3.4.63 \(\int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+(16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9) \log (3 x)+(16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8) \log ^2(3 x)}{8 x} \, dx\)

Optimal. Leaf size=29 \[ \left (1+\left (3-\frac {1}{x^2}\right ) \left (x+\frac {x^2}{2}\right )^2 (x+\log (3 x))\right )^2 \]

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Rubi [B]  time = 0.57, antiderivative size = 251, normalized size of antiderivative = 8.66, number of steps used = 35, number of rules used = 9, integrand size = 153, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 14, 2357, 2295, 2301, 2304, 2356, 2296, 2305} \begin {gather*} \frac {9 x^{10}}{16}+\frac {9 x^9}{2}+\frac {9}{8} x^9 \log (3 x)+\frac {105 x^8}{8}+\frac {9}{16} x^8 \log ^2(3 x)+9 x^8 \log (3 x)+15 x^7+\frac {9}{2} x^7 \log ^2(3 x)+\frac {105}{4} x^7 \log (3 x)+\frac {x^6}{16}+\frac {105}{8} x^6 \log ^2(3 x)+30 x^6 \log (3 x)-10 x^5+15 x^5 \log ^2(3 x)+\frac {1}{8} x^5 \log (3 x)+\frac {3 x^4}{2}+\frac {1}{16} x^4 \log ^2(3 x)-\frac {43}{2} x^4 \log (3 x)+\frac {15 x^3}{2}-\frac {23}{2} x^3 \log ^2(3 x)-3 x^3 \log (3 x)-x^2-\frac {9}{2} x^2 \log ^2(3 x)+\frac {19}{2} x^2 \log (3 x)-2 x+2 x \log ^2(3 x)+\log ^2(3 x)-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 - 16*x + 60*x^2 + 156*x^3 - 124*x^4 - 399*x^5 + 243*x^6 + 1050*x^7 + 912*x^8 + 333*x^9 + 45*x^10 + (1
6 + 32*x + 80*x^2 - 256*x^3 - 687*x^4 + 245*x^5 + 1650*x^6 + 1542*x^7 + 585*x^8 + 81*x^9)*Log[3*x] + (16*x - 7
2*x^2 - 276*x^3 + 2*x^4 + 600*x^5 + 630*x^6 + 252*x^7 + 36*x^8)*Log[3*x]^2)/(8*x),x]

[Out]

-2*x - x^2 + (15*x^3)/2 + (3*x^4)/2 - 10*x^5 + x^6/16 + 15*x^7 + (105*x^8)/8 + (9*x^9)/2 + (9*x^10)/16 - 2*Log
[x] + (19*x^2*Log[3*x])/2 - 3*x^3*Log[3*x] - (43*x^4*Log[3*x])/2 + (x^5*Log[3*x])/8 + 30*x^6*Log[3*x] + (105*x
^7*Log[3*x])/4 + 9*x^8*Log[3*x] + (9*x^9*Log[3*x])/8 + Log[3*x]^2 + 2*x*Log[3*x]^2 - (9*x^2*Log[3*x]^2)/2 - (2
3*x^3*Log[3*x]^2)/2 + (x^4*Log[3*x]^2)/16 + 15*x^5*Log[3*x]^2 + (105*x^6*Log[3*x]^2)/8 + (9*x^7*Log[3*x]^2)/2
+ (9*x^8*Log[3*x]^2)/16

