Optimal. Leaf size=20 \[ x \log \left (5 x^2+\left (\frac {67}{2}+(3+x)^2\right )^2\right ) \]
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Rubi [B] time = 0.78, antiderivative size = 70, normalized size of antiderivative = 3.50, number of steps used = 13, number of rules used = 4, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 6742, 2100, 2523} \begin {gather*} x \log \left (x^4+12 x^3+126 x^2+510 x+\frac {7225}{4}\right )+3 \log \left (x^4+12 x^3+126 x^2+510 x+\frac {7225}{4}\right )-3 \log \left (4 x^4+48 x^3+504 x^2+2040 x+7225\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2100
Rule 2523
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {8 x \left (255+126 x+18 x^2+2 x^3\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )\right ) \, dx\\ &=8 \int \frac {x \left (255+126 x+18 x^2+2 x^3\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx+\int \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right ) \, dx\\ &=x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+8 \int \left (\frac {1}{2}-\frac {7225+1530 x+252 x^2+12 x^3}{2 \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )}\right ) \, dx-\int \frac {x \left (510+252 x+36 x^2+4 x^3\right )}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4} \, dx\\ &=4 x+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-4 \int \frac {7225+1530 x+252 x^2+12 x^3}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx-\int \left (4-\frac {7225+1530 x+252 x^2+12 x^3}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4}\right ) \, dx\\ &=x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )-\frac {1}{4} \int \frac {91120+12384 x+2304 x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx+\int \frac {7225+1530 x+252 x^2+12 x^3}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4} \, dx\\ &=3 \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )+\frac {1}{4} \int \frac {22780+3096 x+576 x^2}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4} \, dx-\frac {1}{4} \int \left (\frac {91120}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {12384 x}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {2304 x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4}\right ) \, dx\\ &=3 \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )+\frac {1}{4} \int \left (\frac {91120}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {12384 x}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {2304 x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4}\right ) \, dx-576 \int \frac {x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx-3096 \int \frac {x}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx-22780 \int \frac {1}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx\\ &=3 \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 23, normalized size = 1.15 \begin {gather*} x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 21, normalized size = 1.05 \begin {gather*} x \log \left (x^{4} + 12 \, x^{3} + 126 \, x^{2} + 510 \, x + \frac {7225}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 21, normalized size = 1.05 \begin {gather*} x \log \left (x^{4} + 12 \, x^{3} + 126 \, x^{2} + 510 \, x + \frac {7225}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 22, normalized size = 1.10
method | result | size |
norman | \(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\) | \(22\) |
risch | \(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\) | \(22\) |
default | \(-2 x \ln \relax (2)+x \ln \left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 29, normalized size = 1.45 \begin {gather*} -2 \, x \log \relax (2) + x \log \left (4 \, x^{4} + 48 \, x^{3} + 504 \, x^{2} + 2040 \, x + 7225\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.88, size = 21, normalized size = 1.05 \begin {gather*} x\,\ln \left (x^4+12\,x^3+126\,x^2+510\,x+\frac {7225}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 22, normalized size = 1.10 \begin {gather*} x \log {\left (x^{4} + 12 x^{3} + 126 x^{2} + 510 x + \frac {7225}{4} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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