3.75.41 \(\int (2 x+190 x \log (3)+4500 x \log ^2(3)+(20 x \log (3)+950 x \log ^2(3)) \log (x)+50 x \log ^2(3) \log ^2(x)) \, dx\)

Optimal. Leaf size=16 \[ x^2 (1+5 \log (3) (9+\log (x)))^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.04, antiderivative size = 82, normalized size of antiderivative = 5.12, number of steps used = 8, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6, 12, 2304, 2305} \begin {gather*} 25 x^2 \log ^2(3) \log ^2(x)-25 x^2 \log ^2(3) \log (x)+x^2 \left (1+2250 \log ^2(3)+95 \log (3)\right )+\frac {25}{2} x^2 \log ^2(3)+5 x^2 \log (3) (2+95 \log (3)) \log (x)-\frac {5}{2} x^2 \log (3) (2+95 \log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*x + 190*x*Log[3] + 4500*x*Log[3]^2 + (20*x*Log[3] + 950*x*Log[3]^2)*Log[x] + 50*x*Log[3]^2*Log[x]^2,x]

[Out]

(25*x^2*Log[3]^2)/2 - (5*x^2*Log[3]*(2 + 95*Log[3]))/2 + x^2*(1 + 95*Log[3] + 2250*Log[3]^2) - 25*x^2*Log[3]^2
*Log[x] + 5*x^2*Log[3]*(2 + 95*Log[3])*Log[x] + 25*x^2*Log[3]^2*Log[x]^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4500 x \log ^2(3)+x (2+190 \log (3))+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx\\ &=\int \left (x \left (2+190 \log (3)+4500 \log ^2(3)\right )+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx\\ &=x^2 \left (1+95 \log (3)+2250 \log ^2(3)\right )+\left (50 \log ^2(3)\right ) \int x \log ^2(x) \, dx+\int \left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x) \, dx\\ &=x^2 \left (1+95 \log (3)+2250 \log ^2(3)\right )+25 x^2 \log ^2(3) \log ^2(x)-\left (50 \log ^2(3)\right ) \int x \log (x) \, dx+\int x \left (20 \log (3)+950 \log ^2(3)\right ) \log (x) \, dx\\ &=\frac {25}{2} x^2 \log ^2(3)+x^2 \left (1+95 \log (3)+2250 \log ^2(3)\right )-25 x^2 \log ^2(3) \log (x)+25 x^2 \log ^2(3) \log ^2(x)+(10 \log (3) (2+95 \log (3))) \int x \log (x) \, dx\\ &=\frac {25}{2} x^2 \log ^2(3)-\frac {5}{2} x^2 \log (3) (2+95 \log (3))+x^2 \left (1+95 \log (3)+2250 \log ^2(3)\right )-25 x^2 \log ^2(3) \log (x)+5 x^2 \log (3) (2+95 \log (3)) \log (x)+25 x^2 \log ^2(3) \log ^2(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 53, normalized size = 3.31 \begin {gather*} x^2+90 x^2 \log (3)+2025 x^2 \log ^2(3)+10 x^2 \log (3) \log (x)+450 x^2 \log ^2(3) \log (x)+25 x^2 \log ^2(3) \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*x + 190*x*Log[3] + 4500*x*Log[3]^2 + (20*x*Log[3] + 950*x*Log[3]^2)*Log[x] + 50*x*Log[3]^2*Log[x]^
2,x]

[Out]

x^2 + 90*x^2*Log[3] + 2025*x^2*Log[3]^2 + 10*x^2*Log[3]*Log[x] + 450*x^2*Log[3]^2*Log[x] + 25*x^2*Log[3]^2*Log
[x]^2

________________________________________________________________________________________

fricas [B]  time = 0.96, size = 53, normalized size = 3.31 \begin {gather*} 25 \, x^{2} \log \relax (3)^{2} \log \relax (x)^{2} + 2025 \, x^{2} \log \relax (3)^{2} + 90 \, x^{2} \log \relax (3) + x^{2} + 10 \, {\left (45 \, x^{2} \log \relax (3)^{2} + x^{2} \log \relax (3)\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*log(3)^2*log(x)^2+(950*x*log(3)^2+20*x*log(3))*log(x)+4500*x*log(3)^2+190*x*log(3)+2*x,x, algor
ithm="fricas")

[Out]

25*x^2*log(3)^2*log(x)^2 + 2025*x^2*log(3)^2 + 90*x^2*log(3) + x^2 + 10*(45*x^2*log(3)^2 + x^2*log(3))*log(x)