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, 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]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{x} \, dx\\ &=\frac {1}{8} \int \left (\frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}}{x}+\frac {(1+x) (2+x) \left (8+4 x+30 x^2-175 x^3-96 x^4+354 x^5+342 x^6+81 x^7\right ) \log (3 x)}{x}+2 (2+x)^3 \left (-1+3 x^2\right ) \left (-1+6 x+6 x^2\right ) \log ^2(3 x)\right ) \, dx\\ &=\frac {1}{8} \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}}{x} \, dx+\frac {1}{8} \int \frac {(1+x) (2+x) \left (8+4 x+30 x^2-175 x^3-96 x^4+354 x^5+342 x^6+81 x^7\right ) \log (3 x)}{x} \, dx+\frac {1}{4} \int (2+x)^3 \left (-1+3 x^2\right ) \left (-1+6 x+6 x^2\right ) \log ^2(3 x) \, dx\\ &=\frac {1}{8} \int \left (-16-\frac {16}{x}+60 x+156 x^2-124 x^3-399 x^4+243 x^5+1050 x^6+912 x^7+333 x^8+45 x^9\right ) \, dx+\frac {1}{8} \int \left (32 \log (3 x)+\frac {16 \log (3 x)}{x}+80 x \log (3 x)-256 x^2 \log (3 x)-687 x^3 \log (3 x)+245 x^4 \log (3 x)+1650 x^5 \log (3 x)+1542 x^6 \log (3 x)+585 x^7 \log (3 x)+81 x^8 \log (3 x)\right ) \, dx+\frac {1}{4} \int \left (8 \log ^2(3 x)-36 x \log ^2(3 x)-138 x^2 \log ^2(3 x)+x^3 \log ^2(3 x)+300 x^4 \log ^2(3 x)+315 x^5 \log ^2(3 x)+126 x^6 \log ^2(3 x)+18 x^7 \log ^2(3 x)\right ) \, dx\\ &=-2 x+\frac {15 x^2}{4}+\frac {13 x^3}{2}-\frac {31 x^4}{8}-\frac {399 x^5}{40}+\frac {81 x^6}{16}+\frac {75 x^7}{4}+\frac {57 x^8}{4}+\frac {37 x^9}{8}+\frac {9 x^{10}}{16}-2 \log (x)+\frac {1}{4} \int x^3 \log ^2(3 x) \, dx+2 \int \frac {\log (3 x)}{x} \, dx+2 \int \log ^2(3 x) \, dx+4 \int \log (3 x) \, dx+\frac {9}{2} \int x^7 \log ^2(3 x) \, dx-9 \int x \log ^2(3 x) \, dx+10 \int x \log (3 x) \, dx+\frac {81}{8} \int x^8 \log (3 x) \, dx+\frac {245}{8} \int x^4 \log (3 x) \, dx+\frac {63}{2} \int x^6 \log ^2(3 x) \, dx-32 \int x^2 \log (3 x) \, dx-\frac {69}{2} \int x^2 \log ^2(3 x) \, dx+\frac {585}{8} \int x^7 \log (3 x) \, dx+75 \int x^4 \log ^2(3 x) \, dx+\frac {315}{4} \int x^5 \log ^2(3 x) \, dx-\frac {687}{8} \int x^3 \log (3 x) \, dx+\frac {771}{4} \int x^6 \log (3 x) \, dx+\frac {825}{4} \int x^5 \log (3 x) \, dx\\ &=-6 x+\frac {5 x^2}{4}+\frac {181 x^3}{18}+\frac {191 x^4}{128}-\frac {56 x^5}{5}-\frac {2 x^6}{3}+\frac {726 x^7}{49}+\frac {6711 x^8}{512}+\frac {9 x^9}{2}+\frac {9 x^{10}}{16}-2 \log (x)+4 x \log (3 x)+5 x^2 \log (3 x)-\frac {32}{3} x^3 \log (3 x)-\frac {687}{32} x^4 \log (3 x)+\frac {49}{8} x^5 \log (3 x)+\frac {275}{8} x^6 \log (3 x)+\frac {771}{28} x^7 \log (3 x)+\frac {585}{64} x^8 \log (3 x)+\frac {9}{8} x^9 \log (3 x)+\log ^2(3 x)+2 x \log ^2(3 x)-\frac {9}{2} x^2 \log ^2(3 x)-\frac {23}{2} x^3 \log ^2(3 x)+\frac {1}{16} x^4 \log ^2(3 x)+15 x^5 \log ^2(3 x)+\frac {105}{8} x^6 \log ^2(3 x)+\frac {9}{2} x^7 \log ^2(3 x)+\frac {9}{16} x^8 \log ^2(3 x)-\frac {1}{8} \int x^3 \log (3 x) \, dx-\frac {9}{8} \int x^7 \log (3 x) \, dx-4 \int \log (3 x) \, dx+9 \int x \log (3 x) \, dx-9 \int x^6 \log (3 x) \, dx+23 \int x^2 \log (3 x) \, dx-\frac {105}{4} \int x^5 \log (3 x) \, dx-30 \int x^4 \log (3 x) \, dx\\ &=-2 x-x^2+\frac {15 x^3}{2}+\frac {3 x^4}{2}-10 x^5+\frac {x^6}{16}+15 x^7+\frac {105 x^8}{8}+\frac {9 x^9}{2}+\frac {9 x^{10}}{16}-2 \log (x)+\frac {19}{2} x^2 \log (3 x)-3 x^3 \log (3 x)-\frac {43}{2} x^4 \log (3 x)+\frac {1}{8} x^5 \log (3 x)+30 x^6 \log (3 x)+\frac {105}{4} x^7 \log (3 x)+9 x^8 \log (3 x)+\frac {9}{8} x^9 \log (3 x)+\log ^2(3 x)+2 x \log ^2(3 x)-\frac {9}{2} x^2 \log ^2(3 x)-\frac {23}{2} x^3 \log ^2(3 x)+\frac {1}{16} x^4 \log ^2(3 x)+15 x^5 \log ^2(3 x)+\frac {105}{8} x^6 \log ^2(3 x)+\frac {9}{2} x^7 \log ^2(3 x)+\frac {9}{16} x^8 \log ^2(3 x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 114, normalized size = 3.93 \begin {gather*} \frac {1}{16} \left (x (2+x)^2 \left (-8+4 x+28 x^2-23 x^3-24 x^4+30 x^5+36 x^6+9 x^7\right )-32 \log (x)+2 x^2 \left (76-24 x-172 x^2+x^3+240 x^4+210 x^5+72 x^6+9 x^7\right ) \log (3 x)+(2+x)^4 \left (1-3 x^2\right )^2 \log ^2(3 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 16*x + 60*x^2 + 156*x^3 - 124*x^4 - 399*x^5 + 243*x^6 + 1050*x^7 + 912*x^8 + 333*x^9 + 45*x^1
0 + (16 + 32*x + 80*x^2 - 256*x^3 - 687*x^4 + 245*x^5 + 1650*x^6 + 1542*x^7 + 585*x^8 + 81*x^9)*Log[3*x] + (16
*x - 72*x^2 - 276*x^3 + 2*x^4 + 600*x^5 + 630*x^6 + 252*x^7 + 36*x^8)*Log[3*x]^2)/(8*x),x]

