3.66.44 \(\int \frac {96-1896 x+6246 x^2-1230 x^3+3072 x^4+(24-384 x+54 x^2-288 x^3) \log (x)+6 x^2 \log ^2(x)}{x} \, dx\)

Optimal. Leaf size=21 \[ 3 \left (4+x^2\right ) \left (-3+2 x+(4-16 x+\log (x))^2\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.09, antiderivative size = 57, normalized size of antiderivative = 2.71, number of steps used = 12, number of rules used = 6, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {14, 2357, 2295, 2301, 2304, 2305} \begin {gather*} 768 x^4-378 x^3-96 x^3 \log (x)+3111 x^2+3 x^2 \log ^2(x)+24 x^2 \log (x)-1512 x+12 \log ^2(x)-384 x \log (x)+96 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(96 - 1896*x + 6246*x^2 - 1230*x^3 + 3072*x^4 + (24 - 384*x + 54*x^2 - 288*x^3)*Log[x] + 6*x^2*Log[x]^2)/x
,x]

[Out]

-1512*x + 3111*x^2 - 378*x^3 + 768*x^4 + 96*Log[x] - 384*x*Log[x] + 24*x^2*Log[x] - 96*x^3*Log[x] + 12*Log[x]^
2 + 3*x^2*Log[x]^2

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 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 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 {6 \left (16-316 x+1041 x^2-205 x^3+512 x^4\right )}{x}-\frac {6 \left (-4+64 x-9 x^2+48 x^3\right ) \log (x)}{x}+6 x \log ^2(x)\right ) \, dx\\ &=6 \int \frac {16-316 x+1041 x^2-205 x^3+512 x^4}{x} \, dx-6 \int \frac {\left (-4+64 x-9 x^2+48 x^3\right ) \log (x)}{x} \, dx+6 \int x \log ^2(x) \, dx\\ &=3 x^2 \log ^2(x)+6 \int \left (-316+\frac {16}{x}+1041 x-205 x^2+512 x^3\right ) \, dx-6 \int x \log (x) \, dx-6 \int \left (64 \log (x)-\frac {4 \log (x)}{x}-9 x \log (x)+48 x^2 \log (x)\right ) \, dx\\ &=-1896 x+\frac {6249 x^2}{2}-410 x^3+768 x^4+96 \log (x)-3 x^2 \log (x)+3 x^2 \log ^2(x)+24 \int \frac {\log (x)}{x} \, dx+54 \int x \log (x) \, dx-288 \int x^2 \log (x) \, dx-384 \int \log (x) \, dx\\ &=-1512 x+3111 x^2-378 x^3+768 x^4+96 \log (x)-384 x \log (x)+24 x^2 \log (x)-96 x^3 \log (x)+12 \log ^2(x)+3 x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 57, normalized size = 2.71 \begin {gather*} -1512 x+3111 x^2-378 x^3+768 x^4+96 \log (x)-384 x \log (x)+24 x^2 \log (x)-96 x^3 \log (x)+12 \log ^2(x)+3 x^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(96 - 1896*x + 6246*x^2 - 1230*x^3 + 3072*x^4 + (24 - 384*x + 54*x^2 - 288*x^3)*Log[x] + 6*x^2*Log[x
]^2)/x,x]

[Out]

-1512*x + 3111*x^2 - 378*x^3 + 768*x^4 + 96*Log[x] - 384*x*Log[x] + 24*x^2*Log[x] - 96*x^3*Log[x] + 12*Log[x]^
2 + 3*x^2*Log[x]^2

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 49, normalized size = 2.33 \begin {gather*} 768 \, x^{4} - 378 \, x^{3} + 3 \, {\left (x^{2} + 4\right )} \log \relax (x)^{2} + 3111 \, x^{2} - 24 \, {\left (4 \, x^{3} - x^{2} + 16 \, x - 4\right )} \log \relax (x) - 1512 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2*log(x)^2+(-288*x^3+54*x^2-384*x+24)*log(x)+3072*x^4-1230*x^3+6246*x^2-1896*x+96)/x,x, algorit
hm="fricas")

[Out]

768*x^4 - 378*x^3 + 3*(x^2 + 4)*log(x)^2 + 3111*x^2 - 24*(4*x^3 - x^2 + 16*x - 4)*log(x) - 1512*x

