3.82.59 \(\int \frac {45-52 x+19 x^2+38 x^3-8 x^4+(-9+39 x+12 x^2-12 x^3) \log (x)+(-6 x-4 x^2) \log ^2(x)}{2 x} \, dx\)

Optimal. Leaf size=25 \[ 3+x-\left (-3+\frac {1}{2} (3-2 x)\right )^2 (-5+x+\log (x))^2 \]

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Rubi [B]  time = 0.12, antiderivative size = 74, normalized size of antiderivative = 2.96, number of steps used = 17, number of rules used = 9, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {12, 14, 2357, 2295, 2301, 2304, 2330, 2296, 2305} \begin {gather*} -x^4+7 x^3-2 x^3 \log (x)+\frac {11 x^2}{4}-x^2 \log ^2(x)+4 x^2 \log (x)-\frac {103 x}{2}-3 x \log ^2(x)-\frac {9 \log ^2(x)}{4}+\frac {51}{2} x \log (x)+\frac {45 \log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(45 - 52*x + 19*x^2 + 38*x^3 - 8*x^4 + (-9 + 39*x + 12*x^2 - 12*x^3)*Log[x] + (-6*x - 4*x^2)*Log[x]^2)/(2*
x),x]

[Out]

(-103*x)/2 + (11*x^2)/4 + 7*x^3 - x^4 + (45*Log[x])/2 + (51*x*Log[x])/2 + 4*x^2*Log[x] - 2*x^3*Log[x] - (9*Log
[x]^2)/4 - 3*x*Log[x]^2 - x^2*Log[x]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, 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 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]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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} &=\frac {1}{2} \int \frac {45-52 x+19 x^2+38 x^3-8 x^4+\left (-9+39 x+12 x^2-12 x^3\right ) \log (x)+\left (-6 x-4 x^2\right ) \log ^2(x)}{x} \, dx\\ &=\frac {1}{2} \int \left (\frac {45-52 x+19 x^2+38 x^3-8 x^4}{x}-\frac {3 (3+2 x) \left (1-5 x+2 x^2\right ) \log (x)}{x}-2 (3+2 x) \log ^2(x)\right ) \, dx\\ &=\frac {1}{2} \int \frac {45-52 x+19 x^2+38 x^3-8 x^4}{x} \, dx-\frac {3}{2} \int \frac {(3+2 x) \left (1-5 x+2 x^2\right ) \log (x)}{x} \, dx-\int (3+2 x) \log ^2(x) \, dx\\ &=\frac {1}{2} \int \left (-52+\frac {45}{x}+19 x+38 x^2-8 x^3\right ) \, dx-\frac {3}{2} \int \left (-13 \log (x)+\frac {3 \log (x)}{x}-4 x \log (x)+4 x^2 \log (x)\right ) \, dx-\int \left (3 \log ^2(x)+2 x \log ^2(x)\right ) \, dx\\ &=-26 x+\frac {19 x^2}{4}+\frac {19 x^3}{3}-x^4+\frac {45 \log (x)}{2}-2 \int x \log ^2(x) \, dx-3 \int \log ^2(x) \, dx-\frac {9}{2} \int \frac {\log (x)}{x} \, dx+6 \int x \log (x) \, dx-6 \int x^2 \log (x) \, dx+\frac {39}{2} \int \log (x) \, dx\\ &=-\frac {91 x}{2}+\frac {13 x^2}{4}+7 x^3-x^4+\frac {45 \log (x)}{2}+\frac {39}{2} x \log (x)+3 x^2 \log (x)-2 x^3 \log (x)-\frac {9 \log ^2(x)}{4}-3 x \log ^2(x)-x^2 \log ^2(x)+2 \int x \log (x) \, dx+6 \int \log (x) \, dx\\ &=-\frac {103 x}{2}+\frac {11 x^2}{4}+7 x^3-x^4+\frac {45 \log (x)}{2}+\frac {51}{2} x \log (x)+4 x^2 \log (x)-2 x^3 \log (x)-\frac {9 \log ^2(x)}{4}-3 x \log ^2(x)-x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 74, normalized size = 2.96 \begin {gather*} -\frac {103 x}{2}+\frac {11 x^2}{4}+7 x^3-x^4+\frac {45 \log (x)}{2}+\frac {51}{2} x \log (x)+4 x^2 \log (x)-2 x^3 \log (x)-\frac {9 \log ^2(x)}{4}-3 x \log ^2(x)-x^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(45 - 52*x + 19*x^2 + 38*x^3 - 8*x^4 + (-9 + 39*x + 12*x^2 - 12*x^3)*Log[x] + (-6*x - 4*x^2)*Log[x]^
2)/(2*x),x]

[Out]

(-103*x)/2 + (11*x^2)/4 + 7*x^3 - x^4 + (45*Log[x])/2 + (51*x*Log[x])/2 + 4*x^2*Log[x] - 2*x^3*Log[x] - (9*Log
[x]^2)/4 - 3*x*Log[x]^2 - x^2*Log[x]^2

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fricas [B]  time = 0.55, size = 54, normalized size = 2.16 \begin {gather*} -x^{4} + 7 \, x^{3} - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \relax (x)^{2} + \frac {11}{4} \, x^{2} - \frac {1}{2} \, {\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \log \relax (x) - \frac {103}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-4*x^2-6*x)*log(x)^2+(-12*x^3+12*x^2+39*x-9)*log(x)-8*x^4+38*x^3+19*x^2-52*x+45)/x,x, algorith
m="fricas")

[Out]

-x^4 + 7*x^3 - 1/4*(4*x^2 + 12*x + 9)*log(x)^2 + 11/4*x^2 - 1/2*(4*x^3 - 8*x^2 - 51*x - 45)*log(x) - 103/2*x