[Out]

(x*(2 + x)^2*(-8 + 4*x + 28*x^2 - 23*x^3 - 24*x^4 + 30*x^5 + 36*x^6 + 9*x^7) - 32*Log[x] + 2*x^2*(76 - 24*x -
172*x^2 + x^3 + 240*x^4 + 210*x^5 + 72*x^6 + 9*x^7)*Log[3*x] + (2 + x)^4*(1 - 3*x^2)^2*Log[3*x]^2)/16

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fricas [B]  time = 0.64, size = 141, normalized size = 4.86 \begin {gather*} \frac {9}{16} \, x^{10} + \frac {9}{2} \, x^{9} + \frac {105}{8} \, x^{8} + 15 \, x^{7} + \frac {1}{16} \, x^{6} - 10 \, x^{5} + \frac {3}{2} \, x^{4} + \frac {15}{2} \, x^{3} + \frac {1}{16} \, {\left (9 \, x^{8} + 72 \, x^{7} + 210 \, x^{6} + 240 \, x^{5} + x^{4} - 184 \, x^{3} - 72 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x\right )^{2} - x^{2} + \frac {1}{8} \, {\left (9 \, x^{9} + 72 \, x^{8} + 210 \, x^{7} + 240 \, x^{6} + x^{5} - 172 \, x^{4} - 24 \, x^{3} + 76 \, x^{2} - 16\right )} \log \left (3 \, x\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)*log(3*x)^2+(81*x^9+585*x^8+1542*x^7+
1650*x^6+245*x^5-687*x^4-256*x^3+80*x^2+32*x+16)*log(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124
*x^4+156*x^3+60*x^2-16*x-16)/x,x, algorithm="fricas")

