3.25.63 \(\int (4620 x^2+2160 x^3+240 x^4+(816 x^2+192 x^3) \log (x)+36 x^2 \log ^2(x)) \, dx\) [2463]

3.25.63.1 Optimal result

Integrand size = 39, antiderivative size = 18 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=12 x (3 x+x (8+2 x+\log (x)))^2 \]

output

12*x*(3*x+x*(8+2*x+ln(x)))^2
 

3.25.63.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=1452 x^3+528 x^4+48 x^5+264 x^3 \log (x)+48 x^4 \log (x)+12 x^3 \log ^2(x) \]

input

Integrate[4620*x^2 + 2160*x^3 + 240*x^4 + (816*x^2 + 192*x^3)*Log[x] + 36* 
x^2*Log[x]^2,x]
 

output

1452*x^3 + 528*x^4 + 48*x^5 + 264*x^3*Log[x] + 48*x^4*Log[x] + 12*x^3*Log[ 
x]^2
 

3.25.63.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(18)=36\).

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.61, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (240 x^4+2160 x^3+4620 x^2+36 x^2 \log ^2(x)+\left (192 x^3+816 x^2\right ) \log (x)\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 48 x^5+528 x^4+1452 x^3+12 x^3 \log ^2(x)-8 x^3 \log (x)+16 \left (3 x^4+17 x^3\right ) \log (x)\)

input

Int[4620*x^2 + 2160*x^3 + 240*x^4 + (816*x^2 + 192*x^3)*Log[x] + 36*x^2*Lo 
g[x]^2,x]
 

output

1452*x^3 + 528*x^4 + 48*x^5 - 8*x^3*Log[x] + 16*(17*x^3 + 3*x^4)*Log[x] + 
12*x^3*Log[x]^2
 

3.25.63.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.25.63.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).

Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.22

method	result	size
default	\(48 x^{4} \ln \left (x \right )+528 x^{4}+264 x^{3} \ln \left (x \right )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \left (x \right )^{2}\)	\(40\)
norman	\(48 x^{4} \ln \left (x \right )+528 x^{4}+264 x^{3} \ln \left (x \right )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \left (x \right )^{2}\)	\(40\)
risch	\(48 x^{4} \ln \left (x \right )+528 x^{4}+264 x^{3} \ln \left (x \right )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \left (x \right )^{2}\)	\(40\)
parallelrisch	\(48 x^{4} \ln \left (x \right )+528 x^{4}+264 x^{3} \ln \left (x \right )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \left (x \right )^{2}\)	\(40\)
parts	\(48 x^{4} \ln \left (x \right )+528 x^{4}+264 x^{3} \ln \left (x \right )+1452 x^{3}+48 x^{5}+12 x^{3} \ln \left (x \right )^{2}\)	\(40\)

input

int(36*x^2*ln(x)^2+(192*x^3+816*x^2)*ln(x)+240*x^4+2160*x^3+4620*x^2,x,met 
hod=_RETURNVERBOSE)
 

output

48*x^4*ln(x)+528*x^4+264*x^3*ln(x)+1452*x^3+48*x^5+12*x^3*ln(x)^2
 

3.25.63.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.22 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=48 \, x^{5} + 12 \, x^{3} \log \left (x\right )^{2} + 528 \, x^{4} + 1452 \, x^{3} + 24 \, {\left (2 \, x^{4} + 11 \, x^{3}\right )} \log \left (x\right ) \]

input

integrate(36*x^2*log(x)^2+(192*x^3+816*x^2)*log(x)+240*x^4+2160*x^3+4620*x 
^2,x, algorithm=\
 

output

48*x^5 + 12*x^3*log(x)^2 + 528*x^4 + 1452*x^3 + 24*(2*x^4 + 11*x^3)*log(x)
 

3.25.63.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).

Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=48 x^{5} + 528 x^{4} + 12 x^{3} \log {\left (x \right )}^{2} + 1452 x^{3} + \left (48 x^{4} + 264 x^{3}\right ) \log {\left (x \right )} \]

input

integrate(36*x**2*ln(x)**2+(192*x**3+816*x**2)*ln(x)+240*x**4+2160*x**3+46 
20*x**2,x)
 

output

48*x**5 + 528*x**4 + 12*x**3*log(x)**2 + 1452*x**3 + (48*x**4 + 264*x**3)* 
log(x)
 

3.25.63.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.67 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=48 \, x^{5} + \frac {4}{3} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} + 528 \, x^{4} + \frac {4348}{3} \, x^{3} + 16 \, {\left (3 \, x^{4} + 17 \, x^{3}\right )} \log \left (x\right ) \]

input

integrate(36*x^2*log(x)^2+(192*x^3+816*x^2)*log(x)+240*x^4+2160*x^3+4620*x 
^2,x, algorithm=\
 

output

48*x^5 + 4/3*(9*log(x)^2 - 6*log(x) + 2)*x^3 + 528*x^4 + 4348/3*x^3 + 16*( 
3*x^4 + 17*x^3)*log(x)
 

3.25.63.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=48 \, x^{5} + 48 \, x^{4} \log \left (x\right ) + 12 \, x^{3} \log \left (x\right )^{2} + 528 \, x^{4} + 264 \, x^{3} \log \left (x\right ) + 1452 \, x^{3} \]

input

integrate(36*x^2*log(x)^2+(192*x^3+816*x^2)*log(x)+240*x^4+2160*x^3+4620*x 
^2,x, algorithm=\
 

output

48*x^5 + 48*x^4*log(x) + 12*x^3*log(x)^2 + 528*x^4 + 264*x^3*log(x) + 1452 
*x^3
 

3.25.63.9 Mupad [B] (verification not implemented)

Time = 14.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (4620 x^2+2160 x^3+240 x^4+\left (816 x^2+192 x^3\right ) \log (x)+36 x^2 \log ^2(x)\right ) \, dx=12\,x^3\,{\left (2\,x+\ln \left (x\right )+11\right )}^2 \]

input

int(log(x)*(816*x^2 + 192*x^3) + 36*x^2*log(x)^2 + 4620*x^2 + 2160*x^3 + 2 
40*x^4,x)
 

output

12*x^3*(2*x + log(x) + 11)^2