3.62.50 \(\int \frac {-240 x^3-48 x^4+40 x^6+(3600+1440 x+144 x^2-480 x^3-144 x^4) \log (x)+(-1800+72 x^2) \log ^2(x)}{5 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {2}{5} x \left (-2 x^2+\frac {6 (5+x) \log (x)}{x}\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.18, antiderivative size = 48, normalized size of antiderivative = 2.09, number of steps used = 18, number of rules used = 8, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 14, 2357, 2295, 2304, 2301, 2296, 2305} \begin {gather*} \frac {8 x^5}{5}-\frac {48}{5} x^3 \log (x)-48 x^2 \log (x)+\frac {72}{5} x \log ^2(x)+144 \log ^2(x)+\frac {360 \log ^2(x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-240*x^3 - 48*x^4 + 40*x^6 + (3600 + 1440*x + 144*x^2 - 480*x^3 - 144*x^4)*Log[x] + (-1800 + 72*x^2)*Log[
x]^2)/(5*x^2),x]

[Out]

(8*x^5)/5 - 48*x^2*Log[x] - (48*x^3*Log[x])/5 + 144*Log[x]^2 + (360*Log[x]^2)/x + (72*x*Log[x]^2)/5

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 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}{5} \int \frac {-240 x^3-48 x^4+40 x^6+\left (3600+1440 x+144 x^2-480 x^3-144 x^4\right ) \log (x)+\left (-1800+72 x^2\right ) \log ^2(x)}{x^2} \, dx\\ &=\frac {1}{5} \int \left (8 x \left (-30-6 x+5 x^3\right )-\frac {48 \left (-75-30 x-3 x^2+10 x^3+3 x^4\right ) \log (x)}{x^2}+\frac {72 (-5+x) (5+x) \log ^2(x)}{x^2}\right ) \, dx\\ &=\frac {8}{5} \int x \left (-30-6 x+5 x^3\right ) \, dx-\frac {48}{5} \int \frac {\left (-75-30 x-3 x^2+10 x^3+3 x^4\right ) \log (x)}{x^2} \, dx+\frac {72}{5} \int \frac {(-5+x) (5+x) \log ^2(x)}{x^2} \, dx\\ &=\frac {8}{5} \int \left (-30 x-6 x^2+5 x^4\right ) \, dx-\frac {48}{5} \int \left (-3 \log (x)-\frac {75 \log (x)}{x^2}-\frac {30 \log (x)}{x}+10 x \log (x)+3 x^2 \log (x)\right ) \, dx+\frac {72}{5} \int \left (\log ^2(x)-\frac {25 \log ^2(x)}{x^2}\right ) \, dx\\ &=-24 x^2-\frac {16 x^3}{5}+\frac {8 x^5}{5}+\frac {72}{5} \int \log ^2(x) \, dx+\frac {144}{5} \int \log (x) \, dx-\frac {144}{5} \int x^2 \log (x) \, dx-96 \int x \log (x) \, dx+288 \int \frac {\log (x)}{x} \, dx-360 \int \frac {\log ^2(x)}{x^2} \, dx+720 \int \frac {\log (x)}{x^2} \, dx\\ &=-\frac {720}{x}-\frac {144 x}{5}+\frac {8 x^5}{5}-\frac {720 \log (x)}{x}+\frac {144}{5} x \log (x)-48 x^2 \log (x)-\frac {48}{5} x^3 \log (x)+144 \log ^2(x)+\frac {360 \log ^2(x)}{x}+\frac {72}{5} x \log ^2(x)-\frac {144}{5} \int \log (x) \, dx-720 \int \frac {\log (x)}{x^2} \, dx\\ &=\frac {8 x^5}{5}-48 x^2 \log (x)-\frac {48}{5} x^3 \log (x)+144 \log ^2(x)+\frac {360 \log ^2(x)}{x}+\frac {72}{5} x \log ^2(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 48, normalized size = 2.09 \begin {gather*} \frac {8 x^5}{5}-48 x^2 \log (x)-\frac {48}{5} x^3 \log (x)+144 \log ^2(x)+\frac {360 \log ^2(x)}{x}+\frac {72}{5} x \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-240*x^3 - 48*x^4 + 40*x^6 + (3600 + 1440*x + 144*x^2 - 480*x^3 - 144*x^4)*Log[x] + (-1800 + 72*x^2
)*Log[x]^2)/(5*x^2),x]

[Out]

(8*x^5)/5 - 48*x^2*Log[x] - (48*x^3*Log[x])/5 + 144*Log[x]^2 + (360*Log[x]^2)/x + (72*x*Log[x]^2)/5

________________________________________________________________________________________

fricas [A]  time = 1.07, size = 36, normalized size = 1.57 \begin {gather*} \frac {8 \, {\left (x^{6} + 9 \, {\left (x^{2} + 10 \, x + 25\right )} \log \relax (x)^{2} - 6 \, {\left (x^{4} + 5 \, x^{3}\right )} \log \relax (x)\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+3600)*log(x)+40*x^6-48*x^4-240*x^3)/x^2
,x, algorithm="fricas")

