3.78.63 \(\int (1250 x+4500 x^2+4200 x^3+900 x^4+54 x^5+(-300 x-990 x^2-792 x^3-90 x^4) \log (2)+(18 x+54 x^2+36 x^3) \log ^2(2)+(-300 x-540 x^2-72 x^3+(36 x+54 x^2) \log (2)) \log (4)+18 x \log ^2(4)) \, dx\)

Optimal. Leaf size=24 \[ (2 x+3 x (-((1+x) (9+x-\log (2)))+\log (4)))^2 \]

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Rubi [B]  time = 0.04, antiderivative size = 127, normalized size of antiderivative = 5.29, number of steps used = 6, number of rules used = 1, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {6} \begin {gather*} 9 x^6+180 x^5-18 x^5 \log (2)+1050 x^4+9 x^4 \log ^2(2)-18 x^4 \log (4)-198 x^4 \log (2)+1500 x^3+18 x^3 \log ^2(2)+18 x^3 \log (2) \log (4)-180 x^3 \log (4)-330 x^3 \log (2)+x^2 \left (625+9 \log ^2(4)\right )+9 x^2 \log ^2(2)+18 x^2 \log (2) \log (4)-150 x^2 \log (4)-150 x^2 \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1250*x + 4500*x^2 + 4200*x^3 + 900*x^4 + 54*x^5 + (-300*x - 990*x^2 - 792*x^3 - 90*x^4)*Log[2] + (18*x + 5
4*x^2 + 36*x^3)*Log[2]^2 + (-300*x - 540*x^2 - 72*x^3 + (36*x + 54*x^2)*Log[2])*Log[4] + 18*x*Log[4]^2,x]

[Out]

1500*x^3 + 1050*x^4 + 180*x^5 + 9*x^6 - 150*x^2*Log[2] - 330*x^3*Log[2] - 198*x^4*Log[2] - 18*x^5*Log[2] + 9*x
^2*Log[2]^2 + 18*x^3*Log[2]^2 + 9*x^4*Log[2]^2 - 150*x^2*Log[4] - 180*x^3*Log[4] - 18*x^4*Log[4] + 18*x^2*Log[
2]*Log[4] + 18*x^3*Log[2]*Log[4] + x^2*(625 + 9*Log[4]^2)

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4500 x^2+4200 x^3+900 x^4+54 x^5+\left (-300 x-990 x^2-792 x^3-90 x^4\right ) \log (2)+\left (18 x+54 x^2+36 x^3\right ) \log ^2(2)+\left (-300 x-540 x^2-72 x^3+\left (36 x+54 x^2\right ) \log (2)\right ) \log (4)+x \left (1250+18 \log ^2(4)\right )\right ) \, dx\\ &=1500 x^3+1050 x^4+180 x^5+9 x^6+x^2 \left (625+9 \log ^2(4)\right )+\log (2) \int \left (-300 x-990 x^2-792 x^3-90 x^4\right ) \, dx+\log ^2(2) \int \left (18 x+54 x^2+36 x^3\right ) \, dx+\log (4) \int \left (-300 x-540 x^2-72 x^3+\left (36 x+54 x^2\right ) \log (2)\right ) \, dx\\ &=1500 x^3+1050 x^4+180 x^5+9 x^6-150 x^2 \log (2)-330 x^3 \log (2)-198 x^4 \log (2)-18 x^5 \log (2)+9 x^2 \log ^2(2)+18 x^3 \log ^2(2)+9 x^4 \log ^2(2)-150 x^2 \log (4)-180 x^3 \log (4)-18 x^4 \log (4)+x^2 \left (625+9 \log ^2(4)\right )+(\log (2) \log (4)) \int \left (36 x+54 x^2\right ) \, dx\\ &=1500 x^3+1050 x^4+180 x^5+9 x^6-150 x^2 \log (2)-330 x^3 \log (2)-198 x^4 \log (2)-18 x^5 \log (2)+9 x^2 \log ^2(2)+18 x^3 \log ^2(2)+9 x^4 \log ^2(2)-150 x^2 \log (4)-180 x^3 \log (4)-18 x^4 \log (4)+18 x^2 \log (2) \log (4)+18 x^3 \log (2) \log (4)+x^2 \left (625+9 \log ^2(4)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 57, normalized size = 2.38 \begin {gather*} x^2 \left (9 x^4-18 x^3 (-10+\log (2))+3 x^2 \left (350-66 \log (2)+3 \log ^2(2)-6 \log (4)\right )+6 x (-10+\log (2)) (-25+\log (512))+(-25+\log (512))^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1250*x + 4500*x^2 + 4200*x^3 + 900*x^4 + 54*x^5 + (-300*x - 990*x^2 - 792*x^3 - 90*x^4)*Log[2] + (18
*x + 54*x^2 + 36*x^3)*Log[2]^2 + (-300*x - 540*x^2 - 72*x^3 + (36*x + 54*x^2)*Log[2])*Log[4] + 18*x*Log[4]^2,x
]

