3.83.64 \(\int \frac {1}{9} (-9+60 x^2-414 x^3+(200 x-2820 x^2+9450 x^3) \log (2 x)+(200 x-4500 x^2+22500 x^3) \log ^2(2 x)) \, dx\)

Optimal. Leaf size=24 \[ -x+x^2 \left (x+\left (\frac {10}{3}-25 x\right ) \log (2 x)\right )^2 \]

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Rubi [B]  time = 0.15, antiderivative size = 64, normalized size of antiderivative = 2.67, number of steps used = 17, number of rules used = 5, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 1594, 2356, 2304, 2305} \begin {gather*} x^4+625 x^4 \log ^2(2 x)-50 x^4 \log (2 x)-\frac {500}{3} x^3 \log ^2(2 x)+\frac {20}{3} x^3 \log (2 x)+\frac {100}{9} x^2 \log ^2(2 x)-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 + 60*x^2 - 414*x^3 + (200*x - 2820*x^2 + 9450*x^3)*Log[2*x] + (200*x - 4500*x^2 + 22500*x^3)*Log[2*x]^
2)/9,x]

[Out]

-x + x^4 + (20*x^3*Log[2*x])/3 - 50*x^4*Log[2*x] + (100*x^2*Log[2*x]^2)/9 - (500*x^3*Log[2*x]^2)/3 + 625*x^4*L
og[2*x]^2

Rule 12

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

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \left (-9+60 x^2-414 x^3+\left (200 x-2820 x^2+9450 x^3\right ) \log (2 x)+\left (200 x-4500 x^2+22500 x^3\right ) \log ^2(2 x)\right ) \, dx\\ &=-x+\frac {20 x^3}{9}-\frac {23 x^4}{2}+\frac {1}{9} \int \left (200 x-2820 x^2+9450 x^3\right ) \log (2 x) \, dx+\frac {1}{9} \int \left (200 x-4500 x^2+22500 x^3\right ) \log ^2(2 x) \, dx\\ &=-x+\frac {20 x^3}{9}-\frac {23 x^4}{2}+\frac {1}{9} \int x \left (200-2820 x+9450 x^2\right ) \log (2 x) \, dx+\frac {1}{9} \int x \left (200-4500 x+22500 x^2\right ) \log ^2(2 x) \, dx\\ &=-x+\frac {20 x^3}{9}-\frac {23 x^4}{2}+\frac {1}{9} \int \left (200 x \log (2 x)-2820 x^2 \log (2 x)+9450 x^3 \log (2 x)\right ) \, dx+\frac {1}{9} \int \left (200 x \log ^2(2 x)-4500 x^2 \log ^2(2 x)+22500 x^3 \log ^2(2 x)\right ) \, dx\\ &=-x+\frac {20 x^3}{9}-\frac {23 x^4}{2}+\frac {200}{9} \int x \log (2 x) \, dx+\frac {200}{9} \int x \log ^2(2 x) \, dx-\frac {940}{3} \int x^2 \log (2 x) \, dx-500 \int x^2 \log ^2(2 x) \, dx+1050 \int x^3 \log (2 x) \, dx+2500 \int x^3 \log ^2(2 x) \, dx\\ &=-x-\frac {50 x^2}{9}+\frac {1000 x^3}{27}-\frac {617 x^4}{8}+\frac {100}{9} x^2 \log (2 x)-\frac {940}{9} x^3 \log (2 x)+\frac {525}{2} x^4 \log (2 x)+\frac {100}{9} x^2 \log ^2(2 x)-\frac {500}{3} x^3 \log ^2(2 x)+625 x^4 \log ^2(2 x)-\frac {200}{9} \int x \log (2 x) \, dx+\frac {1000}{3} \int x^2 \log (2 x) \, dx-1250 \int x^3 \log (2 x) \, dx\\ &=-x+x^4+\frac {20}{3} x^3 \log (2 x)-50 x^4 \log (2 x)+\frac {100}{9} x^2 \log ^2(2 x)-\frac {500}{3} x^3 \log ^2(2 x)+625 x^4 \log ^2(2 x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 64, normalized size = 2.67 \begin {gather*} -x+x^4+\frac {20}{3} x^3 \log (2 x)-50 x^4 \log (2 x)+\frac {100}{9} x^2 \log ^2(2 x)-\frac {500}{3} x^3 \log ^2(2 x)+625 x^4 \log ^2(2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 + 60*x^2 - 414*x^3 + (200*x - 2820*x^2 + 9450*x^3)*Log[2*x] + (200*x - 4500*x^2 + 22500*x^3)*Log
[2*x]^2)/9,x]