[Out]

8/5*(x^6 + 9*(x^2 + 10*x + 25)*log(x)^2 - 6*(x^4 + 5*x^3)*log(x))/x

________________________________________________________________________________________

giac [A]  time = 0.21, size = 33, normalized size = 1.43 \begin {gather*} \frac {8}{5} \, x^{5} + \frac {72}{5} \, {\left (x + \frac {25}{x} + 10\right )} \log \relax (x)^{2} - \frac {48}{5} \, {\left (x^{3} + 5 \, x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+3600)*log(x)+40*x^6-48*x^4-240*x^3)/x^2
,x, algorithm="giac")

[Out]

8/5*x^5 + 72/5*(x + 25/x + 10)*log(x)^2 - 48/5*(x^3 + 5*x^2)*log(x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 39, normalized size = 1.70

method	result	size
risch	\(\frac {72 \left (x^{2}+10 x +25\right ) \ln \relax (x )^{2}}{5 x}+\frac {\left (-48 x^{3}-240 x^{2}\right ) \ln \relax (x )}{5}+\frac {8 x^{5}}{5}\)	\(39\)
default	\(\frac {8 x^{5}}{5}-\frac {48 x^{3} \ln \relax (x )}{5}+\frac {72 x \ln \relax (x )^{2}}{5}-48 x^{2} \ln \relax (x )+\frac {360 \ln \relax (x )^{2}}{x}+144 \ln \relax (x )^{2}\)	\(43\)
norman	\(\frac {\frac {8 x^{6}}{5}+360 \ln \relax (x )^{2}+144 x \ln \relax (x )^{2}+\frac {72 x^{2} \ln \relax (x )^{2}}{5}-48 x^{3} \ln \relax (x )-\frac {48 x^{4} \ln \relax (x )}{5}}{x}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((72*x^2-1800)*ln(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+3600)*ln(x)+40*x^6-48*x^4-240*x^3)/x^2,x,metho
d=_RETURNVERBOSE)

[Out]

72/5*(x^2+10*x+25)/x*ln(x)^2+1/5*(-48*x^3-240*x^2)*ln(x)+8/5*x^5

________________________________________________________________________________________

maxima [B]  time = 0.35, size = 74, normalized size = 3.22 \begin {gather*} \frac {8}{5} \, x^{5} - \frac {48}{5} \, x^{3} \log \relax (x) - 48 \, x^{2} \log \relax (x) + \frac {72}{5} \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + \frac {144}{5} \, x \log \relax (x) + 144 \, \log \relax (x)^{2} - \frac {144}{5} \, x + \frac {360 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 2\right )}}{x} - \frac {720 \, \log \relax (x)}{x} - \frac {720}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((72*x^2-1800)*log(x)^2+(-144*x^4-480*x^3+144*x^2+1440*x+3600)*log(x)+40*x^6-48*x^4-240*x^3)/x^2
,x, algorithm="maxima")

[Out]

8/5*x^5 - 48/5*x^3*log(x) - 48*x^2*log(x) + 72/5*(log(x)^2 - 2*log(x) + 2)*x + 144/5*x*log(x) + 144*log(x)^2 -
144/5*x + 360*(log(x)^2 + 2*log(x) + 2)/x - 720*log(x)/x - 720/x

________________________________________________________________________________________

mupad [B]  time = 4.11, size = 22, normalized size = 0.96 \begin {gather*} \frac {8\,{\left (15\,\ln \relax (x)+3\,x\,\ln \relax (x)-x^3\right )}^2}{5\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(x)*(1440*x + 144*x^2 - 480*x^3 - 144*x^4 + 3600))/5 + (log(x)^2*(72*x^2 - 1800))/5 - 48*x^3 - (48*x^
4)/5 + 8*x^6)/x^2,x)

[Out]

(8*(15*log(x) + 3*x*log(x) - x^3)^2)/(5*x)

________________________________________________________________________________________

sympy [B]  time = 0.17, size = 41, normalized size = 1.78 \begin {gather*} \frac {8 x^{5}}{5} + \left (- \frac {48 x^{3}}{5} - 48 x^{2}\right ) \log {\relax (x )} + \frac {\left (72 x^{2} + 720 x + 1800\right ) \log {\relax (x )}^{2}}{5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((72*x**2-1800)*ln(x)**2+(-144*x**4-480*x**3+144*x**2+1440*x+3600)*ln(x)+40*x**6-48*x**4-240*x**
3)/x**2,x)

[Out]

8*x**5/5 + (-48*x**3/5 - 48*x**2)*log(x) + (72*x**2 + 720*x + 1800)*log(x)**2/(5*x)

________________________________________________________________________________________