Integrand size = 65, antiderivative size = 23 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log ^2\left (\frac {1}{3+\frac {x}{(25+x)^2 \left (-3 x+x^2\right )}}\right ) \]
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\[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-3+x}+\frac {4 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{25+x}-\frac {6 \left (475+94 x+3 x^2\right ) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3}\right ) \, dx \\ & = 2 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-3+x} \, dx+4 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{25+x} \, dx-6 \int \frac {\left (475+94 x+3 x^2\right ) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx \\ & = 2 \log (-3+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \log (25+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )-2 \int \frac {\left (-5624+1425 x+141 x^2+3 x^3\right ) \left (-\frac {\left (1425+282 x+9 x^2\right ) \left (-1875+475 x+47 x^2+x^3\right )}{\left (-5624+1425 x+141 x^2+3 x^3\right )^2}+\frac {475+94 x+3 x^2}{-5624+1425 x+141 x^2+3 x^3}\right ) \log (-3+x)}{-1875+475 x+47 x^2+x^3} \, dx-4 \int \frac {\left (-5624+1425 x+141 x^2+3 x^3\right ) \left (-\frac {\left (1425+282 x+9 x^2\right ) \left (-1875+475 x+47 x^2+x^3\right )}{\left (-5624+1425 x+141 x^2+3 x^3\right )^2}+\frac {475+94 x+3 x^2}{-5624+1425 x+141 x^2+3 x^3}\right ) \log (25+x)}{-1875+475 x+47 x^2+x^3} \, dx-6 \int \left (\frac {475 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3}+\frac {94 x \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3}+\frac {3 x^2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3}\right ) \, dx \\ & = 2 \log (-3+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \log (25+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )-2 \int \left (\frac {\log (-3+x)}{-3+x}+\frac {2 \log (-3+x)}{25+x}-\frac {3 \left (475+94 x+3 x^2\right ) \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3}\right ) \, dx-4 \int \left (\frac {\log (25+x)}{-3+x}+\frac {2 \log (25+x)}{25+x}-\frac {3 \left (475+94 x+3 x^2\right ) \log (25+x)}{-5624+1425 x+141 x^2+3 x^3}\right ) \, dx-18 \int \frac {x^2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx-564 \int \frac {x \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx-2850 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx \\ & = 2 \log (-3+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \log (25+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )-2 \int \frac {\log (-3+x)}{-3+x} \, dx-4 \int \frac {\log (-3+x)}{25+x} \, dx-4 \int \frac {\log (25+x)}{-3+x} \, dx+6 \int \frac {\left (475+94 x+3 x^2\right ) \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-8 \int \frac {\log (25+x)}{25+x} \, dx+12 \int \frac {\left (475+94 x+3 x^2\right ) \log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-18 \int \frac {x^2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx-564 \int \frac {x \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx-2850 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx \\ & = -4 \log (-3+x) \log \left (\frac {25+x}{28}\right )-4 \log \left (\frac {3-x}{28}\right ) \log (25+x)+2 \log (-3+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \log (25+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )-2 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-3+x\right )+4 \int \frac {\log \left (\frac {3-x}{28}\right )}{25+x} \, dx+4 \int \frac {\log \left (\frac {25+x}{28}\right )}{-3+x} \, dx+6 \int \left (\frac {475 \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3}+\frac {94 x \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3}+\frac {3 x^2 \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3}\right ) \, dx-8 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25+x\right )+12 \int \left (\frac {475 \log (25+x)}{-5624+1425 x+141 x^2+3 x^3}+\frac {94 x \log (25+x)}{-5624+1425 x+141 x^2+3 x^3}+\frac {3 x^2 \log (25+x)}{-5624+1425 x+141 x^2+3 x^3}\right ) \, dx-18 \int \frac {x^2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx-564 \int \frac {x \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx-2850 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx \\ & = -\log ^2(-3+x)-4 \log (-3+x) \log \left (\frac {25+x}{28}\right )-4 \log \left (\frac {3-x}{28}\right ) \log (25+x)-4 \log ^2(25+x)+2 \log (-3+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \log (25+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{28}\right )}{x} \, dx,x,25+x\right )+4 \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{28}\right )}{x} \, dx,x,-3+x\right )+18 \int \frac {x^2 \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-18 \int \frac {x^2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx+36 \int \frac {x^2 \log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx+564 \int \frac {x \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-564 \int \frac {x \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx+1128 \int \frac {x \log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx+2850 \int \frac {\log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-2850 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx+5700 \int \frac {\log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx \\ & = -\log ^2(-3+x)-4 \log (-3+x) \log \left (\frac {25+x}{28}\right )-4 \log \left (\frac {3-x}{28}\right ) \log (25+x)-4 \log ^2(25+x)+2 \log (-3+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )+4 \log (25+x) \log \left (\frac {1875-475 x-47 x^2-x^3}{5624-1425 x-141 x^2-3 x^3}\right )-4 \operatorname {PolyLog}\left (2,\frac {3-x}{28}\right )-4 \operatorname {PolyLog}\left (2,\frac {25+x}{28}\right )+18 \int \frac {x^2 \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-18 \int \frac {x^2 \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx+36 \int \frac {x^2 \log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx+564 \int \frac {x \log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-564 \int \frac {x \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx+1128 \int \frac {x \log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx+2850 \int \frac {\log (-3+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx-2850 \int \frac {\log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{-5624+1425 x+141 x^2+3 x^3} \, dx+5700 \int \frac {\log (25+x)}{-5624+1425 x+141 x^2+3 x^3} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log ^2\left (\frac {(-3+x) (25+x)^2}{-5624+1425 x+141 x^2+3 x^3}\right ) \]
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Time = 1.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
method | result | size |
norman | \(\ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right )^{2}\) | \(35\) |
parts | \(-2 \ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )+4 \ln \left (x +25\right ) \ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right )+2 \ln \left (-3+x \right ) \ln \left (\frac {x^{3}+47 x^{2}+475 x -1875}{3 x^{3}+141 x^{2}+1425 x -5624}\right )-\ln \left (-3+x \right )^{2}-4 \ln \left (-3+x \right ) \ln \left (\frac {25}{28}+\frac {x}{28}\right )+2 \ln \left (-3+x \right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}+141 \textit {\_Z}^{2}+1425 \textit {\_Z} -5624\right )}{\sum }\left (-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )+\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47-\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}+2 x}{3 \underline {\hspace {1.25 ex}}\alpha +47+\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}-94 \underline {\hspace {1.25 ex}}\alpha +309}}\right )+\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2}\right )\right )+4 \ln \left (x +25\right ) \ln \left (3 x^{3}+141 x^{2}+1425 x -5624\right )-4 \ln \left (x +25\right )^{2}-4 \left (\ln \left (x +25\right )-\ln \left (\frac {25}{28}+\frac {x}{28}\right )\right ) \ln \left (-\frac {x}{28}+\frac {3}{28}\right )\) | \(455\) |
default | \(\text {Expression too large to display}\) | \(1016\) |
risch | \(\text {Expression too large to display}\) | \(6661\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log \left (\frac {x^{3} + 47 \, x^{2} + 475 \, x - 1875}{3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624}\right )^{2} \]
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Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log {\left (\frac {x^{3} + 47 x^{2} + 475 x - 1875}{3 x^{3} + 141 x^{2} + 1425 x - 5624} \right )}^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 6.04 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=2 \, {\left (2 \, \log \left (x + 25\right ) + \log \left (x - 3\right )\right )} \log \left (3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624\right ) - \log \left (3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624\right )^{2} - 4 \, \log \left (x + 25\right )^{2} - 4 \, \log \left (x + 25\right ) \log \left (x - 3\right ) - \log \left (x - 3\right )^{2} - 2 \, {\left (\log \left (3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624\right ) - 2 \, \log \left (x + 25\right ) - \log \left (x - 3\right )\right )} \log \left (\frac {x^{3} + 47 \, x^{2} + 475 \, x - 1875}{3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx=\log \left (\frac {x^{3} + 47 \, x^{2} + 475 \, x - 1875}{3 \, x^{3} + 141 \, x^{2} + 1425 \, x - 5624}\right )^{2} \]
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Time = 13.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {(38+6 x) \log \left (\frac {-1875+475 x+47 x^2+x^3}{-5624+1425 x+141 x^2+3 x^3}\right )}{421800-230603 x+15151 x^2+4302 x^3+207 x^4+3 x^5} \, dx={\ln \left (\frac {1}{3}-\frac {1}{3\,\left (3\,x^3+141\,x^2+1425\,x-5624\right )}\right )}^2 \]
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