3.14.72 \(\int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+(-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)) \log (x)}{x^2} \, dx\)

Optimal. Leaf size=22 \[ (-2+x) x \left (4+x-x^2-\frac {\log (5)}{x^2}\right ) \log (x) \]

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Rubi [B]  time = 0.11, antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 11, number of rules used = 5, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {14, 1620, 2357, 2295, 2304} \begin {gather*} x^4 (-\log (x))+3 x^3 \log (x)+2 x^2 \log (x)-8 x \log (x)-\log (5) \log (x)+\frac {\log (25) \log (x)}{x}+\frac {\log (25)}{x}-\frac {2 \log (5)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8*x^2 + 2*x^3 + 3*x^4 - x^5 + (2 - x)*Log[5] + (-8*x^2 + 4*x^3 + 9*x^4 - 4*x^5 - 2*Log[5])*Log[x])/x^2,x
]

[Out]

(-2*Log[5])/x + Log[25]/x - 8*x*Log[x] + 2*x^2*Log[x] + 3*x^3*Log[x] - x^4*Log[x] - Log[5]*Log[x] + (Log[25]*L
og[x])/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 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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+x) \left (-4 x^2-x^3+x^4+\log (5)\right )}{x^2}-\frac {\left (8 x^2-4 x^3-9 x^4+4 x^5+\log (25)\right ) \log (x)}{x^2}\right ) \, dx\\ &=-\int \frac {(-2+x) \left (-4 x^2-x^3+x^4+\log (5)\right )}{x^2} \, dx-\int \frac {\left (8 x^2-4 x^3-9 x^4+4 x^5+\log (25)\right ) \log (x)}{x^2} \, dx\\ &=-\int \left (8-2 x-3 x^2+x^3-\frac {2 \log (5)}{x^2}+\frac {\log (5)}{x}\right ) \, dx-\int \left (8 \log (x)-4 x \log (x)-9 x^2 \log (x)+4 x^3 \log (x)+\frac {\log (25) \log (x)}{x^2}\right ) \, dx\\ &=-8 x+x^2+x^3-\frac {x^4}{4}-\frac {2 \log (5)}{x}-\log (5) \log (x)+4 \int x \log (x) \, dx-4 \int x^3 \log (x) \, dx-8 \int \log (x) \, dx+9 \int x^2 \log (x) \, dx-\log (25) \int \frac {\log (x)}{x^2} \, dx\\ &=-\frac {2 \log (5)}{x}+\frac {\log (25)}{x}-8 x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)-x^4 \log (x)-\log (5) \log (x)+\frac {\log (25) \log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 42, normalized size = 1.91 \begin {gather*} -8 x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)-x^4 \log (x)-\log (5) \log (x)+\frac {2 \log (5) \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x^2 + 2*x^3 + 3*x^4 - x^5 + (2 - x)*Log[5] + (-8*x^2 + 4*x^3 + 9*x^4 - 4*x^5 - 2*Log[5])*Log[x])
/x^2,x]

[Out]

-8*x*Log[x] + 2*x^2*Log[x] + 3*x^3*Log[x] - x^4*Log[x] - Log[5]*Log[x] + (2*Log[5]*Log[x])/x

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fricas [A]  time = 1.30, size = 32, normalized size = 1.45 \begin {gather*} -\frac {{\left (x^{5} - 3 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + {\left (x - 2\right )} \log \relax (5)\right )} \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(5)-4*x^5+9*x^4+4*x^3-8*x^2)*log(x)+(2-x)*log(5)-x^5+3*x^4+2*x^3-8*x^2)/x^2,x, algorithm="fr
icas")

[Out]