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giac [B]  time = 0.22, size = 57, normalized size = 2.28 \begin {gather*} -x^{4} + 7 \, x^{3} - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \relax (x)^{2} + \frac {11}{4} \, x^{2} - \frac {1}{2} \, {\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x\right )} \log \relax (x) - \frac {103}{2} \, x + \frac {45}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-4*x^2-6*x)*log(x)^2+(-12*x^3+12*x^2+39*x-9)*log(x)-8*x^4+38*x^3+19*x^2-52*x+45)/x,x, algorith
m="giac")

[Out]

-x^4 + 7*x^3 - 1/4*(4*x^2 + 12*x + 9)*log(x)^2 + 11/4*x^2 - 1/2*(4*x^3 - 8*x^2 - 51*x)*log(x) - 103/2*x + 45/2
*log(x)

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maple [B]  time = 0.02, size = 58, normalized size = 2.32

method	result	size
risch	\(\frac {\left (-2 x^{2}-6 x -\frac {9}{2}\right ) \ln \relax (x )^{2}}{2}+\frac {\left (-4 x^{3}+8 x^{2}+51 x \right ) \ln \relax (x )}{2}-x^{4}+7 x^{3}+\frac {11 x^{2}}{4}-\frac {103 x}{2}+\frac {45 \ln \relax (x )}{2}\)	\(58\)
default	\(-x^{2} \ln \relax (x )^{2}+4 x^{2} \ln \relax (x )+\frac {11 x^{2}}{4}-2 x^{3} \ln \relax (x )+7 x^{3}-x^{4}-3 x \ln \relax (x )^{2}+\frac {51 x \ln \relax (x )}{2}-\frac {103 x}{2}-\frac {9 \ln \relax (x )^{2}}{4}+\frac {45 \ln \relax (x )}{2}\)	\(65\)
norman	\(-x^{2} \ln \relax (x )^{2}+4 x^{2} \ln \relax (x )+\frac {11 x^{2}}{4}-2 x^{3} \ln \relax (x )+7 x^{3}-x^{4}-3 x \ln \relax (x )^{2}+\frac {51 x \ln \relax (x )}{2}-\frac {103 x}{2}-\frac {9 \ln \relax (x )^{2}}{4}+\frac {45 \ln \relax (x )}{2}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-4*x^2-6*x)*ln(x)^2+(-12*x^3+12*x^2+39*x-9)*ln(x)-8*x^4+38*x^3+19*x^2-52*x+45)/x,x,method=_RETURNVER
BOSE)

[Out]

1/2*(-2*x^2-6*x-9/2)*ln(x)^2+1/2*(-4*x^3+8*x^2+51*x)*ln(x)-x^4+7*x^3+11/4*x^2-103/2*x+45/2*ln(x)

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maxima [B]  time = 0.36, size = 78, normalized size = 3.12 \begin {gather*} -x^{4} - 2 \, x^{3} \log \relax (x) - \frac {1}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + 7 \, x^{3} + 3 \, x^{2} \log \relax (x) - 3 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + \frac {13}{4} \, x^{2} + \frac {39}{2} \, x \log \relax (x) - \frac {9}{4} \, \log \relax (x)^{2} - \frac {91}{2} \, x + \frac {45}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-4*x^2-6*x)*log(x)^2+(-12*x^3+12*x^2+39*x-9)*log(x)-8*x^4+38*x^3+19*x^2-52*x+45)/x,x, algorith
m="maxima")

[Out]

-x^4 - 2*x^3*log(x) - 1/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 7*x^3 + 3*x^2*log(x) - 3*(log(x)^2 - 2*log(x) + 2)
*x + 13/4*x^2 + 39/2*x*log(x) - 9/4*log(x)^2 - 91/2*x + 45/2*log(x)

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mupad [B]  time = 5.41, size = 64, normalized size = 2.56 \begin {gather*} -x^4-2\,x^3\,\ln \relax (x)+7\,x^3-x^2\,{\ln \relax (x)}^2+4\,x^2\,\ln \relax (x)+\frac {11\,x^2}{4}-3\,x\,{\ln \relax (x)}^2+\frac {51\,x\,\ln \relax (x)}{2}-\frac {103\,x}{2}-\frac {9\,{\ln \relax (x)}^2}{4}+\frac {45\,\ln \relax (x)}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((19*x^2)/2 - (log(x)^2*(6*x + 4*x^2))/2 - 26*x + 19*x^3 - 4*x^4 + (log(x)*(39*x + 12*x^2 - 12*x^3 - 9))/2
+ 45/2)/x,x)

[Out]

(45*log(x))/2 - (103*x)/2 - 3*x*log(x)^2 + 4*x^2*log(x) - 2*x^3*log(x) - (9*log(x)^2)/4 - x^2*log(x)^2 + (51*x
*log(x))/2 + (11*x^2)/4 + 7*x^3 - x^4

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sympy [B]  time = 0.19, size = 61, normalized size = 2.44 \begin {gather*} - x^{4} + 7 x^{3} + \frac {11 x^{2}}{4} - \frac {103 x}{2} + \left (- x^{2} - 3 x - \frac {9}{4}\right ) \log {\relax (x )}^{2} + \left (- 2 x^{3} + 4 x^{2} + \frac {51 x}{2}\right ) \log {\relax (x )} + \frac {45 \log {\relax (x )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-4*x**2-6*x)*ln(x)**2+(-12*x**3+12*x**2+39*x-9)*ln(x)-8*x**4+38*x**3+19*x**2-52*x+45)/x,x)

[Out]

-x**4 + 7*x**3 + 11*x**2/4 - 103*x/2 + (-x**2 - 3*x - 9/4)*log(x)**2 + (-2*x**3 + 4*x**2 + 51*x/2)*log(x) + 45
*log(x)/2

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