[Out]

9/16*x^10 + 9/2*x^9 + 105/8*x^8 + 15*x^7 + 1/16*x^6 - 10*x^5 + 3/2*x^4 + 15/2*x^3 + 1/16*(9*x^8 + 72*x^7 + 210
*x^6 + 240*x^5 + x^4 - 184*x^3 - 72*x^2 + 32*x + 16)*log(3*x)^2 - x^2 + 1/8*(9*x^9 + 72*x^8 + 210*x^7 + 240*x^
6 + x^5 - 172*x^4 - 24*x^3 + 76*x^2 - 16)*log(3*x) - 2*x

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giac [B]  time = 0.42, size = 144, normalized size = 4.97 \begin {gather*} \frac {9}{16} \, x^{10} + \frac {9}{2} \, x^{9} + \frac {105}{8} \, x^{8} + 15 \, x^{7} + \frac {1}{16} \, x^{6} - 10 \, x^{5} + \frac {3}{2} \, x^{4} + \frac {15}{2} \, x^{3} + \frac {1}{16} \, {\left (9 \, x^{8} + 72 \, x^{7} + 210 \, x^{6} + 240 \, x^{5} + x^{4} - 184 \, x^{3} - 72 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x\right )^{2} - x^{2} + \frac {1}{8} \, {\left (9 \, x^{9} + 72 \, x^{8} + 210 \, x^{7} + 240 \, x^{6} + x^{5} - 172 \, x^{4} - 24 \, x^{3} + 76 \, x^{2}\right )} \log \left (3 \, x\right ) - 2 \, x - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)*log(3*x)^2+(81*x^9+585*x^8+1542*x^7+
1650*x^6+245*x^5-687*x^4-256*x^3+80*x^2+32*x+16)*log(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124
*x^4+156*x^3+60*x^2-16*x-16)/x,x, algorithm="giac")

[Out]

9/16*x^10 + 9/2*x^9 + 105/8*x^8 + 15*x^7 + 1/16*x^6 - 10*x^5 + 3/2*x^4 + 15/2*x^3 + 1/16*(9*x^8 + 72*x^7 + 210
*x^6 + 240*x^5 + x^4 - 184*x^3 - 72*x^2 + 32*x + 16)*log(3*x)^2 - x^2 + 1/8*(9*x^9 + 72*x^8 + 210*x^7 + 240*x^
6 + x^5 - 172*x^4 - 24*x^3 + 76*x^2)*log(3*x) - 2*x - 2*log(x)

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maple [B]  time = 0.05, size = 147, normalized size = 5.07