[Out]

-x + x^4 + (20*x^3*Log[2*x])/3 - 50*x^4*Log[2*x] + (100*x^2*Log[2*x]^2)/9 - (500*x^3*Log[2*x]^2)/3 + 625*x^4*L
og[2*x]^2

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fricas [A]  time = 0.62, size = 48, normalized size = 2.00 \begin {gather*} x^{4} + \frac {25}{9} \, {\left (225 \, x^{4} - 60 \, x^{3} + 4 \, x^{2}\right )} \log \left (2 \, x\right )^{2} - \frac {10}{3} \, {\left (15 \, x^{4} - 2 \, x^{3}\right )} \log \left (2 \, x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(22500*x^3-4500*x^2+200*x)*log(2*x)^2+1/9*(9450*x^3-2820*x^2+200*x)*log(2*x)-46*x^3+20/3*x^2-1,x
, algorithm="fricas")

[Out]

x^4 + 25/9*(225*x^4 - 60*x^3 + 4*x^2)*log(2*x)^2 - 10/3*(15*x^4 - 2*x^3)*log(2*x) - x

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giac [B]  time = 0.13, size = 58, normalized size = 2.42 \begin {gather*} 625 \, x^{4} \log \left (2 \, x\right )^{2} - 50 \, x^{4} \log \left (2 \, x\right ) - \frac {500}{3} \, x^{3} \log \left (2 \, x\right )^{2} + x^{4} + \frac {20}{3} \, x^{3} \log \left (2 \, x\right ) + \frac {100}{9} \, x^{2} \log \left (2 \, x\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(22500*x^3-4500*x^2+200*x)*log(2*x)^2+1/9*(9450*x^3-2820*x^2+200*x)*log(2*x)-46*x^3+20/3*x^2-1,x
, algorithm="giac")

[Out]

625*x^4*log(2*x)^2 - 50*x^4*log(2*x) - 500/3*x^3*log(2*x)^2 + x^4 + 20/3*x^3*log(2*x) + 100/9*x^2*log(2*x)^2 -
x

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maple [B]  time = 0.04, size = 59, normalized size = 2.46

method	result	size
derivativedivides	\(-x +x^{4}-50 x^{4} \ln \left (2 x \right )+\frac {20 x^{3} \ln \left (2 x \right )}{3}+625 x^{4} \ln \left (2 x \right )^{2}-\frac {500 x^{3} \ln \left (2 x \right )^{2}}{3}+\frac {100 x^{2} \ln \left (2 x \right )^{2}}{9}\)	\(59\)
default	\(-x +x^{4}-50 x^{4} \ln \left (2 x \right )+\frac {20 x^{3} \ln \left (2 x \right )}{3}+625 x^{4} \ln \left (2 x \right )^{2}-\frac {500 x^{3} \ln \left (2 x \right )^{2}}{3}+\frac {100 x^{2} \ln \left (2 x \right )^{2}}{9}\)	\(59\)
norman	\(-x +x^{4}-50 x^{4} \ln \left (2 x \right )+\frac {20 x^{3} \ln \left (2 x \right )}{3}+625 x^{4} \ln \left (2 x \right )^{2}-\frac {500 x^{3} \ln \left (2 x \right )^{2}}{3}+\frac {100 x^{2} \ln \left (2 x \right )^{2}}{9}\)	\(59\)
risch	\(-x +x^{4}-50 x^{4} \ln \left (2 x \right )+\frac {20 x^{3} \ln \left (2 x \right )}{3}+625 x^{4} \ln \left (2 x \right )^{2}-\frac {500 x^{3} \ln \left (2 x \right )^{2}}{3}+\frac {100 x^{2} \ln \left (2 x \right )^{2}}{9}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(22500*x^3-4500*x^2+200*x)*ln(2*x)^2+1/9*(9450*x^3-2820*x^2+200*x)*ln(2*x)-46*x^3+20/3*x^2-1,x,method=
_RETURNVERBOSE)

