\(\int \frac {1}{25} (25+(900+1050 x+450 x^2+100 x^3+(60+60 x+30 x^2) \log (2)+2 x \log ^2(2)) \log ^2(9)) \, dx\) [8885]

Optimal result

Integrand size = 46, antiderivative size = 28 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=x+x^2 \left (x+\frac {3 (2+x)}{x}+\frac {\log (2)}{5}\right )^2 \log ^2(9) \]

[Out]

4*x^2*(x+1/5*ln(2)+3*(2+x)/x)^2*ln(3)^2+x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(28)=56\).

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 6} \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=x^4 \log ^2(9)+\frac {2}{5} x^3 \log (2) \log ^2(9)+6 x^3 \log ^2(9)+\frac {1}{25} x^2 \left (525+\log ^2(2)\right ) \log ^2(9)+\frac {6}{5} x^2 \log (2) \log ^2(9)+x+\frac {12}{5} x \log (2) \log ^2(9)+36 x \log ^2(9) \]

[In]

Int[(25 + (900 + 1050*x + 450*x^2 + 100*x^3 + (60 + 60*x + 30*x^2)*Log[2] + 2*x*Log[2]^2)*Log[9]^2)/25,x]

[Out]

x + 36*x*Log[9]^2 + 6*x^3*Log[9]^2 + x^4*Log[9]^2 + (12*x*Log[2]*Log[9]^2)/5 + (6*x^2*Log[2]*Log[9]^2)/5 + (2*
x^3*Log[2]*Log[9]^2)/5 + (x^2*(525 + Log[2]^2)*Log[9]^2)/25

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx \\ & = x+\frac {1}{25} \log ^2(9) \int \left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \, dx \\ & = x+\frac {1}{25} \log ^2(9) \int \left (900+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+x \left (1050+2 \log ^2(2)\right )\right ) \, dx \\ & = x+36 x \log ^2(9)+6 x^3 \log ^2(9)+x^4 \log ^2(9)+\frac {1}{25} x^2 \left (525+\log ^2(2)\right ) \log ^2(9)+\frac {1}{25} \left (\log (2) \log ^2(9)\right ) \int \left (60+60 x+30 x^2\right ) \, dx \\ & = x+36 x \log ^2(9)+6 x^3 \log ^2(9)+x^4 \log ^2(9)+\frac {12}{5} x \log (2) \log ^2(9)+\frac {6}{5} x^2 \log (2) \log ^2(9)+\frac {2}{5} x^3 \log (2) \log ^2(9)+\frac {1}{25} x^2 \left (525+\log ^2(2)\right ) \log ^2(9) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(28)=56\).

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=x+x^4 \log ^2(9)+\frac {12}{5} x (15+\log (2)) \log ^2(9)+\frac {2}{5} x^3 (15+\log (2)) \log ^2(9)+\frac {1}{25} x^2 \left (525+30 \log (2)+\log ^2(2)\right ) \log ^2(9) \]

[In]

Integrate[(25 + (900 + 1050*x + 450*x^2 + 100*x^3 + (60 + 60*x + 30*x^2)*Log[2] + 2*x*Log[2]^2)*Log[9]^2)/25,x
]

[Out]

x + x^4*Log[9]^2 + (12*x*(15 + Log[2])*Log[9]^2)/5 + (2*x^3*(15 + Log[2])*Log[9]^2)/5 + (x^2*(525 + 30*Log[2]
+ Log[2]^2)*Log[9]^2)/25