[Out]

x^2*(9*x^4 - 18*x^3*(-10 + Log[2]) + 3*x^2*(350 - 66*Log[2] + 3*Log[2]^2 - 6*Log[4]) + 6*x*(-10 + Log[2])*(-25
+ Log[512]) + (-25 + Log[512])^2)

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fricas [B]  time = 0.61, size = 71, normalized size = 2.96 \begin {gather*} 9 \, x^{6} + 180 \, x^{5} + 1050 \, x^{4} + 1500 \, x^{3} + 9 \, {\left (x^{4} + 6 \, x^{3} + 9 \, x^{2}\right )} \log \relax (2)^{2} + 625 \, x^{2} - 6 \, {\left (3 \, x^{5} + 39 \, x^{4} + 115 \, x^{3} + 75 \, x^{2}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(72*x*log(2)^2+2*((54*x^2+36*x)*log(2)-72*x^3-540*x^2-300*x)*log(2)+(36*x^3+54*x^2+18*x)*log(2)^2+(-9
0*x^4-792*x^3-990*x^2-300*x)*log(2)+54*x^5+900*x^4+4200*x^3+4500*x^2+1250*x,x, algorithm="fricas")

[Out]

9*x^6 + 180*x^5 + 1050*x^4 + 1500*x^3 + 9*(x^4 + 6*x^3 + 9*x^2)*log(2)^2 + 625*x^2 - 6*(3*x^5 + 39*x^4 + 115*x
^3 + 75*x^2)*log(2)

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giac [B]  time = 0.13, size = 109, normalized size = 4.54 \begin {gather*} 9 \, x^{6} + 180 \, x^{5} + 1050 \, x^{4} + 36 \, x^{2} \log \relax (2)^{2} + 1500 \, x^{3} + 9 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + 625 \, x^{2} - 6 \, {\left (3 \, x^{5} + 33 \, x^{4} + 55 \, x^{3} + 25 \, x^{2}\right )} \log \relax (2) - 12 \, {\left (3 \, x^{4} + 30 \, x^{3} + 25 \, x^{2} - 3 \, {\left (x^{3} + x^{2}\right )} \log \relax (2)\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(72*x*log(2)^2+2*((54*x^2+36*x)*log(2)-72*x^3-540*x^2-300*x)*log(2)+(36*x^3+54*x^2+18*x)*log(2)^2+(-9
0*x^4-792*x^3-990*x^2-300*x)*log(2)+54*x^5+900*x^4+4200*x^3+4500*x^2+1250*x,x, algorithm="giac")

[Out]

9*x^6 + 180*x^5 + 1050*x^4 + 36*x^2*log(2)^2 + 1500*x^3 + 9*(x^4 + 2*x^3 + x^2)*log(2)^2 + 625*x^2 - 6*(3*x^5
+ 33*x^4 + 55*x^3 + 25*x^2)*log(2) - 12*(3*x^4 + 30*x^3 + 25*x^2 - 3*(x^3 + x^2)*log(2))*log(2)

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maple [A]  time = 0.04, size = 26, normalized size = 1.08

method	result	size
gosper	\(x^{2} \left (3 x \ln \relax (2)-3 x^{2}+9 \ln \relax (2)-30 x -25\right )^{2}\)	\(26\)
norman	\(\left (-18 \ln \relax (2)+180\right ) x^{5}+\left (9 \ln \relax (2)^{2}-234 \ln \relax (2)+1050\right ) x^{4}+\left (54 \ln \relax (2)^{2}-690 \ln \relax (2)+1500\right ) x^{3}+\left (81 \ln \relax (2)^{2}-450 \ln \relax (2)+625\right ) x^{2}+9 x^{6}\)	\(65\)
risch	\(9 x^{4} \ln \relax (2)^{2}-18 x^{5} \ln \relax (2)+9 x^{6}+54 x^{3} \ln \relax (2)^{2}-234 x^{4} \ln \relax (2)+180 x^{5}+81 x^{2} \ln \relax (2)^{2}-690 x^{3} \ln \relax (2)+1050 x^{4}-450 x^{2} \ln \relax (2)+1500 x^{3}+625 x^{2}\)	\(82\)
default	\(36 x^{2} \ln \relax (2)^{2}+2 \ln \relax (2) \left (\ln \relax (2) \left (18 x^{3}+18 x^{2}\right )-18 x^{4}-180 x^{3}-150 x^{2}\right )+\ln \relax (2)^{2} \left (9 x^{4}+18 x^{3}+9 x^{2}\right )+\ln \relax (2) \left (-18 x^{5}-198 x^{4}-330 x^{3}-150 x^{2}\right )+9 x^{6}+180 x^{5}+1050 x^{4}+1500 x^{3}+625 x^{2}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(72*x*ln(2)^2+2*((54*x^2+36*x)*ln(2)-72*x^3-540*x^2-300*x)*ln(2)+(36*x^3+54*x^2+18*x)*ln(2)^2+(-90*x^4-792*
x^3-990*x^2-300*x)*ln(2)+54*x^5+900*x^4+4200*x^3+4500*x^2+1250*x,x,method=_RETURNVERBOSE)

