3.27.47 \(\int \frac {1}{4} (2500 x^3-1250 x^4+150 x^5+(-500 x^3+125 x^4) \log (4)+25 x^3 \log ^2(4)+(15000 x^2-8000 x^3+1000 x^4+(-3000 x^2+800 x^3) \log (4)+150 x^2 \log ^2(4)) \log ^2(5)+(20000 x-12000 x^2+1600 x^3+(-4000 x+1200 x^2) \log (4)+200 x \log ^2(4)) \log ^4(5)) \, dx\)

Optimal. Leaf size=28 \[ \frac {25}{16} x^2 (10-2 x-\log (4))^2 \left (x+4 \log ^2(5)\right )^2 \]

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Rubi [B]  time = 0.08, antiderivative size = 158, normalized size of antiderivative = 5.64, number of steps used = 10, number of rules used = 2, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6, 12} \begin {gather*} \frac {25 x^6}{4}-\frac {125 x^5}{2}+50 x^5 \log ^2(5)+\frac {25}{4} x^5 \log (4)+100 x^4 \log ^4(5)+50 x^4 \log (4) \log ^2(5)-500 x^4 \log ^2(5)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )-\frac {125}{4} x^4 \log (4)+100 x^3 \log (4) \log ^4(5)-1000 x^3 \log ^4(5)+\frac {25}{2} x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-250 x^3 \log (4) \log ^2(5)-500 x^2 \log (4) \log ^4(5)+25 x^2 \left (100+\log ^2(4)\right ) \log ^4(5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2500*x^3 - 1250*x^4 + 150*x^5 + (-500*x^3 + 125*x^4)*Log[4] + 25*x^3*Log[4]^2 + (15000*x^2 - 8000*x^3 + 1
000*x^4 + (-3000*x^2 + 800*x^3)*Log[4] + 150*x^2*Log[4]^2)*Log[5]^2 + (20000*x - 12000*x^2 + 1600*x^3 + (-4000
*x + 1200*x^2)*Log[4] + 200*x*Log[4]^2)*Log[5]^4)/4,x]

[Out]

(-125*x^5)/2 + (25*x^6)/4 - (125*x^4*Log[4])/4 + (25*x^5*Log[4])/4 + (25*x^4*(100 + Log[4]^2))/16 - 500*x^4*Lo
g[5]^2 + 50*x^5*Log[5]^2 - 250*x^3*Log[4]*Log[5]^2 + 50*x^4*Log[4]*Log[5]^2 + (25*x^3*(100 + Log[4]^2)*Log[5]^
2)/2 - 1000*x^3*Log[5]^4 + 100*x^4*Log[5]^4 - 500*x^2*Log[4]*Log[5]^4 + 100*x^3*Log[4]*Log[5]^4 + 25*x^2*(100
+ Log[4]^2)*Log[5]^4

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{4} \left (-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+x^3 \left (2500+25 \log ^2(4)\right )+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx\\ &=\frac {1}{4} \int \left (-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+x^3 \left (2500+25 \log ^2(4)\right )+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx\\ &=-\frac {125 x^5}{2}+\frac {25 x^6}{4}+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )+\frac {1}{4} \log (4) \int \left (-500 x^3+125 x^4\right ) \, dx+\frac {1}{4} \log ^2(5) \int \left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \, dx+\frac {1}{4} \log ^4(5) \int \left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \, dx\\ &=-\frac {125 x^5}{2}+\frac {25 x^6}{4}-\frac {125}{4} x^4 \log (4)+\frac {25}{4} x^5 \log (4)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )+\frac {1}{4} \log ^2(5) \int \left (-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+x^2 \left (15000+150 \log ^2(4)\right )\right ) \, dx+\frac {1}{4} \log ^4(5) \int \left (-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+x \left (20000+200 \log ^2(4)\right )\right ) \, dx\\ &=-\frac {125 x^5}{2}+\frac {25 x^6}{4}-\frac {125}{4} x^4 \log (4)+\frac {25}{4} x^5 \log (4)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )-500 x^4 \log ^2(5)+50 x^5 \log ^2(5)+\frac {25}{2} x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-1000 x^3 \log ^4(5)+100 x^4 \log ^4(5)+25 x^2 \left (100+\log ^2(4)\right ) \log ^4(5)+\frac {1}{4} \left (\log (4) \log ^2(5)\right ) \int \left (-3000 x^2+800 x^3\right ) \, dx+\frac {1}{4} \left (\log (4) \log ^4(5)\right ) \int \left (-4000 x+1200 x^2\right ) \, dx\\ &=-\frac {125 x^5}{2}+\frac {25 x^6}{4}-\frac {125}{4} x^4 \log (4)+\frac {25}{4} x^5 \log (4)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )-500 x^4 \log ^2(5)+50 x^5 \log ^2(5)-250 x^3 \log (4) \log ^2(5)+50 x^4 \log (4) \log ^2(5)+\frac {25}{2} x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-1000 x^3 \log ^4(5)+100 x^4 \log ^4(5)-500 x^2 \log (4) \log ^4(5)+100 x^3 \log (4) \log ^4(5)+25 x^2 \left (100+\log ^2(4)\right ) \log ^4(5)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 26, normalized size = 0.93 \begin {gather*} \frac {25}{16} x^2 (-10+2 x+\log (4))^2 \left (x+4 \log ^2(5)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2500*x^3 - 1250*x^4 + 150*x^5 + (-500*x^3 + 125*x^4)*Log[4] + 25*x^3*Log[4]^2 + (15000*x^2 - 8000*x
^3 + 1000*x^4 + (-3000*x^2 + 800*x^3)*Log[4] + 150*x^2*Log[4]^2)*Log[5]^2 + (20000*x - 12000*x^2 + 1600*x^3 +
(-4000*x + 1200*x^2)*Log[4] + 200*x*Log[4]^2)*Log[5]^4)/4,x]