method	result	size
risch	\(\frac {\left (\frac {9}{2} x^{8}+36 x^{7}+105 x^{6}+120 x^{5}+\frac {1}{2} x^{4}-92 x^{3}-36 x^{2}+16 x +8\right ) \ln \left (3 x \right )^{2}}{8}+\frac {\left (9 x^{9}+72 x^{8}+210 x^{7}+240 x^{6}+x^{5}-172 x^{4}-24 x^{3}+76 x^{2}\right ) \ln \left (3 x \right )}{8}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+\frac {105 x^{8}}{8}+15 x^{7}+\frac {x^{6}}{16}-10 x^{5}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}-2 x -2 \ln \relax (x )\)	\(147\)
derivativedivides	\(-2 x -2 \ln \left (3 x \right )+\ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )^{2}-3 x^{3} \ln \left (3 x \right )+15 x^{7}+\frac {105 x^{8}}{8}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+\frac {x^{6}}{16}-10 x^{5}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}+\frac {19 x^{2} \ln \left (3 x \right )}{2}-\frac {23 x^{3} \ln \left (3 x \right )^{2}}{2}-\frac {9 x^{2} \ln \left (3 x \right )^{2}}{2}+\frac {9 \ln \left (3 x \right )^{2} x^{8}}{16}+\frac {9 \ln \left (3 x \right ) x^{9}}{8}+\frac {9 \ln \left (3 x \right )^{2} x^{7}}{2}+9 \ln \left (3 x \right ) x^{8}+\frac {105 \ln \left (3 x \right )^{2} x^{6}}{8}+\frac {105 \ln \left (3 x \right ) x^{7}}{4}+15 \ln \left (3 x \right )^{2} x^{5}+30 \ln \left (3 x \right ) x^{6}+\frac {\ln \left (3 x \right )^{2} x^{4}}{16}+\frac {\ln \left (3 x \right ) x^{5}}{8}-\frac {43 \ln \left (3 x \right ) x^{4}}{2}\)	\(220\)
default	\(-2 x -2 \ln \left (3 x \right )+\ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )^{2}-3 x^{3} \ln \left (3 x \right )+15 x^{7}+\frac {105 x^{8}}{8}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+\frac {x^{6}}{16}-10 x^{5}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}+\frac {19 x^{2} \ln \left (3 x \right )}{2}-\frac {23 x^{3} \ln \left (3 x \right )^{2}}{2}-\frac {9 x^{2} \ln \left (3 x \right )^{2}}{2}+\frac {9 \ln \left (3 x \right )^{2} x^{8}}{16}+\frac {9 \ln \left (3 x \right ) x^{9}}{8}+\frac {9 \ln \left (3 x \right )^{2} x^{7}}{2}+9 \ln \left (3 x \right ) x^{8}+\frac {105 \ln \left (3 x \right )^{2} x^{6}}{8}+\frac {105 \ln \left (3 x \right ) x^{7}}{4}+15 \ln \left (3 x \right )^{2} x^{5}+30 \ln \left (3 x \right ) x^{6}+\frac {\ln \left (3 x \right )^{2} x^{4}}{16}+\frac {\ln \left (3 x \right ) x^{5}}{8}-\frac {43 \ln \left (3 x \right ) x^{4}}{2}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)*ln(3*x)^2+(81*x^9+585*x^8+1542*x^7+1650*x^
6+245*x^5-687*x^4-256*x^3+80*x^2+32*x+16)*ln(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124*x^4+156
*x^3+60*x^2-16*x-16)/x,x,method=_RETURNVERBOSE)

[Out]

1/8*(9/2*x^8+36*x^7+105*x^6+120*x^5+1/2*x^4-92*x^3-36*x^2+16*x+8)*ln(3*x)^2+1/8*(9*x^9+72*x^8+210*x^7+240*x^6+
x^5-172*x^4-24*x^3+76*x^2)*ln(3*x)+9/16*x^10+9/2*x^9+105/8*x^8+15*x^7+1/16*x^6-10*x^5+3/2*x^4+15/2*x^3-x^2-2*x
-2*ln(x)

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maxima [B]  time = 0.42, size = 302, normalized size = 10.41 \begin {gather*} \frac {9}{16} \, x^{10} + \frac {9}{8} \, x^{9} \log \left (3 \, x\right ) + \frac {9}{512} \, {\left (32 \, \log \left (3 \, x\right )^{2} - 8 \, \log \left (3 \, x\right ) + 1\right )} x^{8} + \frac {9}{2} \, x^{9} + \frac {585}{64} \, x^{8} \log \left (3 \, x\right ) + \frac {9}{98} \, {\left (49 \, \log \left (3 \, x\right )^{2} - 14 \, \log \left (3 \, x\right ) + 2\right )} x^{7} + \frac {6711}{512} \, x^{8} + \frac {771}{28} \, x^{7} \log \left (3 \, x\right ) + \frac {35}{48} \, {\left (18 \, \log \left (3 \, x\right )^{2} - 6 \, \log \left (3 \, x\right ) + 1\right )} x^{6} + \frac {726}{49} \, x^{7} + \frac {275}{8} \, x^{6} \log \left (3 \, x\right ) + \frac {3}{5} \, {\left (25 \, \log \left (3 \, x\right )^{2} - 10 \, \log \left (3 \, x\right ) + 2\right )} x^{5} - \frac {2}{3} \, x^{6} + \frac {49}{8} \, x^{5} \log \left (3 \, x\right ) + \frac {1}{128} \, {\left (8 \, \log \left (3 \, x\right )^{2} - 4 \, \log \left (3 \, x\right ) + 1\right )} x^{4} - \frac {56}{5} \, x^{5} - \frac {687}{32} \, x^{4} \log \left (3 \, x\right ) - \frac {23}{18} \, {\left (9 \, \log \left (3 \, x\right )^{2} - 6 \, \log \left (3 \, x\right ) + 2\right )} x^{3} + \frac {191}{128} \, x^{4} - \frac {32}{3} \, x^{3} \log \left (3 \, x\right ) - \frac {9}{4} \, {\left (2 \, \log \left (3 \, x\right )^{2} - 2 \, \log \left (3 \, x\right ) + 1\right )} x^{2} + \frac {181}{18} \, x^{3} + 5 \, x^{2} \log \left (3 \, x\right ) + 2 \, {\left (\log \left (3 \, x\right )^{2} - 2 \, \log \left (3 \, x\right ) + 2\right )} x + \frac {5}{4} \, x^{2} + 4 \, x \log \left (3 \, x\right ) + \log \left (3 \, x\right )^{2} - 6 \, x - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)*log(3*x)^2+(81*x^9+585*x^8+1542*x^7+
1650*x^6+245*x^5-687*x^4-256*x^3+80*x^2+32*x+16)*log(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124
*x^4+156*x^3+60*x^2-16*x-16)/x,x, algorithm="maxima")