[Out]

x^2*(3*x*ln(2)-3*x^2+9*ln(2)-30*x-25)^2

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maxima [B]  time = 0.48, size = 109, normalized size = 4.54 \begin {gather*} 9 \, x^{6} + 180 \, x^{5} + 1050 \, x^{4} + 36 \, x^{2} \log \relax (2)^{2} + 1500 \, x^{3} + 9 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + 625 \, x^{2} - 6 \, {\left (3 \, x^{5} + 33 \, x^{4} + 55 \, x^{3} + 25 \, x^{2}\right )} \log \relax (2) - 12 \, {\left (3 \, x^{4} + 30 \, x^{3} + 25 \, x^{2} - 3 \, {\left (x^{3} + x^{2}\right )} \log \relax (2)\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(72*x*log(2)^2+2*((54*x^2+36*x)*log(2)-72*x^3-540*x^2-300*x)*log(2)+(36*x^3+54*x^2+18*x)*log(2)^2+(-9
0*x^4-792*x^3-990*x^2-300*x)*log(2)+54*x^5+900*x^4+4200*x^3+4500*x^2+1250*x,x, algorithm="maxima")

[Out]

9*x^6 + 180*x^5 + 1050*x^4 + 36*x^2*log(2)^2 + 1500*x^3 + 9*(x^4 + 2*x^3 + x^2)*log(2)^2 + 625*x^2 - 6*(3*x^5
+ 33*x^4 + 55*x^3 + 25*x^2)*log(2) - 12*(3*x^4 + 30*x^3 + 25*x^2 - 3*(x^3 + x^2)*log(2))*log(2)

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mupad [B]  time = 5.34, size = 59, normalized size = 2.46 \begin {gather*} 9\,x^6+\left (180-18\,\ln \relax (2)\right )\,x^5+\left (9\,{\ln \relax (2)}^2-234\,\ln \relax (2)+1050\right )\,x^4+\left (54\,{\ln \relax (2)}^2-690\,\ln \relax (2)+1500\right )\,x^3+{\left (\ln \left (512\right )-25\right )}^2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1250*x + 72*x*log(2)^2 - log(2)*(300*x + 990*x^2 + 792*x^3 + 90*x^4) - 2*log(2)*(300*x - log(2)*(36*x + 54
*x^2) + 540*x^2 + 72*x^3) + log(2)^2*(18*x + 54*x^2 + 36*x^3) + 4500*x^2 + 4200*x^3 + 900*x^4 + 54*x^5,x)

[Out]

x^2*(log(512) - 25)^2 - x^5*(18*log(2) - 180) + x^4*(9*log(2)^2 - 234*log(2) + 1050) + x^3*(54*log(2)^2 - 690*
log(2) + 1500) + 9*x^6

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sympy [B]  time = 0.09, size = 65, normalized size = 2.71 \begin {gather*} 9 x^{6} + x^{5} \left (180 - 18 \log {\relax (2 )}\right ) + x^{4} \left (- 234 \log {\relax (2 )} + 9 \log {\relax (2 )}^{2} + 1050\right ) + x^{3} \left (- 690 \log {\relax (2 )} + 54 \log {\relax (2 )}^{2} + 1500\right ) + x^{2} \left (- 450 \log {\relax (2 )} + 81 \log {\relax (2 )}^{2} + 625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(72*x*ln(2)**2+2*((54*x**2+36*x)*ln(2)-72*x**3-540*x**2-300*x)*ln(2)+(36*x**3+54*x**2+18*x)*ln(2)**2+
(-90*x**4-792*x**3-990*x**2-300*x)*ln(2)+54*x**5+900*x**4+4200*x**3+4500*x**2+1250*x,x)

[Out]

9*x**6 + x**5*(180 - 18*log(2)) + x**4*(-234*log(2) + 9*log(2)**2 + 1050) + x**3*(-690*log(2) + 54*log(2)**2 +
1500) + x**2*(-450*log(2) + 81*log(2)**2 + 625)

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