[Out]

(25*x^2*(-10 + 2*x + Log[4])^2*(x + 4*Log[5]^2)^2)/16

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fricas [B]  time = 0.65, size = 120, normalized size = 4.29 \begin {gather*} \frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \relax (2)^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \relax (2)^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (2)\right )} \log \relax (5)^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \relax (2)^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \relax (2)\right )} \log \relax (5)^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^2+20000*x)*log(5)^4+1/4*(600*x^2*log
(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000*x^4-8000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*
log(2)+75/2*x^5-625/2*x^4+625*x^3,x, algorithm="fricas")

[Out]

25/4*x^6 + 25/4*x^4*log(2)^2 - 125/2*x^5 + 100*(x^4 + x^2*log(2)^2 - 10*x^3 + 25*x^2 + 2*(x^3 - 5*x^2)*log(2))
*log(5)^4 + 625/4*x^4 + 50*(x^5 + x^3*log(2)^2 - 10*x^4 + 25*x^3 + 2*(x^4 - 5*x^3)*log(2))*log(5)^2 + 25/2*(x^
5 - 5*x^4)*log(2)

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giac [B]  time = 0.14, size = 120, normalized size = 4.29 \begin {gather*} \frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \relax (2)^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \relax (2)^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (2)\right )} \log \relax (5)^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \relax (2)^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \relax (2)\right )} \log \relax (5)^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^2+20000*x)*log(5)^4+1/4*(600*x^2*log
(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000*x^4-8000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*
log(2)+75/2*x^5-625/2*x^4+625*x^3,x, algorithm="giac")

[Out]

25/4*x^6 + 25/4*x^4*log(2)^2 - 125/2*x^5 + 100*(x^4 + x^2*log(2)^2 - 10*x^3 + 25*x^2 + 2*(x^3 - 5*x^2)*log(2))
*log(5)^4 + 625/4*x^4 + 50*(x^5 + x^3*log(2)^2 - 10*x^4 + 25*x^3 + 2*(x^4 - 5*x^3)*log(2))*log(5)^2 + 25/2*(x^
5 - 5*x^4)*log(2)

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maple [B]  time = 0.04, size = 93, normalized size = 3.32