[Out]

9/16*x^10 + 9/8*x^9*log(3*x) + 9/512*(32*log(3*x)^2 - 8*log(3*x) + 1)*x^8 + 9/2*x^9 + 585/64*x^8*log(3*x) + 9/
98*(49*log(3*x)^2 - 14*log(3*x) + 2)*x^7 + 6711/512*x^8 + 771/28*x^7*log(3*x) + 35/48*(18*log(3*x)^2 - 6*log(3
*x) + 1)*x^6 + 726/49*x^7 + 275/8*x^6*log(3*x) + 3/5*(25*log(3*x)^2 - 10*log(3*x) + 2)*x^5 - 2/3*x^6 + 49/8*x^
5*log(3*x) + 1/128*(8*log(3*x)^2 - 4*log(3*x) + 1)*x^4 - 56/5*x^5 - 687/32*x^4*log(3*x) - 23/18*(9*log(3*x)^2
- 6*log(3*x) + 2)*x^3 + 191/128*x^4 - 32/3*x^3*log(3*x) - 9/4*(2*log(3*x)^2 - 2*log(3*x) + 1)*x^2 + 181/18*x^3
+ 5*x^2*log(3*x) + 2*(log(3*x)^2 - 2*log(3*x) + 2)*x + 5/4*x^2 + 4*x*log(3*x) + log(3*x)^2 - 6*x - 2*log(x)

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mupad [B]  time = 1.61, size = 217, normalized size = 7.48 \begin {gather*} 2\,x\,{\ln \left (3\,x\right )}^2-2\,\ln \relax (x)-2\,x+\frac {19\,x^2\,\ln \left (3\,x\right )}{2}-3\,x^3\,\ln \left (3\,x\right )-\frac {43\,x^4\,\ln \left (3\,x\right )}{2}+\frac {x^5\,\ln \left (3\,x\right )}{8}+30\,x^6\,\ln \left (3\,x\right )+\frac {105\,x^7\,\ln \left (3\,x\right )}{4}+9\,x^8\,\ln \left (3\,x\right )+\frac {9\,x^9\,\ln \left (3\,x\right )}{8}+{\ln \left (3\,x\right )}^2-x^2+\frac {15\,x^3}{2}+\frac {3\,x^4}{2}-10\,x^5+\frac {x^6}{16}+15\,x^7+\frac {105\,x^8}{8}+\frac {9\,x^9}{2}+\frac {9\,x^{10}}{16}-\frac {9\,x^2\,{\ln \left (3\,x\right )}^2}{2}-\frac {23\,x^3\,{\ln \left (3\,x\right )}^2}{2}+\frac {x^4\,{\ln \left (3\,x\right )}^2}{16}+15\,x^5\,{\ln \left (3\,x\right )}^2+\frac {105\,x^6\,{\ln \left (3\,x\right )}^2}{8}+\frac {9\,x^7\,{\ln \left (3\,x\right )}^2}{2}+\frac {9\,x^8\,{\ln \left (3\,x\right )}^2}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(3*x)*(32*x + 80*x^2 - 256*x^3 - 687*x^4 + 245*x^5 + 1650*x^6 + 1542*x^7 + 585*x^8 + 81*x^9 + 16))/8
- 2*x + (log(3*x)^2*(16*x - 72*x^2 - 276*x^3 + 2*x^4 + 600*x^5 + 630*x^6 + 252*x^7 + 36*x^8))/8 + (15*x^2)/2 +
(39*x^3)/2 - (31*x^4)/2 - (399*x^5)/8 + (243*x^6)/8 + (525*x^7)/4 + 114*x^8 + (333*x^9)/8 + (45*x^10)/8 - 2)/
x,x)

