Optimal. Leaf size=23 \[ \frac {\left (x+\log \left (4+\frac {9+x}{5+x}\right )\right )^2}{2 x} \]
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Rubi [C] time = 0.85, antiderivative size = 246, normalized size of antiderivative = 10.70, number of steps used = 29, number of rules used = 17, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1594, 6728, 1657, 616, 31, 2514, 2494, 2317, 2391, 2488, 2411, 2343, 2333, 2315, 2490, 2503, 2502} \begin {gather*} -\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {5}{29} \text {Li}_2\left (\frac {5 (x+5)}{5 x+29}\right )-\frac {1}{5} \text {Li}_2\left (1+\frac {4}{5 (x+5)}\right )-\frac {4}{145} \text {Li}_2\left (\frac {4 x}{5 (5 x+29)}+1\right )+\frac {x}{2}+\frac {(x+5) \log ^2\left (\frac {5 x+29}{x+5}\right )}{10 x}-\frac {4}{145} \log (x) \log \left (\frac {5 x+29}{x+5}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (x+5)}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {5}{29} \log \left (\frac {4}{5 x+29}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (5 x+29)}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {4}{145} \log \left (\frac {5 x}{29}+1\right ) \log (x)-\frac {4}{145} \log \left (\frac {x}{5}+1\right ) \log (x)-\log (x+5)+\log (5 x+29) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 1594
Rule 1657
Rule 2315
Rule 2317
Rule 2333
Rule 2343
Rule 2391
Rule 2411
Rule 2488
Rule 2490
Rule 2494
Rule 2502
Rule 2503
Rule 2514
Rule 6728
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{x^2 \left (290+108 x+10 x^2\right )} \, dx\\ &=\int \left (\frac {137+54 x+5 x^2}{2 \left (145+54 x+5 x^2\right )}-\frac {4 \log \left (\frac {29+5 x}{5+x}\right )}{x (5+x) (29+5 x)}-\frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{2 x^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {137+54 x+5 x^2}{145+54 x+5 x^2} \, dx-\frac {1}{2} \int \frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{x^2} \, dx-4 \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x (5+x) (29+5 x)} \, dx\\ &=\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}+\frac {1}{2} \int \left (1-\frac {8}{145+54 x+5 x^2}\right ) \, dx+\frac {4}{5} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x (29+5 x)} \, dx-4 \int \left (\frac {\log \left (\frac {29+5 x}{5+x}\right )}{145 x}-\frac {\log \left (\frac {29+5 x}{5+x}\right )}{20 (5+x)}+\frac {25 \log \left (\frac {29+5 x}{5+x}\right )}{116 (29+5 x)}\right ) \, dx\\ &=\frac {x}{2}+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4}{145} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x} \, dx+\frac {16}{145} \int \frac {\log \left (-\frac {4 x}{5 (29+5 x)}\right )}{(5+x) (29+5 x)} \, dx+\frac {1}{5} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{5+x} \, dx-\frac {25}{29} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{29+5 x} \, dx-4 \int \frac {1}{145+54 x+5 x^2} \, dx\\ &=\frac {x}{2}-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}+\frac {16}{725} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4 x}{5}\right )}{1+\frac {4 x}{5}} \, dx,x,\frac {x}{29+5 x}\right )-\frac {4}{145} \int \frac {\log (x)}{5+x} \, dx+\frac {4}{29} \int \frac {\log (x)}{29+5 x} \, dx+\frac {20}{29} \int \frac {\log \left (\frac {4}{29+5 x}\right )}{(5+x) (29+5 x)} \, dx-\frac {4}{5} \int \frac {\log \left (-\frac {4}{5 (5+x)}\right )}{(5+x) (29+5 x)} \, dx-5 \int \frac {1}{25+5 x} \, dx+5 \int \frac {1}{29+5 x} \, dx\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )-\frac {4}{145} \int \frac {\log \left (1+\frac {5 x}{29}\right )}{x} \, dx+\frac {4}{145} \int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx+\frac {4}{29} \operatorname {Subst}\left (\int \frac {\log \left (\frac {4}{x}\right )}{\left (-\frac {4}{5}+\frac {x}{5}\right ) x} \, dx,x,29+5 x\right )-\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4}{5 x}\right )}{x (4+5 x)} \, dx,x,5+x\right )\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )-\frac {4}{29} \operatorname {Subst}\left (\int \frac {\log (4 x)}{\left (-\frac {4}{5}+\frac {1}{5 x}\right ) x} \, dx,x,\frac {1}{29+5 x}\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4 x}{5}\right )}{\left (4+\frac {5}{x}\right ) x} \, dx,x,\frac {1}{5+x}\right )\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )-\frac {4}{29} \operatorname {Subst}\left (\int \frac {\log (4 x)}{\frac {1}{5}-\frac {4 x}{5}} \, dx,x,\frac {1}{29+5 x}\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4 x}{5}\right )}{5+4 x} \, dx,x,\frac {1}{5+x}\right )\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {1}{5} \text {Li}_2\left (1+\frac {4}{5 (5+x)}\right )-\frac {5}{29} \text {Li}_2\left (1-\frac {4}{29+5 x}\right )-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.14, size = 152, normalized size = 6.61 \begin {gather*} \frac {1}{2} \left (x-\frac {1}{5} \log ^2\left (-\frac {4}{5 (5+x)}\right )-2 \log (5+x)+\frac {1}{5} \log ^2(5+x)-\frac {2}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {1}{4} (29+5 x)\right )-\frac {2}{5} \log (5+x) \log \left (\frac {1}{4} (29+5 x)\right )+2 \log (29+5 x)+\frac {2}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {2}{5} \log (5+x) \log \left (\frac {29+5 x}{5+x}\right )+\frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 38, normalized size = 1.65 \begin {gather*} \frac {x^{2} + 2 \, x \log \left (\frac {5 \, x + 29}{x + 5}\right ) + \log \left (\frac {5 \, x + 29}{x + 5}\right )^{2}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 66, normalized size = 2.87 \begin {gather*} -\frac {1}{10} \, {\left (\frac {4}{\frac {5 \, {\left (5 \, x + 29\right )}}{x + 5} - 29} + 1\right )} \log \left (\frac {5 \, x + 29}{x + 5}\right )^{2} + \frac {2}{\frac {5 \, x + 29}{x + 5} - 5} + \log \left (\frac {5 \, x + 29}{x + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 36, normalized size = 1.57
method | result | size |
risch | \(\frac {\ln \left (\frac {5 x +29}{5+x}\right )^{2}}{2 x}+\frac {x}{2}+\ln \left (5 x +29\right )-\ln \left (5+x \right )\) | \(36\) |
norman | \(\frac {x \ln \left (\frac {5 x +29}{5+x}\right )+\frac {x^{2}}{2}+\frac {\ln \left (\frac {5 x +29}{5+x}\right )^{2}}{2}}{x}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 48, normalized size = 2.09 \begin {gather*} \frac {1}{2} \, x + \frac {\log \left (5 \, x + 29\right )^{2} - 2 \, \log \left (5 \, x + 29\right ) \log \left (x + 5\right ) + \log \left (x + 5\right )^{2}}{2 \, x} + \log \left (5 \, x + 29\right ) - \log \left (x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 31, normalized size = 1.35 \begin {gather*} \frac {x}{2}+2\,\mathrm {atanh}\left (\frac {5\,x}{2}+\frac {27}{2}\right )+\frac {{\ln \left (\frac {5\,x+29}{x+5}\right )}^2}{2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 29, normalized size = 1.26 \begin {gather*} \frac {x}{2} - \log {\left (x + 5 \right )} + \log {\left (x + \frac {29}{5} \right )} + \frac {\log {\left (\frac {5 x + 29}{x + 5} \right )}^{2}}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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