-(x^5 - 3*x^4 - 2*x^3 + 8*x^2 + (x - 2)*log(5))*log(x)/x

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giac [A]  time = 0.54, size = 35, normalized size = 1.59 \begin {gather*} -{\left (x^{4} - 3 \, x^{3} - 2 \, x^{2} + 8 \, x - \frac {2 \, \log \relax (5)}{x}\right )} \log \relax (x) - \log \relax (5) \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(5)-4*x^5+9*x^4+4*x^3-8*x^2)*log(x)+(2-x)*log(5)-x^5+3*x^4+2*x^3-8*x^2)/x^2,x, algorithm="gi
ac")

[Out]

-(x^4 - 3*x^3 - 2*x^2 + 8*x - 2*log(5)/x)*log(x) - log(5)*log(x)

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maple [A]  time = 0.04, size = 39, normalized size = 1.77

method	result	size
risch	\(\frac {\left (-x^{5}+3 x^{4}+2 x^{3}-8 x^{2}+2 \ln \relax (5)\right ) \ln \relax (x )}{x}-\ln \relax (5) \ln \relax (x )\)	\(39\)
norman	\(\frac {-x \ln \relax (5) \ln \relax (x )-8 x^{2} \ln \relax (x )+2 x^{3} \ln \relax (x )+3 x^{4} \ln \relax (x )-x^{5} \ln \relax (x )+2 \ln \relax (5) \ln \relax (x )}{x}\)	\(47\)
default	\(-x^{4} \ln \relax (x )+3 x^{3} \ln \relax (x )+2 x^{2} \ln \relax (x )-8 x \ln \relax (x )-2 \ln \relax (5) \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )-\ln \relax (5) \ln \relax (x )-\frac {2 \ln \relax (5)}{x}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*ln(5)-4*x^5+9*x^4+4*x^3-8*x^2)*ln(x)+(2-x)*ln(5)-x^5+3*x^4+2*x^3-8*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-x^5+3*x^4+2*x^3-8*x^2+2*ln(5))/x*ln(x)-ln(5)*ln(x)

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maxima [B]  time = 0.51, size = 54, normalized size = 2.45 \begin {gather*} -x^{4} \log \relax (x) + 3 \, x^{3} \log \relax (x) + 2 \, x^{2} \log \relax (x) + 2 \, {\left (\frac {\log \relax (x)}{x} + \frac {1}{x}\right )} \log \relax (5) - 8 \, x \log \relax (x) - \log \relax (5) \log \relax (x) - \frac {2 \, \log \relax (5)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(5)-4*x^5+9*x^4+4*x^3-8*x^2)*log(x)+(2-x)*log(5)-x^5+3*x^4+2*x^3-8*x^2)/x^2,x, algorithm="ma
xima")

[Out]

-x^4*log(x) + 3*x^3*log(x) + 2*x^2*log(x) + 2*(log(x)/x + 1/x)*log(5) - 8*x*log(x) - log(5)*log(x) - 2*log(5)/
x

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mupad [B]  time = 1.07, size = 26, normalized size = 1.18 \begin {gather*} -\frac {\ln \relax (x)\,\left (x-2\right )\,\left (x^4-x^3-4\,x^2+\ln \relax (5)\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(2*log(5) + 8*x^2 - 4*x^3 - 9*x^4 + 4*x^5) + log(5)*(x - 2) + 8*x^2 - 2*x^3 - 3*x^4 + x^5)/x^2,x)

[Out]

-(log(x)*(x - 2)*(log(5) - 4*x^2 - x^3 + x^4))/x

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sympy [A]  time = 0.16, size = 34, normalized size = 1.55 \begin {gather*} - \log {\relax (5 )} \log {\relax (x )} + \frac {\left (- x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} + 2 \log {\relax (5 )}\right ) \log {\relax (x )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*ln(5)-4*x**5+9*x**4+4*x**3-8*x**2)*ln(x)+(2-x)*ln(5)-x**5+3*x**4+2*x**3-8*x**2)/x**2,x)

[Out]

-log(5)*log(x) + (-x**5 + 3*x**4 + 2*x**3 - 8*x**2 + 2*log(5))*log(x)/x

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