Optimal. Leaf size=19 \[ \frac {4}{9} (8+x)^2 (-x+\log (2 x))^2 \]
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Rubi [B] time = 0.14, antiderivative size = 87, normalized size of antiderivative = 4.58, number of steps used = 16, number of rules used = 9, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.145, Rules used = {12, 14, 2357, 2295, 2301, 2304, 2330, 2296, 2305} \begin {gather*} \frac {4 x^4}{9}+\frac {64 x^3}{9}-\frac {8}{9} x^3 \log (2 x)+\frac {256 x^2}{9}+\frac {4}{9} x^2 \log ^2(2 x)-\frac {128}{9} x^2 \log (2 x)+\frac {64}{9} x \log ^2(2 x)+\frac {256}{9} \log ^2(2 x)-\frac {512}{9} x \log (2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2295
Rule 2296
Rule 2301
Rule 2304
Rule 2305
Rule 2330
Rule 2357
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{x} \, dx\\ &=\frac {1}{9} \int \left (8 \left (-64+48 x+23 x^2+2 x^3\right )-\frac {8 (8+x) \left (-8+7 x+3 x^2\right ) \log (2 x)}{x}+8 (8+x) \log ^2(2 x)\right ) \, dx\\ &=\frac {8}{9} \int \left (-64+48 x+23 x^2+2 x^3\right ) \, dx-\frac {8}{9} \int \frac {(8+x) \left (-8+7 x+3 x^2\right ) \log (2 x)}{x} \, dx+\frac {8}{9} \int (8+x) \log ^2(2 x) \, dx\\ &=-\frac {512 x}{9}+\frac {64 x^2}{3}+\frac {184 x^3}{27}+\frac {4 x^4}{9}-\frac {8}{9} \int \left (48 \log (2 x)-\frac {64 \log (2 x)}{x}+31 x \log (2 x)+3 x^2 \log (2 x)\right ) \, dx+\frac {8}{9} \int \left (8 \log ^2(2 x)+x \log ^2(2 x)\right ) \, dx\\ &=-\frac {512 x}{9}+\frac {64 x^2}{3}+\frac {184 x^3}{27}+\frac {4 x^4}{9}+\frac {8}{9} \int x \log ^2(2 x) \, dx-\frac {8}{3} \int x^2 \log (2 x) \, dx+\frac {64}{9} \int \log ^2(2 x) \, dx-\frac {248}{9} \int x \log (2 x) \, dx-\frac {128}{3} \int \log (2 x) \, dx+\frac {512}{9} \int \frac {\log (2 x)}{x} \, dx\\ &=-\frac {128 x}{9}+\frac {254 x^2}{9}+\frac {64 x^3}{9}+\frac {4 x^4}{9}-\frac {128}{3} x \log (2 x)-\frac {124}{9} x^2 \log (2 x)-\frac {8}{9} x^3 \log (2 x)+\frac {256}{9} \log ^2(2 x)+\frac {64}{9} x \log ^2(2 x)+\frac {4}{9} x^2 \log ^2(2 x)-\frac {8}{9} \int x \log (2 x) \, dx-\frac {128}{9} \int \log (2 x) \, dx\\ &=\frac {256 x^2}{9}+\frac {64 x^3}{9}+\frac {4 x^4}{9}-\frac {512}{9} x \log (2 x)-\frac {128}{9} x^2 \log (2 x)-\frac {8}{9} x^3 \log (2 x)+\frac {256}{9} \log ^2(2 x)+\frac {64}{9} x \log ^2(2 x)+\frac {4}{9} x^2 \log ^2(2 x)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.07, size = 77, normalized size = 4.05 \begin {gather*} \frac {8}{9} \left (32 x^2+8 x^3+\frac {x^4}{2}-64 x \log (2 x)-16 x^2 \log (2 x)-x^3 \log (2 x)+32 \log ^2(2 x)+8 x \log ^2(2 x)+\frac {1}{2} x^2 \log ^2(2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 50, normalized size = 2.63 \begin {gather*} \frac {4}{9} \, x^{4} + \frac {64}{9} \, x^{3} + \frac {4}{9} \, {\left (x^{2} + 16 \, x + 64\right )} \log \left (2 \, x\right )^{2} + \frac {256}{9} \, x^{2} - \frac {8}{9} \, {\left (x^{3} + 16 \, x^{2} + 64 \, x\right )} \log \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 50, normalized size = 2.63 \begin {gather*} \frac {4}{9} \, x^{4} + \frac {64}{9} \, x^{3} + \frac {4}{9} \, {\left (x^{2} + 16 \, x + 64\right )} \log \left (2 \, x\right )^{2} + \frac {256}{9} \, x^{2} - \frac {8}{9} \, {\left (x^{3} + 16 \, x^{2} + 64 \, x\right )} \log \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 55, normalized size = 2.89
method | result | size |
risch | \(\frac {\left (4 x^{2}+64 x +256\right ) \ln \left (2 x \right )^{2}}{9}+\frac {\left (-8 x^{3}-128 x^{2}-512 x \right ) \ln \left (2 x \right )}{9}+\frac {4 x^{4}}{9}+\frac {64 x^{3}}{9}+\frac {256 x^{2}}{9}\) | \(55\) |
derivativedivides | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 x^{3} \ln \left (2 x \right )}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
default | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 x^{3} \ln \left (2 x \right )}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
norman | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 x^{3} \ln \left (2 x \right )}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 90, normalized size = 4.74 \begin {gather*} \frac {4}{9} \, x^{4} - \frac {8}{9} \, x^{3} \log \left (2 \, x\right ) + \frac {2}{9} \, {\left (2 \, \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 1\right )} x^{2} + \frac {64}{9} \, x^{3} - \frac {124}{9} \, x^{2} \log \left (2 \, x\right ) + \frac {64}{9} \, {\left (\log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 2\right )} x + \frac {254}{9} \, x^{2} - \frac {128}{3} \, x \log \left (2 \, x\right ) + \frac {256}{9} \, \log \left (2 \, x\right )^{2} - \frac {128}{9} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 17, normalized size = 0.89 \begin {gather*} \frac {4\,{\left (x-\ln \left (2\,x\right )\right )}^2\,{\left (x+8\right )}^2}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 66, normalized size = 3.47 \begin {gather*} \frac {4 x^{4}}{9} + \frac {64 x^{3}}{9} + \frac {256 x^{2}}{9} + \left (\frac {4 x^{2}}{9} + \frac {64 x}{9} + \frac {256}{9}\right ) \log {\left (2 x \right )}^{2} + \left (- \frac {8 x^{3}}{9} - \frac {128 x^{2}}{9} - \frac {512 x}{9}\right ) \log {\left (2 x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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