method	result	size
gosper	\(\frac {25 \left (4 \ln \relax (5)^{2}+x \right ) \left (4 \ln \relax (2)^{2} \ln \relax (5)^{2}+8 x \ln \relax (2) \ln \relax (5)^{2}+4 x^{2} \ln \relax (5)^{2}-40 \ln \relax (2) \ln \relax (5)^{2}-40 x \ln \relax (5)^{2}+x \ln \relax (2)^{2}+2 x^{2} \ln \relax (2)+x^{3}+100 \ln \relax (5)^{2}-10 x \ln \relax (2)-10 x^{2}+25 x \right ) x^{2}}{4}\)	\(93\)
norman	\(\left (50 \ln \relax (5)^{2}+\frac {25 \ln \relax (2)}{2}-\frac {125}{2}\right ) x^{5}+\left (100 \ln \relax (5)^{4} \ln \relax (2)^{2}-1000 \ln \relax (5)^{4} \ln \relax (2)+2500 \ln \relax (5)^{4}\right ) x^{2}+\left (200 \ln \relax (5)^{4} \ln \relax (2)-1000 \ln \relax (5)^{4}+50 \ln \relax (2)^{2} \ln \relax (5)^{2}-500 \ln \relax (2) \ln \relax (5)^{2}+1250 \ln \relax (5)^{2}\right ) x^{3}+\left (100 \ln \relax (5)^{4}+100 \ln \relax (2) \ln \relax (5)^{2}-500 \ln \relax (5)^{2}+\frac {25 \ln \relax (2)^{2}}{4}-\frac {125 \ln \relax (2)}{2}+\frac {625}{4}\right ) x^{4}+\frac {25 x^{6}}{4}\)	\(131\)
default	\(\frac {\ln \relax (5)^{4} \left (400 x^{2} \ln \relax (2)^{2}+2 \ln \relax (2) \left (400 x^{3}-2000 x^{2}\right )+400 x^{4}-4000 x^{3}+10000 x^{2}\right )}{4}+\frac {\ln \relax (5)^{2} \left (200 x^{3} \ln \relax (2)^{2}+2 \ln \relax (2) \left (200 x^{4}-1000 x^{3}\right )+200 x^{5}-2000 x^{4}+5000 x^{3}\right )}{4}+\frac {25 x^{4} \ln \relax (2)^{2}}{4}+\frac {\ln \relax (2) \left (25 x^{5}-125 x^{4}\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\)	\(133\)
risch	\(100 x^{4} \ln \relax (5)^{4}+200 \ln \relax (5)^{4} x^{3} \ln \relax (2)-1000 x^{3} \ln \relax (5)^{4}+100 \ln \relax (5)^{4} \ln \relax (2)^{2} x^{2}-1000 \ln \relax (5)^{4} \ln \relax (2) x^{2}+2500 \ln \relax (5)^{4} x^{2}+50 x^{5} \ln \relax (5)^{2}+100 \ln \relax (5)^{2} x^{4} \ln \relax (2)-500 x^{4} \ln \relax (5)^{2}+50 \ln \relax (5)^{2} \ln \relax (2)^{2} x^{3}-500 \ln \relax (5)^{2} \ln \relax (2) x^{3}+1250 x^{3} \ln \relax (5)^{2}+\frac {25 x^{4} \ln \relax (2)^{2}}{4}+\frac {25 x^{5} \ln \relax (2)}{2}-\frac {125 x^{4} \ln \relax (2)}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(800*x*ln(2)^2+2*(1200*x^2-4000*x)*ln(2)+1600*x^3-12000*x^2+20000*x)*ln(5)^4+1/4*(600*x^2*ln(2)^2+2*(8
00*x^3-3000*x^2)*ln(2)+1000*x^4-8000*x^3+15000*x^2)*ln(5)^2+25*x^3*ln(2)^2+1/2*(125*x^4-500*x^3)*ln(2)+75/2*x^
5-625/2*x^4+625*x^3,x,method=_RETURNVERBOSE)

[Out]

25/4*(4*ln(5)^2+x)*(4*ln(2)^2*ln(5)^2+8*x*ln(2)*ln(5)^2+4*x^2*ln(5)^2-40*ln(2)*ln(5)^2-40*x*ln(5)^2+x*ln(2)^2+
2*x^2*ln(2)+x^3+100*ln(5)^2-10*x*ln(2)-10*x^2+25*x)*x^2

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maxima [B]  time = 0.68, size = 120, normalized size = 4.29 \begin {gather*} \frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \relax (2)^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \relax (2)^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \relax (2)\right )} \log \relax (5)^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \relax (2)^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \relax (2)\right )} \log \relax (5)^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^2+20000*x)*log(5)^4+1/4*(600*x^2*log
(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000*x^4-8000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*
log(2)+75/2*x^5-625/2*x^4+625*x^3,x, algorithm="maxima")

[Out]

25/4*x^6 + 25/4*x^4*log(2)^2 - 125/2*x^5 + 100*(x^4 + x^2*log(2)^2 - 10*x^3 + 25*x^2 + 2*(x^3 - 5*x^2)*log(2))
*log(5)^4 + 625/4*x^4 + 50*(x^5 + x^3*log(2)^2 - 10*x^4 + 25*x^3 + 2*(x^4 - 5*x^3)*log(2))*log(5)^2 + 25/2*(x^
5 - 5*x^4)*log(2)