[Out]

-x+x^4-50*x^4*ln(2*x)+20/3*x^3*ln(2*x)+625*x^4*ln(2*x)^2-500/3*x^3*ln(2*x)^2+100/9*x^2*ln(2*x)^2

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maxima [B]  time = 0.41, size = 104, normalized size = 4.33 \begin {gather*} \frac {625}{8} \, {\left (8 \, \log \left (2 \, x\right )^{2} - 4 \, \log \left (2 \, x\right ) + 1\right )} x^{4} - \frac {500}{27} \, {\left (9 \, \log \left (2 \, x\right )^{2} - 6 \, \log \left (2 \, x\right ) + 2\right )} x^{3} - \frac {617}{8} \, x^{4} + \frac {50}{9} \, {\left (2 \, \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 1\right )} x^{2} + \frac {1000}{27} \, x^{3} - \frac {50}{9} \, x^{2} + \frac {5}{18} \, {\left (945 \, x^{4} - 376 \, x^{3} + 40 \, x^{2}\right )} \log \left (2 \, x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(22500*x^3-4500*x^2+200*x)*log(2*x)^2+1/9*(9450*x^3-2820*x^2+200*x)*log(2*x)-46*x^3+20/3*x^2-1,x
, algorithm="maxima")

[Out]

625/8*(8*log(2*x)^2 - 4*log(2*x) + 1)*x^4 - 500/27*(9*log(2*x)^2 - 6*log(2*x) + 2)*x^3 - 617/8*x^4 + 50/9*(2*l
og(2*x)^2 - 2*log(2*x) + 1)*x^2 + 1000/27*x^3 - 50/9*x^2 + 5/18*(945*x^4 - 376*x^3 + 40*x^2)*log(2*x) - x

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mupad [B]  time = 5.00, size = 54, normalized size = 2.25 \begin {gather*} x^3\,\left (\frac {20\,\ln \left (2\,x\right )}{3}-\frac {500\,{\ln \left (2\,x\right )}^2}{3}\right )-x+x^4\,\left (625\,{\ln \left (2\,x\right )}^2-50\,\ln \left (2\,x\right )+1\right )+\frac {100\,x^2\,{\ln \left (2\,x\right )}^2}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x)*(200*x - 2820*x^2 + 9450*x^3))/9 + (log(2*x)^2*(200*x - 4500*x^2 + 22500*x^3))/9 + (20*x^2)/3 -
46*x^3 - 1,x)

[Out]

x^3*((20*log(2*x))/3 - (500*log(2*x)^2)/3) - x + x^4*(625*log(2*x)^2 - 50*log(2*x) + 1) + (100*x^2*log(2*x)^2)
/9

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sympy [B]  time = 0.21, size = 46, normalized size = 1.92 \begin {gather*} x^{4} - x + \left (- 50 x^{4} + \frac {20 x^{3}}{3}\right ) \log {\left (2 x \right )} + \left (625 x^{4} - \frac {500 x^{3}}{3} + \frac {100 x^{2}}{9}\right ) \log {\left (2 x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(22500*x**3-4500*x**2+200*x)*ln(2*x)**2+1/9*(9450*x**3-2820*x**2+200*x)*ln(2*x)-46*x**3+20/3*x**
2-1,x)

[Out]

x**4 - x + (-50*x**4 + 20*x**3/3)*log(2*x) + (625*x**4 - 500*x**3/3 + 100*x**2/9)*log(2*x)**2

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