[Out]

2*x*log(3*x)^2 - 2*log(x) - 2*x + (19*x^2*log(3*x))/2 - 3*x^3*log(3*x) - (43*x^4*log(3*x))/2 + (x^5*log(3*x))/
8 + 30*x^6*log(3*x) + (105*x^7*log(3*x))/4 + 9*x^8*log(3*x) + (9*x^9*log(3*x))/8 + log(3*x)^2 - x^2 + (15*x^3)
/2 + (3*x^4)/2 - 10*x^5 + x^6/16 + 15*x^7 + (105*x^8)/8 + (9*x^9)/2 + (9*x^10)/16 - (9*x^2*log(3*x)^2)/2 - (23
*x^3*log(3*x)^2)/2 + (x^4*log(3*x)^2)/16 + 15*x^5*log(3*x)^2 + (105*x^6*log(3*x)^2)/8 + (9*x^7*log(3*x)^2)/2 +
(9*x^8*log(3*x)^2)/16

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sympy [B]  time = 0.32, size = 168, normalized size = 5.79 \begin {gather*} \frac {9 x^{10}}{16} + \frac {9 x^{9}}{2} + \frac {105 x^{8}}{8} + 15 x^{7} + \frac {x^{6}}{16} - 10 x^{5} + \frac {3 x^{4}}{2} + \frac {15 x^{3}}{2} - x^{2} - 2 x + \left (\frac {9 x^{9}}{8} + 9 x^{8} + \frac {105 x^{7}}{4} + 30 x^{6} + \frac {x^{5}}{8} - \frac {43 x^{4}}{2} - 3 x^{3} + \frac {19 x^{2}}{2}\right ) \log {\left (3 x \right )} + \left (\frac {9 x^{8}}{16} + \frac {9 x^{7}}{2} + \frac {105 x^{6}}{8} + 15 x^{5} + \frac {x^{4}}{16} - \frac {23 x^{3}}{2} - \frac {9 x^{2}}{2} + 2 x + 1\right ) \log {\left (3 x \right )}^{2} - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((36*x**8+252*x**7+630*x**6+600*x**5+2*x**4-276*x**3-72*x**2+16*x)*ln(3*x)**2+(81*x**9+585*x**8+
1542*x**7+1650*x**6+245*x**5-687*x**4-256*x**3+80*x**2+32*x+16)*ln(3*x)+45*x**10+333*x**9+912*x**8+1050*x**7+2
43*x**6-399*x**5-124*x**4+156*x**3+60*x**2-16*x-16)/x,x)

[Out]

9*x**10/16 + 9*x**9/2 + 105*x**8/8 + 15*x**7 + x**6/16 - 10*x**5 + 3*x**4/2 + 15*x**3/2 - x**2 - 2*x + (9*x**9
/8 + 9*x**8 + 105*x**7/4 + 30*x**6 + x**5/8 - 43*x**4/2 - 3*x**3 + 19*x**2/2)*log(3*x) + (9*x**8/16 + 9*x**7/2
+ 105*x**6/8 + 15*x**5 + x**4/16 - 23*x**3/2 - 9*x**2/2 + 2*x + 1)*log(3*x)**2 - 2*log(x)

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