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96

method	result	size
parallelrisch	\(\frac {4 \ln \left (3\right )^{2} \left (x^{2} \ln \left (2\right )^{2}+10 x^{3} \ln \left (2\right )+25 x^{4}+30 x^{2} \ln \left (2\right )+150 x^{3}+60 x \ln \left (2\right )+525 x^{2}+900 x \right )}{25}+x\)	\(55\)
gosper	\(\frac {x \left (4 \ln \left (2\right )^{2} \ln \left (3\right )^{2} x +40 \ln \left (2\right ) \ln \left (3\right )^{2} x^{2}+100 x^{3} \ln \left (3\right )^{2}+120 x \ln \left (2\right ) \ln \left (3\right )^{2}+600 x^{2} \ln \left (3\right )^{2}+240 \ln \left (2\right ) \ln \left (3\right )^{2}+2100 x \ln \left (3\right )^{2}+3600 \ln \left (3\right )^{2}+25\right )}{25}\)	\(76\)
norman	\(\left (\frac {8 \ln \left (2\right ) \ln \left (3\right )^{2}}{5}+24 \ln \left (3\right )^{2}\right ) x^{3}+\left (\frac {48 \ln \left (2\right ) \ln \left (3\right )^{2}}{5}+144 \ln \left (3\right )^{2}+1\right ) x +\left (\frac {4 \ln \left (3\right )^{2} \ln \left (2\right )^{2}}{25}+\frac {24 \ln \left (2\right ) \ln \left (3\right )^{2}}{5}+84 \ln \left (3\right )^{2}\right ) x^{2}+4 x^{4} \ln \left (3\right )^{2}\)	\(77\)
default	\(\frac {4 x^{2} \ln \left (3\right )^{2} \ln \left (2\right )^{2}}{25}+\frac {8 \ln \left (2\right ) \ln \left (3\right )^{2} x^{3}}{5}+4 x^{4} \ln \left (3\right )^{2}+\frac {24 \ln \left (2\right ) \ln \left (3\right )^{2} x^{2}}{5}+24 x^{3} \ln \left (3\right )^{2}+\frac {48 x \ln \left (2\right ) \ln \left (3\right )^{2}}{5}+84 x^{2} \ln \left (3\right )^{2}+144 x \ln \left (3\right )^{2}+x\)	\(81\)
parts	\(\frac {4 x^{2} \ln \left (3\right )^{2} \ln \left (2\right )^{2}}{25}+\frac {8 \ln \left (2\right ) \ln \left (3\right )^{2} x^{3}}{5}+4 x^{4} \ln \left (3\right )^{2}+\frac {24 \ln \left (2\right ) \ln \left (3\right )^{2} x^{2}}{5}+24 x^{3} \ln \left (3\right )^{2}+\frac {48 x \ln \left (2\right ) \ln \left (3\right )^{2}}{5}+84 x^{2} \ln \left (3\right )^{2}+144 x \ln \left (3\right )^{2}+x\)	\(81\)
risch	\(\frac {4 x^{2} \ln \left (3\right )^{2} \ln \left (2\right )^{2}}{25}+\frac {8 \ln \left (2\right ) \ln \left (3\right )^{2} x^{3}}{5}+4 x^{4} \ln \left (3\right )^{2}+\frac {24 \ln \left (2\right ) \ln \left (3\right )^{2} x^{2}}{5}+24 x^{3} \ln \left (3\right )^{2}+\frac {48 x \ln \left (2\right ) \ln \left (3\right )^{2}}{5}+84 x^{2} \ln \left (3\right )^{2}+144 x \ln \left (3\right )^{2}+144 \ln \left (3\right )^{2}+x\)	\(87\)

[In]

int(4/25*(2*x*ln(2)^2+(30*x^2+60*x+60)*ln(2)+100*x^3+450*x^2+1050*x+900)*ln(3)^2+1,x,method=_RETURNVERBOSE)

[Out]

4/25*ln(3)^2*(x^2*ln(2)^2+10*x^3*ln(2)+25*x^4+30*x^2*ln(2)+150*x^3+60*x*ln(2)+525*x^2+900*x)+x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=\frac {4}{25} \, {\left (25 \, x^{4} + x^{2} \log \left (2\right )^{2} + 150 \, x^{3} + 525 \, x^{2} + 10 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x\right )} \log \left (2\right ) + 900 \, x\right )} \log \left (3\right )^{2} + x \]

[In]

integrate(4/25*(2*x*log(2)^2+(30*x^2+60*x+60)*log(2)+100*x^3+450*x^2+1050*x+900)*log(3)^2+1,x, algorithm="fric
as")

[Out]

4/25*(25*x^4 + x^2*log(2)^2 + 150*x^3 + 525*x^2 + 10*(x^3 + 3*x^2 + 6*x)*log(2) + 900*x)*log(3)^2 + x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (26) = 52\).