________________________________________________________________________________________

giac [B]  time = 0.34, size = 52, normalized size = 2.48 \begin {gather*} 768 \, x^{4} - 378 \, x^{3} + 3 \, {\left (x^{2} + 4\right )} \log \relax (x)^{2} + 3111 \, x^{2} - 24 \, {\left (4 \, x^{3} - x^{2} + 16 \, x\right )} \log \relax (x) - 1512 \, x + 96 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2*log(x)^2+(-288*x^3+54*x^2-384*x+24)*log(x)+3072*x^4-1230*x^3+6246*x^2-1896*x+96)/x,x, algorit
hm="giac")

[Out]

768*x^4 - 378*x^3 + 3*(x^2 + 4)*log(x)^2 + 3111*x^2 - 24*(4*x^3 - x^2 + 16*x)*log(x) - 1512*x + 96*log(x)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 53, normalized size = 2.52

method	result	size
risch	\(\left (3 x^{2}+12\right ) \ln \relax (x )^{2}+\left (-96 x^{3}+24 x^{2}-384 x \right ) \ln \relax (x )+768 x^{4}-378 x^{3}+3111 x^{2}-1512 x +96 \ln \relax (x )\)	\(53\)
default	\(3 x^{2} \ln \relax (x )^{2}+24 x^{2} \ln \relax (x )+3111 x^{2}-96 x^{3} \ln \relax (x )-378 x^{3}+768 x^{4}-384 x \ln \relax (x )-1512 x +12 \ln \relax (x )^{2}+96 \ln \relax (x )\)	\(58\)
norman	\(3 x^{2} \ln \relax (x )^{2}+24 x^{2} \ln \relax (x )+3111 x^{2}-96 x^{3} \ln \relax (x )-378 x^{3}+768 x^{4}-384 x \ln \relax (x )-1512 x +12 \ln \relax (x )^{2}+96 \ln \relax (x )\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2*ln(x)^2+(-288*x^3+54*x^2-384*x+24)*ln(x)+3072*x^4-1230*x^3+6246*x^2-1896*x+96)/x,x,method=_RETURNVE
RBOSE)

[Out]

(3*x^2+12)*ln(x)^2+(-96*x^3+24*x^2-384*x)*ln(x)+768*x^4-378*x^3+3111*x^2-1512*x+96*ln(x)

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 65, normalized size = 3.10 \begin {gather*} 768 \, x^{4} - 96 \, x^{3} \log \relax (x) + \frac {3}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} - 378 \, x^{3} + 27 \, x^{2} \log \relax (x) + \frac {6219}{2} \, x^{2} - 384 \, x \log \relax (x) + 12 \, \log \relax (x)^{2} - 1512 \, x + 96 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2*log(x)^2+(-288*x^3+54*x^2-384*x+24)*log(x)+3072*x^4-1230*x^3+6246*x^2-1896*x+96)/x,x, algorit
hm="maxima")

[Out]

768*x^4 - 96*x^3*log(x) + 3/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 - 378*x^3 + 27*x^2*log(x) + 6219/2*x^2 - 384*x*l
og(x) + 12*log(x)^2 - 1512*x + 96*log(x)

________________________________________________________________________________________

mupad [B]  time = 4.11, size = 57, normalized size = 2.71 \begin {gather*} 768\,x^4-96\,x^3\,\ln \relax (x)-378\,x^3+3\,x^2\,{\ln \relax (x)}^2+24\,x^2\,\ln \relax (x)+3111\,x^2-384\,x\,\ln \relax (x)-1512\,x+12\,{\ln \relax (x)}^2+96\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2*log(x)^2 - 1896*x + 6246*x^2 - 1230*x^3 + 3072*x^4 - log(x)*(384*x - 54*x^2 + 288*x^3 - 24) + 96)/x
,x)

[Out]

96*log(x) - 1512*x + 24*x^2*log(x) - 96*x^3*log(x) + 12*log(x)^2 + 3*x^2*log(x)^2 - 384*x*log(x) + 3111*x^2 -
378*x^3 + 768*x^4

________________________________________________________________________________________

sympy [B]  time = 0.16, size = 51, normalized size = 2.43 \begin {gather*} 768 x^{4} - 378 x^{3} + 3111 x^{2} - 1512 x + \left (3 x^{2} + 12\right ) \log {\relax (x )}^{2} + \left (- 96 x^{3} + 24 x^{2} - 384 x\right ) \log {\relax (x )} + 96 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**2*ln(x)**2+(-288*x**3+54*x**2-384*x+24)*ln(x)+3072*x**4-1230*x**3+6246*x**2-1896*x+96)/x,x)

[Out]

768*x**4 - 378*x**3 + 3111*x**2 - 1512*x + (3*x**2 + 12)*log(x)**2 + (-96*x**3 + 24*x**2 - 384*x)*log(x) + 96*
log(x)

________________________________________________________________________________________