 + 64*x^4 + (-3600 + 3264*x^2 - 1536*x^3 + 192*x^4)*Log[x])/x^2,x]

[Out]

10/x - 5760*Log[x] + (3600*Log[x])/x + 3264*x*Log[x] - 768*x^2*Log[x] + 64*x^3*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (1795-2880 x+1632 x^2-384 x^3+32 x^4\right )}{x^2}+\frac {48 \left (15-12 x+2 x^2\right ) \left (-5-4 x+2 x^2\right ) \log (x)}{x^2}\right ) \, dx\\ &=2 \int \frac {1795-2880 x+1632 x^2-384 x^3+32 x^4}{x^2} \, dx+48 \int \frac {\left (15-12 x+2 x^2\right ) \left (-5-4 x+2 x^2\right ) \log (x)}{x^2} \, dx\\ &=2 \int \left (1632+\frac {1795}{x^2}-\frac {2880}{x}-384 x+32 x^2\right ) \, dx+48 \int \left (68 \log (x)-\frac {75 \log (x)}{x^2}-32 x \log (x)+4 x^2 \log (x)\right ) \, dx\\ &=-\frac {3590}{x}+3264 x-384 x^2+\frac {64 x^3}{3}-5760 \log (x)+192 \int x^2 \log (x) \, dx-1536 \int x \log (x) \, dx+3264 \int \log (x) \, dx-3600 \int \frac {\log (x)}{x^2} \, dx\\ &=\frac {10}{x}-5760 \log (x)+\frac {3600 \log (x)}{x}+3264 x \log (x)-768 x^2 \log (x)+64 x^3 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.44 \begin {gather*} \frac {10}{x}-5760 \log (x)+\frac {3600 \log (x)}{x}+3264 x \log (x)-768 x^2 \log (x)+64 x^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3590 - 5760*x + 3264*x^2 - 768*x^3 + 64*x^4 + (-3600 + 3264*x^2 - 1536*x^3 + 192*x^4)*Log[x])/x^2,x
]

[Out]

10/x - 5760*Log[x] + (3600*Log[x])/x + 3264*x*Log[x] - 768*x^2*Log[x] + 64*x^3*Log[x]

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fricas [A]  time = 0.69, size = 31, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (8 \, {\left (4 \, x^{4} - 48 \, x^{3} + 204 \, x^{2} - 360 \, x + 225\right )} \log \relax (x) + 5\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^4-1536*x^3+3264*x^2-3600)*log(x)+64*x^4-768*x^3+3264*x^2-5760*x+3590)/x^2,x, algorithm="fric
as")

[Out]

2*(8*(4*x^4 - 48*x^3 + 204*x^2 - 360*x + 225)*log(x) + 5)/x

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giac [A]  time = 0.23, size = 33, normalized size = 1.32 \begin {gather*} 16 \, {\left (4 \, x^{3} - 48 \, x^{2} + 204 \, x + \frac {225}{x}\right )} \log \relax (x) + \frac {10}{x} - 5760 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^4-1536*x^3+3264*x^2-3600)*log(x)+64*x^4-768*x^3+3264*x^2-5760*x+3590)/x^2,x, algorithm="giac
")

[Out]

16*(4*x^3 - 48*x^2 + 204*x + 225/x)*log(x) + 10/x - 5760*log(x)

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maple [A]  time = 0.02, size = 37, normalized size = 1.48

method	result	size
default	\(64 x^{3} \ln \relax (x )-768 x^{2} \ln \relax (x )+3264 x \ln \relax (x )+\frac {3600 \ln \relax (x )}{x}+\frac {10}{x}-5760 \ln \relax (x )\)	\(37\)
norman	\(\frac {10-5760 x \ln \relax (x )+3264 x^{2} \ln \relax (x )-768 x^{3} \ln \relax (x )+64 x^{4} \ln \relax (x )+3600 \ln \relax (x )}{x}\)	\(37\)
risch	\(\frac {16 \left (4 x^{4}-48 x^{3}+204 x^{2}+225\right ) \ln \relax (x )}{x}-\frac {10 \left (576 x \ln \relax (x )-1\right )}{x}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((192*x^4-1536*x^3+3264*x^2-3600)*ln(x)+64*x^4-768*x^3+3264*x^2-5760*x+3590)/x^2,x,method=_RETURNVERBOSE)

[Out]

64*x^3*ln(x)-768*x^2*ln(x)+3264*x*ln(x)+3600*ln(x)/x+10/x-5760*ln(x)

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maxima [A]  time = 0.41, size = 36, normalized size = 1.44 \begin {gather*} 64 \, x^{3} \log \relax (x) - 768 \, x^{2} \log \relax (x) + 3264 \, x \log \relax (x) + \frac {3600 \, \log \relax (x)}{x} + \frac {10}{x} - 5760 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^4-1536*x^3+3264*x^2-3600)*log(x)+64*x^4-768*x^3+3264*x^2-5760*x+3590)/x^2,x, algorithm="maxi
ma")

[Out]

64*x^3*log(x) - 768*x^2*log(x) + 3264*x*log(x) + 3600*log(x)/x + 10/x - 5760*log(x)

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mupad [B]  time = 1.95, size = 34, normalized size = 1.36 \begin {gather*} 64\,x^3\,\ln \relax (x)-768\,x^2\,\ln \relax (x)-5760\,\ln \relax (x)+3264\,x\,\ln \relax (x)+\frac {3600\,\ln \relax (x)+10}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(3264*x^2 - 1536*x^3 + 192*x^4 - 3600) - 5760*x + 3264*x^2 - 768*x^3 + 64*x^4 + 3590)/x^2,x)

[Out]

64*x^3*log(x) - 768*x^2*log(x) - 5760*log(x) + 3264*x*log(x) + (3600*log(x) + 10)/x

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sympy [A]  time = 0.18, size = 29, normalized size = 1.16 \begin {gather*} - 5760 \log {\relax (x )} + \frac {\left (64 x^{4} - 768 x^{3} + 3264 x^{2} + 3600\right ) \log {\relax (x )}}{x} + \frac {10}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x**4-1536*x**3+3264*x**2-3600)*ln(x)+64*x**4-768*x**3+3264*x**2-5760*x+3590)/x**2,x)

[Out]

-5760*log(x) + (64*x**4 - 768*x**3 + 3264*x**2 + 3600)*log(x)/x + 10/x

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