Optimal. Leaf size=25 \[ \frac {x}{\left (4+\frac {1+4 x+\left (16 x^2+\log (5)\right )^2}{x}\right )^2} \]
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Rubi [F] time = 0.89, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3 x^2+8 x^3-1280 x^6-32 x^4 \log (5)+3 x^2 \log ^2(5)}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x^3-1280 x^6-32 x^4 \log (5)+x^2 \left (3+3 \log ^2(5)\right )}{1+24 x+192 x^2+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+96 x^2 \log ^5(5)+\log ^6(5)} \, dx\\ &=\int \frac {8 x^3-1280 x^6-32 x^4 \log (5)+x^2 \left (3+3 \log ^2(5)\right )}{1+24 x+512 x^3+768 x^4+12288 x^5+49152 x^6+196608 x^8+1572864 x^9+16777216 x^{12}+\left (96 x^2+1536 x^3+6144 x^4+49152 x^6+393216 x^7+6291456 x^{10}\right ) \log (5)+\left (3+48 x+192 x^2+4608 x^4+36864 x^5+983040 x^8\right ) \log ^2(5)+\left (192 x^2+1536 x^3+81920 x^6\right ) \log ^3(5)+\left (3+24 x+3840 x^4\right ) \log ^4(5)+\log ^6(5)+x^2 \left (192+96 \log ^5(5)\right )} \, dx\\ &=\int \left (\frac {-10 x^2+\log (5)}{2 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {96 x^3-8 x \log (5)+16 x^2 \left (1-\log ^2(5)\right )-\log (5) \left (1+\log ^2(5)\right )}{2 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-10 x^2+\log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \int \frac {96 x^3-8 x \log (5)+16 x^2 \left (1-\log ^2(5)\right )-\log (5) \left (1+\log ^2(5)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx\\ &=-\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\int \frac {-14336 x \log (5)+16384 x^2 \left (1-\log ^2(5)\right )-256 \left (3+4 \log ^3(5)+\log (625)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx}{2048}+\frac {1}{2} \int \left (-\frac {10 x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}\right ) \, dx\\ &=-\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}+\frac {\int \left (-\frac {14336 x \log (5)}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}-\frac {16384 x^2 \left (-1+\log ^2(5)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}-\frac {256 \left (3+4 \log ^3(5)+\log (625)\right )}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3}\right ) \, dx}{2048}-5 \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx\\ &=-\frac {3}{128 \left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2}-5 \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \, dx-(7 \log (5)) \int \frac {x}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx+\left (8 \left (1-\log ^2(5)\right )\right ) \int \frac {x^2}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx+\frac {1}{8} \left (-3-4 \log ^3(5)-\log (625)\right ) \int \frac {1}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^3}{\left (1+8 x+256 x^4+32 x^2 \log (5)+\log ^2(5)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 78, normalized size = 3.12 \begin {gather*} \frac {x^{3}}{65536 \, x^{8} + 4096 \, x^{5} + 64 \, x^{2} \log \relax (5)^{3} + 512 \, x^{4} + \log \relax (5)^{4} + 2 \, {\left (768 \, x^{4} + 8 \, x + 1\right )} \log \relax (5)^{2} + 64 \, x^{2} + 64 \, {\left (256 \, x^{6} + 8 \, x^{3} + x^{2}\right )} \log \relax (5) + 16 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^{3}}{{\left (256 \, x^{4} + 32 \, x^{2} \log \relax (5) + \log \relax (5)^{2} + 8 \, x + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 28, normalized size = 1.12
method | result | size |
default | \(\frac {x^{3}}{\left (256 x^{4}+32 x^{2} \ln \relax (5)+\ln \relax (5)^{2}+8 x +1\right )^{2}}\) | \(28\) |
norman | \(\frac {x^{3}}{\left (256 x^{4}+32 x^{2} \ln \relax (5)+\ln \relax (5)^{2}+8 x +1\right )^{2}}\) | \(28\) |
gosper | \(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \relax (5)+1536 x^{4} \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)^{3}+4096 x^{5}+\ln \relax (5)^{4}+512 x^{3} \ln \relax (5)+512 x^{4}+16 x \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)+2 \ln \relax (5)^{2}+64 x^{2}+16 x +1}\) | \(88\) |
risch | \(\frac {x^{3}}{65536 x^{8}+16384 x^{6} \ln \relax (5)+1536 x^{4} \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)^{3}+4096 x^{5}+\ln \relax (5)^{4}+512 x^{3} \ln \relax (5)+512 x^{4}+16 x \ln \relax (5)^{2}+64 x^{2} \ln \relax (5)+2 \ln \relax (5)^{2}+64 x^{2}+16 x +1}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 77, normalized size = 3.08 \begin {gather*} \frac {x^{3}}{65536 \, x^{8} + 16384 \, x^{6} \log \relax (5) + 512 \, {\left (3 \, \log \relax (5)^{2} + 1\right )} x^{4} + 4096 \, x^{5} + 512 \, x^{3} \log \relax (5) + \log \relax (5)^{4} + 64 \, {\left (\log \relax (5)^{3} + \log \relax (5) + 1\right )} x^{2} + 16 \, {\left (\log \relax (5)^{2} + 1\right )} x + 2 \, \log \relax (5)^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.55, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^3}{{\left (256\,x^4+32\,\ln \relax (5)\,x^2+8\,x+{\ln \relax (5)}^2+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.91, size = 82, normalized size = 3.28 \begin {gather*} \frac {x^{3}}{65536 x^{8} + 16384 x^{6} \log {\relax (5 )} + 4096 x^{5} + x^{4} \left (512 + 1536 \log {\relax (5 )}^{2}\right ) + 512 x^{3} \log {\relax (5 )} + x^{2} \left (64 + 64 \log {\relax (5 )} + 64 \log {\relax (5 )}^{3}\right ) + x \left (16 + 16 \log {\relax (5 )}^{2}\right ) + 1 + 2 \log {\relax (5 )}^{2} + \log {\relax (5 )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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