Optimal. Leaf size=60 \[ \frac {(x-3) (4 x-1) \left (-\frac {13}{340} \log \left (x^2+1\right )+\frac {9}{110} \log (x-3)-\frac {1}{187} \log (4 x-1)+\frac {1}{170} \tan ^{-1}(x)\right )}{\sqrt [5]{\left (4 x^2-13 x+3\right )^5}} \]
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Rubi [B] time = 0.95, antiderivative size = 141, normalized size of antiderivative = 2.35, number of steps used = 9, number of rules used = 8, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used = {6688, 6720, 1075, 632, 31, 635, 203, 260} \begin {gather*} -\frac {\left (4 x^2-13 x+3\right ) \log (1-4 x)}{187 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}+\frac {9 \left (4 x^2-13 x+3\right ) \log (3-x)}{110 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}-\frac {13 \left (4 x^2-13 x+3\right ) \log \left (x^2+1\right )}{340 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}}+\frac {\left (4 x^2-13 x+3\right ) \tan ^{-1}(x)}{170 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 203
Rule 260
Rule 632
Rule 635
Rule 1075
Rule 6688
Rule 6720
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx &=\int \frac {x^2}{\left (1+x^2\right ) \sqrt [5]{\left (3-13 x+4 x^2\right )^5}} \, dx\\ &=\frac {\left (3-13 x+4 x^2\right ) \int \frac {x^2}{\left (1+x^2\right ) \left (3-13 x+4 x^2\right )} \, dx}{\sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (3-13 x+4 x^2\right ) \int \frac {1-13 x}{1+x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}+\frac {\left (3-13 x+4 x^2\right ) \int \frac {-3+52 x}{3-13 x+4 x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (3-13 x+4 x^2\right ) \int \frac {1}{1+x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (4 \left (3-13 x+4 x^2\right )\right ) \int \frac {1}{-1+4 x} \, dx}{187 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (13 \left (3-13 x+4 x^2\right )\right ) \int \frac {x}{1+x^2} \, dx}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}+\frac {\left (18 \left (3-13 x+4 x^2\right )\right ) \int \frac {1}{-12+4 x} \, dx}{55 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (3-13 x+4 x^2\right ) \tan ^{-1}(x)}{170 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (3-13 x+4 x^2\right ) \log (1-4 x)}{187 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}+\frac {9 \left (3-13 x+4 x^2\right ) \log (3-x)}{110 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}-\frac {13 \left (3-13 x+4 x^2\right ) \log \left (1+x^2\right )}{340 \sqrt [5]{\left (3-13 x+4 x^2\right )^5}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.98 \begin {gather*} \frac {\left (4 x^2-13 x+3\right ) \left (-143 \log \left (x^2+1\right )-20 \log (1-4 x)+306 \log (3-x)+22 \tan ^{-1}(x)\right )}{3740 \sqrt [5]{\left (4 x^2-13 x+3\right )^5}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 19.27, size = 60, normalized size = 1.00 \begin {gather*} \frac {(-3+x) (-1+4 x) \left (\frac {1}{170} \tan ^{-1}(x)+\frac {9}{110} \log (-3+x)-\frac {1}{187} \log (-1+4 x)-\frac {13}{340} \log \left (1+x^2\right )\right )}{\sqrt [5]{\left (3-13 x+4 x^2\right )^5}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 27, normalized size = 0.45 \begin {gather*} \frac {1}{170} \, \arctan \relax (x) - \frac {13}{340} \, \log \left (x^{2} + 1\right ) - \frac {1}{187} \, \log \left (4 \, x - 1\right ) + \frac {9}{110} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{5}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 130, normalized size = 2.17
method | result | size |
risch | \(\frac {\left (-\frac {13}{340}+\frac {i}{340}\right ) \left (4 x^{2}-13 x +3\right ) \ln \left (i+x \right )}{\left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}}}+\frac {\left (-\frac {13}{340}-\frac {i}{340}\right ) \left (4 x^{2}-13 x +3\right ) \ln \left (-i+x \right )}{\left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}}}-\frac {\left (4 x^{2}-13 x +3\right ) \ln \left (-1+4 x \right )}{187 \left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}}}+\frac {9 \left (4 x^{2}-13 x +3\right ) \ln \left (-3+x \right )}{110 \left (\left (4 x^{2}-13 x +3\right )^{5}\right )^{\frac {1}{5}}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 27, normalized size = 0.45 \begin {gather*} \frac {1}{170} \, \arctan \relax (x) - \frac {13}{340} \, \log \left (x^{2} + 1\right ) - \frac {1}{187} \, \log \left (4 \, x - 1\right ) + \frac {9}{110} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{\left (x^2+1\right )\,{\left (1024\,x^{10}-16640\,x^9+112000\,x^8-401440\,x^7+820340\,x^6-954733\,x^5+615255\,x^4-225810\,x^3+47250\,x^2-5265\,x+243\right )}^{1/5}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt [5]{\left (x - 3\right )^{5} \left (4 x - 1\right )^{5}} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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