3.89.85 $\int \frac{1}{25}(25+(900+1050x+450x^2+100x^3+(60+60x+30x^2)\log (2)+2x{\log }^2(2)){\log }^2(9))dx$

Optimal. Leaf size=28 $$x+x^2{(x+\frac{3(2+x)}{x}+\frac{\log (2)}{5})}^2{\log }^2(9)$$

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Rubi [B]  time = 0.04, antiderivative size = 80, normalized size of antiderivative = 2.86, number of steps used = 5, number of rules used = 2, integrand size = 46, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.043, Rules used = {12, 6} $$\begin{array}{c}{x}^4{\log }^2(9)+\frac{2}{5}{x}^3\log (2){\log }^2(9)+6x^3{\log }^2(9)+\frac{1}{25}{x}^2(525+{\log }^2(2)){\log }^2(9)+\frac{6}{5}{x}^2\log (2){\log }^2(9)+x+\frac{12}{5}x\log (2){\log }^2(9)+36x{\log }^2(9)\end{array}$$

Antiderivative was successfully verified.

[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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{25}\int (25+(900+1050x+450x^2+100x^3+(60+60x+30x^2)\log (2)+2x{\log }^2(2)){\log }^2(9))dx\\ & =x+\frac{1}{25}{\log }^2(9)\int (900+1050x+450x^2+100x^3+(60+60x+30x^2)\log (2)+2x{\log }^2(2))dx\\ & =x+\frac{1}{25}{\log }^2(9)\int (900+450x^2+100x^3+(60+60x+30x^2)\log (2)+x(1050+2{\log }^2(2)))dx\\ & =x+36x{\log }^2(9)+6x^3{\log }^2(9)+x^4{\log }^2(9)+\frac{1}{25}{x}^2(525+{\log }^2(2)){\log }^2(9)+\frac{1}{25}(\log (2){\log }^2(9))\int (60+60x+30x^2)dx\\ & =x+36x{\log }^2(9)+6x^3{\log }^2(9)+x^4{\log }^2(9)+\frac{12}{5}x\log (2){\log }^2(9)+\frac{6}{5}{x}^2\log (2){\log }^2(9)+\frac{2}{5}{x}^3\log (2){\log }^2(9)+\frac{1}{25}{x}^2(525+{\log }^2(2)){\log }^2(9)\end{array}\end{array}$$

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Mathematica [B]  time = 0.03, size = 59, normalized size = 2.11 $$\begin{array}{c}x+x^4{\log }^2(9)+\frac{12}{5}x(15+\log (2)){\log }^2(9)+\frac{2}{5}{x}^3(15+\log (2)){\log }^2(9)+\frac{1}{25}{x}^2(525+30\log (2)+{\log }^2(2)){\log }^2(9)\end{array}$$

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.53, size = 51, normalized size = 1.82 $$\begin{array}{c}\frac{4}{25}(25x^4+x^2\log (2{)}^2+150x^3+525x^2+10(x^3+3x^2+6x)\log (2)+900x)\log (3{)}^2+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.19, size = 51, normalized size = 1.82 $$\begin{array}{c}\frac{4}{25}(25x^4+x^2\log (2{)}^2+150x^3+525x^2+10(x^3+3x^2+6x)\log (2)+900x)\log (3{)}^2+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.04, size = 53, normalized size = 1.89

method	result	size
default	$\frac{4\ln (3{)}^2(x^2\ln (2{)}^2+\ln (2)(10x^3+30x^2+60x)+25x^4+150x^3+525x^2+900x)}{25}+x$	$53$
gosper	$\frac{x(4\ln (2{)}^2\ln (3{)}^{2}x+40\ln (2)\ln (3{)}^2{x}^2+100x^3\ln (3{)}^2+120x\ln (2)\ln (3{)}^2+600x^2\ln (3{)}^2+240\ln (2)\ln (3{)}^2+2100x\ln (3{)}^2+3600\ln (3{)}^2+25)}{25}$	$76$
norman	$(\frac{8\ln (2)\ln (3{)}^2}{5}+24\ln (3{)}^2)x^3+(\frac{48\ln (2)\ln (3{)}^2}{5}+144\ln (3{)}^2+1)x+(\frac{4\ln (3{)}^2\ln (2{)}^2}{25}+\frac{24\ln (2)\ln (3{)}^2}{5}+84\ln (3{)}^2)x^2+4x^4\ln (3{)}^2$	$77$
risch	$\frac{4x^2\ln (3{)}^2\ln (2{)}^2}{25}+\frac{8\ln (2)\ln (3{)}^2{x}^3}{5}+4x^4\ln (3{)}^2+\frac{24\ln (2)\ln (3{)}^2{x}^2}{5}+24x^3\ln (3{)}^2+\frac{48x\ln (2)\ln (3{)}^2}{5}+84x^2\ln (3{)}^2+144x\ln (3{)}^2+x$	$81$

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [A]  time = 0.37, size = 51, normalized size = 1.82 $$\begin{array}{c}\frac{4}{25}(25x^4+x^2\log (2{)}^2+150x^3+525x^2+10(x^3+3x^2+6x)\log (2)+900x)\log (3{)}^2+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.07, size = 62, normalized size = 2.21 $$\begin{array}{c}4{\ln (3)}^2{x}^4+\frac{4{\ln (3)}^2(30\ln (2)+450)x^3}{75}+\frac{2{\ln (3)}^2(60\ln (2)+2{\ln (2)}^2+1050)x^2}{25}+(\frac{4{\ln (3)}^2(60\ln (2)+900)}{25}+1)x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.08, size = 88, normalized size = 3.14 $$\begin{array}{c}4x^4\log {(3)}^2+x^3(\frac{8\log (2)\log {(3)}^2}{5}+24\log {(3)}^2)+x^2(\frac{4\log {(2)}^2\log {(3)}^2}{25}+\frac{24\log (2)\log {(3)}^2}{5}+84\log {(3)}^2)+x(1+\frac{48\log (2)\log {(3)}^2}{5}+144\log {(3)}^2)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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