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mupad [B]  time = 1.50, size = 96, normalized size = 3.43 \begin {gather*} \frac {25\,x^6}{4}+\left (\frac {25\,\ln \relax (2)}{2}+50\,{\ln \relax (5)}^2-\frac {125}{2}\right )\,x^5+\left (100\,\ln \relax (2)\,{\ln \relax (5)}^2-\frac {125\,\ln \relax (2)}{2}+\frac {25\,{\ln \relax (2)}^2}{4}-500\,{\ln \relax (5)}^2+100\,{\ln \relax (5)}^4+\frac {625}{4}\right )\,x^4+50\,{\ln \relax (5)}^2\,\left (\ln \relax (2)-5\right )\,\left (\ln \relax (2)+4\,{\ln \relax (5)}^2-5\right )\,x^3+100\,{\ln \relax (5)}^4\,{\left (\ln \relax (2)-5\right )}^2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(25*x^3*log(2)^2 - (log(2)*(500*x^3 - 125*x^4))/2 + (log(5)^4*(20000*x - 2*log(2)*(4000*x - 1200*x^2) + 800
*x*log(2)^2 - 12000*x^2 + 1600*x^3))/4 + (log(5)^2*(600*x^2*log(2)^2 - 2*log(2)*(3000*x^2 - 800*x^3) + 15000*x
^2 - 8000*x^3 + 1000*x^4))/4 + 625*x^3 - (625*x^4)/2 + (75*x^5)/2,x)

[Out]

x^4*(100*log(2)*log(5)^2 - (125*log(2))/2 + (25*log(2)^2)/4 - 500*log(5)^2 + 100*log(5)^4 + 625/4) + x^5*((25*
log(2))/2 + 50*log(5)^2 - 125/2) + (25*x^6)/4 + 100*x^2*log(5)^4*(log(2) - 5)^2 + 50*x^3*log(5)^2*(log(2) - 5)
*(log(2) + 4*log(5)^2 - 5)

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sympy [B]  time = 0.10, size = 153, normalized size = 5.46 \begin {gather*} \frac {25 x^{6}}{4} + x^{5} \left (- \frac {125}{2} + \frac {25 \log {\relax (2 )}}{2} + 50 \log {\relax (5 )}^{2}\right ) + x^{4} \left (- 500 \log {\relax (5 )}^{2} - \frac {125 \log {\relax (2 )}}{2} + \frac {25 \log {\relax (2 )}^{2}}{4} + \frac {625}{4} + 100 \log {\relax (2 )} \log {\relax (5 )}^{2} + 100 \log {\relax (5 )}^{4}\right ) + x^{3} \left (- 1000 \log {\relax (5 )}^{4} - 500 \log {\relax (2 )} \log {\relax (5 )}^{2} + 50 \log {\relax (2 )}^{2} \log {\relax (5 )}^{2} + 200 \log {\relax (2 )} \log {\relax (5 )}^{4} + 1250 \log {\relax (5 )}^{2}\right ) + x^{2} \left (- 1000 \log {\relax (2 )} \log {\relax (5 )}^{4} + 100 \log {\relax (2 )}^{2} \log {\relax (5 )}^{4} + 2500 \log {\relax (5 )}^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(800*x*ln(2)**2+2*(1200*x**2-4000*x)*ln(2)+1600*x**3-12000*x**2+20000*x)*ln(5)**4+1/4*(600*x**2*
ln(2)**2+2*(800*x**3-3000*x**2)*ln(2)+1000*x**4-8000*x**3+15000*x**2)*ln(5)**2+25*x**3*ln(2)**2+1/2*(125*x**4-
500*x**3)*ln(2)+75/2*x**5-625/2*x**4+625*x**3,x)

[Out]

25*x**6/4 + x**5*(-125/2 + 25*log(2)/2 + 50*log(5)**2) + x**4*(-500*log(5)**2 - 125*log(2)/2 + 25*log(2)**2/4
+ 625/4 + 100*log(2)*log(5)**2 + 100*log(5)**4) + x**3*(-1000*log(5)**4 - 500*log(2)*log(5)**2 + 50*log(2)**2*
log(5)**2 + 200*log(2)*log(5)**4 + 1250*log(5)**2) + x**2*(-1000*log(2)*log(5)**4 + 100*log(2)**2*log(5)**4 +
2500*log(5)**4)

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