\(\int (2 x+190 x \log (3)+4500 x \log ^2(3)+(20 x \log (3)+950 x \log ^2(3)) \log (x)+50 x \log ^2(3) \log ^2(x)) \, dx\) [7441]

Optimal result

Integrand size = 43, antiderivative size = 16 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=x^2 (1+5 \log (3) (9+\log (x)))^2 \]

[Out]

x^2*(1+5*(9+ln(x))*ln(3))^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(16)=32\).

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 5.12, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6, 12, 2341, 2342} \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=25 x^2 \log ^2(3) \log ^2(x)-25 x^2 \log ^2(3) \log (x)+x^2 \left (1+2250 \log ^2(3)+95 \log (3)\right )+\frac {25}{2} x^2 \log ^2(3)+5 x^2 \log (3) (2+95 \log (3)) \log (x)-\frac {5}{2} x^2 \log (3) (2+95 \log (3)) \]

[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 2341

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 2342

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{align*} \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{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(16)=32\).

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=x^2+90 x^2 \log (3)+2025 x^2 \log ^2(3)+10 x^2 \log (3) \log (x)+450 x^2 \log ^2(3) \log (x)+25 x^2 \log ^2(3) \log ^2(x) \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(16)=32\).

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00

method	result	size
norman	\(\left (1+2025 \ln \left (3\right )^{2}+90 \ln \left (3\right )\right ) x^{2}+\left (450 \ln \left (3\right )^{2}+10 \ln \left (3\right )\right ) x^{2} \ln \left (x \right )+25 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )^{2}\)	\(48\)
risch	\(25 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )^{2}+450 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )+10 x^{2} \ln \left (3\right ) \ln \left (x \right )+2025 x^{2} \ln \left (3\right )^{2}+90 x^{2} \ln \left (3\right )+x^{2}\)	\(54\)
parallelrisch	\(25 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )^{2}+450 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )+10 x^{2} \ln \left (3\right ) \ln \left (x \right )+2025 x^{2} \ln \left (3\right )^{2}+90 x^{2} \ln \left (3\right )+x^{2}\)	\(54\)
default	\(475 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )+\frac {4025 x^{2} \ln \left (3\right )^{2}}{2}+10 x^{2} \ln \left (3\right ) \ln \left (x \right )+90 x^{2} \ln \left (3\right )+x^{2}+50 \ln \left (3\right )^{2} \left (\frac {x^{2} \ln \left (x \right )^{2}}{2}-\frac {x^{2} \ln \left (x \right )}{2}+\frac {x^{2}}{4}\right )\)	\(69\)
parts	\(475 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )+\frac {4025 x^{2} \ln \left (3\right )^{2}}{2}+10 x^{2} \ln \left (3\right ) \ln \left (x \right )+90 x^{2} \ln \left (3\right )+x^{2}+50 \ln \left (3\right )^{2} \left (\frac {x^{2} \ln \left (x \right )^{2}}{2}-\frac {x^{2} \ln \left (x \right )}{2}+\frac {x^{2}}{4}\right )\)	\(69\)

[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*x^2*ln(3)^2*ln(x)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=25 \, x^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} + 2025 \, x^{2} \log \left (3\right )^{2} + 90 \, x^{2} \log \left (3\right ) + x^{2} + 10 \, {\left (45 \, x^{2} \log \left (3\right )^{2} + x^{2} \log \left (3\right )\right )} \log \left (x\right ) \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (15) = 30\).

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=25 x^{2} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + x^{2} \cdot \left (1 + 90 \log {\left (3 \right )} + 2025 \log {\left (3 \right )}^{2}\right ) + \left (10 x^{2} \log {\left (3 \right )} + 450 x^{2} \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (16) = 32\).

Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.88 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=\frac {25}{2} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} \log \left (3\right )^{2} + 2250 \, x^{2} \log \left (3\right )^{2} - \frac {5}{2} \, {\left (95 \, \log \left (3\right )^{2} + 2 \, \log \left (3\right )\right )} x^{2} + 95 \, x^{2} \log \left (3\right ) + x^{2} + 5 \, {\left (95 \, x^{2} \log \left (3\right )^{2} + 2 \, x^{2} \log \left (3\right )\right )} \log \left (x\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.12 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=475 \, x^{2} \log \left (3\right )^{2} \log \left (x\right ) + \frac {4025}{2} \, x^{2} \log \left (3\right )^{2} + 10 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 90 \, x^{2} \log \left (3\right ) + \frac {25}{2} \, {\left (2 \, x^{2} \log \left (x\right )^{2} - 2 \, x^{2} \log \left (x\right ) + x^{2}\right )} \log \left (3\right )^{2} + x^{2} \]

[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

Mupad [B] (verification not implemented)

Time = 12.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \left (2 x+190 x \log (3)+4500 x \log ^2(3)+\left (20 x \log (3)+950 x \log ^2(3)\right ) \log (x)+50 x \log ^2(3) \log ^2(x)\right ) \, dx=x^2\,{\left (45\,\ln \left (3\right )+5\,\ln \left (3\right )\,\ln \left (x\right )+1\right )}^2 \]

[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