________________________________________________________________________________________

giac [B]  time = 0.19, size = 66, normalized size = 4.12 \begin {gather*} 475 \, x^{2} \log \relax (3)^{2} \log \relax (x) + \frac {4025}{2} \, x^{2} \log \relax (3)^{2} + 10 \, x^{2} \log \relax (3) \log \relax (x) + 90 \, x^{2} \log \relax (3) + \frac {25}{2} \, {\left (2 \, x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) + x^{2}\right )} \log \relax (3)^{2} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*log(3)^2*log(x)^2+(950*x*log(3)^2+20*x*log(3))*log(x)+4500*x*log(3)^2+190*x*log(3)+2*x,x, algor
ithm="giac")

[Out]

475*x^2*log(3)^2*log(x) + 4025/2*x^2*log(3)^2 + 10*x^2*log(3)*log(x) + 90*x^2*log(3) + 25/2*(2*x^2*log(x)^2 -
2*x^2*log(x) + x^2)*log(3)^2 + x^2

________________________________________________________________________________________

maple [B]  time = 0.03, size = 48, normalized size = 3.00

method	result	size
norman	\(\left (1+2025 \ln \relax (3)^{2}+90 \ln \relax (3)\right ) x^{2}+\left (450 \ln \relax (3)^{2}+10 \ln \relax (3)\right ) x^{2} \ln \relax (x )+25 \ln \relax (3)^{2} \ln \relax (x )^{2} x^{2}\)	\(48\)
default	\(450 x^{2} \ln \relax (3)^{2} \ln \relax (x )+2025 x^{2} \ln \relax (3)^{2}+10 x^{2} \ln \relax (3) \ln \relax (x )+90 x^{2} \ln \relax (3)+x^{2}+25 \ln \relax (3)^{2} \ln \relax (x )^{2} x^{2}\)	\(54\)
risch	\(450 x^{2} \ln \relax (3)^{2} \ln \relax (x )+2025 x^{2} \ln \relax (3)^{2}+10 x^{2} \ln \relax (3) \ln \relax (x )+90 x^{2} \ln \relax (3)+x^{2}+25 \ln \relax (3)^{2} \ln \relax (x )^{2} x^{2}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50*x*ln(3)^2*ln(x)^2+(950*x*ln(3)^2+20*x*ln(3))*ln(x)+4500*x*ln(3)^2+190*x*ln(3)+2*x,x,method=_RETURNVERBO
SE)

[Out]

(1+2025*ln(3)^2+90*ln(3))*x^2+(450*ln(3)^2+10*ln(3))*x^2*ln(x)+25*ln(3)^2*ln(x)^2*x^2

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 78, normalized size = 4.88 \begin {gather*} \frac {25}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} \log \relax (3)^{2} + 2250 \, x^{2} \log \relax (3)^{2} - \frac {5}{2} \, {\left (95 \, \log \relax (3)^{2} + 2 \, \log \relax (3)\right )} x^{2} + 95 \, x^{2} \log \relax (3) + x^{2} + 5 \, {\left (95 \, x^{2} \log \relax (3)^{2} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*log(3)^2*log(x)^2+(950*x*log(3)^2+20*x*log(3))*log(x)+4500*x*log(3)^2+190*x*log(3)+2*x,x, algor
ithm="maxima")

[Out]

25/2*(2*log(x)^2 - 2*log(x) + 1)*x^2*log(3)^2 + 2250*x^2*log(3)^2 - 5/2*(95*log(3)^2 + 2*log(3))*x^2 + 95*x^2*
log(3) + x^2 + 5*(95*x^2*log(3)^2 + 2*x^2*log(3))*log(x)

________________________________________________________________________________________

mupad [B]  time = 4.60, size = 18, normalized size = 1.12 \begin {gather*} x^2\,{\left (45\,\ln \relax (3)+5\,\ln \relax (3)\,\ln \relax (x)+1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 190*x*log(3) + log(x)*(20*x*log(3) + 950*x*log(3)^2) + 4500*x*log(3)^2 + 50*x*log(3)^2*log(x)^2,x)

[Out]

x^2*(45*log(3) + 5*log(3)*log(x) + 1)^2

________________________________________________________________________________________

sympy [B]  time = 0.18, size = 53, normalized size = 3.31 \begin {gather*} 25 x^{2} \log {\relax (3 )}^{2} \log {\relax (x )}^{2} + x^{2} \left (1 + 90 \log {\relax (3 )} + 2025 \log {\relax (3 )}^{2}\right ) + \left (10 x^{2} \log {\relax (3 )} + 450 x^{2} \log {\relax (3 )}^{2}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*ln(3)**2*ln(x)**2+(950*x*ln(3)**2+20*x*ln(3))*ln(x)+4500*x*ln(3)**2+190*x*ln(3)+2*x,x)

[Out]

25*x**2*log(3)**2*log(x)**2 + x**2*(1 + 90*log(3) + 2025*log(3)**2) + (10*x**2*log(3) + 450*x**2*log(3)**2)*lo
g(x)

________________________________________________________________________________________