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=4 x^{4} \log {\left (3 \right )}^{2} + x^{3} \cdot \left (\frac {8 \log {\left (2 \right )} \log {\left (3 \right )}^{2}}{5} + 24 \log {\left (3 \right )}^{2}\right ) + x^{2} \cdot \left (\frac {4 \log {\left (2 \right )}^{2} \log {\left (3 \right )}^{2}}{25} + \frac {24 \log {\left (2 \right )} \log {\left (3 \right )}^{2}}{5} + 84 \log {\left (3 \right )}^{2}\right ) + x \left (1 + \frac {48 \log {\left (2 \right )} \log {\left (3 \right )}^{2}}{5} + 144 \log {\left (3 \right )}^{2}\right ) \]

[In]

integrate(4/25*(2*x*ln(2)**2+(30*x**2+60*x+60)*ln(2)+100*x**3+450*x**2+1050*x+900)*ln(3)**2+1,x)

[Out]

4*x**4*log(3)**2 + x**3*(8*log(2)*log(3)**2/5 + 24*log(3)**2) + x**2*(4*log(2)**2*log(3)**2/25 + 24*log(2)*log
(3)**2/5 + 84*log(3)**2) + x*(1 + 48*log(2)*log(3)**2/5 + 144*log(3)**2)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=\frac {4}{25} \, {\left (25 \, x^{4} + x^{2} \log \left (2\right )^{2} + 150 \, x^{3} + 525 \, x^{2} + 10 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x\right )} \log \left (2\right ) + 900 \, x\right )} \log \left (3\right )^{2} + x \]

[In]

integrate(4/25*(2*x*log(2)^2+(30*x^2+60*x+60)*log(2)+100*x^3+450*x^2+1050*x+900)*log(3)^2+1,x, algorithm="maxi
ma")

[Out]

4/25*(25*x^4 + x^2*log(2)^2 + 150*x^3 + 525*x^2 + 10*(x^3 + 3*x^2 + 6*x)*log(2) + 900*x)*log(3)^2 + x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=\frac {4}{25} \, {\left (25 \, x^{4} + x^{2} \log \left (2\right )^{2} + 150 \, x^{3} + 525 \, x^{2} + 10 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x\right )} \log \left (2\right ) + 900 \, x\right )} \log \left (3\right )^{2} + x \]

[In]

integrate(4/25*(2*x*log(2)^2+(30*x^2+60*x+60)*log(2)+100*x^3+450*x^2+1050*x+900)*log(3)^2+1,x, algorithm="giac
")

[Out]

4/25*(25*x^4 + x^2*log(2)^2 + 150*x^3 + 525*x^2 + 10*(x^3 + 3*x^2 + 6*x)*log(2) + 900*x)*log(3)^2 + x

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {1}{25} \left (25+\left (900+1050 x+450 x^2+100 x^3+\left (60+60 x+30 x^2\right ) \log (2)+2 x \log ^2(2)\right ) \log ^2(9)\right ) \, dx=4\,{\ln \left (3\right )}^2\,x^4+\frac {4\,{\ln \left (3\right )}^2\,\left (30\,\ln \left (2\right )+450\right )\,x^3}{75}+\frac {2\,{\ln \left (3\right )}^2\,\left (60\,\ln \left (2\right )+2\,{\ln \left (2\right )}^2+1050\right )\,x^2}{25}+\left (\frac {4\,{\ln \left (3\right )}^2\,\left (60\,\ln \left (2\right )+900\right )}{25}+1\right )\,x \]

[In]

int((4*log(3)^2*(1050*x + log(2)*(60*x + 30*x^2 + 60) + 2*x*log(2)^2 + 450*x^2 + 100*x^3 + 900))/25 + 1,x)

[Out]

4*x^4*log(3)^2 + x*((4*log(3)^2*(60*log(2) + 900))/25 + 1) + (4*x^3*log(3)^2*(30*log(2) + 450))/75 + (2*x^2*lo
g(3)^2*(60*log(2) + 2*log(2)^2 + 1050))/25