Integrand size = 45, antiderivative size = 27 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=4+x-\frac {x \log (2+25 (3-x))}{2 x-5 x^2} \]
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Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6874, 46, 78, 90, 2442, 36, 31} \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=x-\frac {\log (77-25 x)}{2-5 x} \]
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Rule 31
Rule 36
Rule 46
Rule 78
Rule 90
Rule 2442
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {358}{(-2+5 x)^2 (-77+25 x)}+\frac {1765 x}{(-2+5 x)^2 (-77+25 x)}-\frac {2425 x^2}{(-2+5 x)^2 (-77+25 x)}+\frac {625 x^3}{(-2+5 x)^2 (-77+25 x)}-\frac {5 \log (77-25 x)}{(-2+5 x)^2}\right ) \, dx \\ & = -\left (5 \int \frac {\log (77-25 x)}{(-2+5 x)^2} \, dx\right )-358 \int \frac {1}{(-2+5 x)^2 (-77+25 x)} \, dx+625 \int \frac {x^3}{(-2+5 x)^2 (-77+25 x)} \, dx+1765 \int \frac {x}{(-2+5 x)^2 (-77+25 x)} \, dx-2425 \int \frac {x^2}{(-2+5 x)^2 (-77+25 x)} \, dx \\ & = -\frac {\log (77-25 x)}{2-5 x}+25 \int \frac {1}{(77-25 x) (-2+5 x)} \, dx-358 \int \left (-\frac {1}{67 (-2+5 x)^2}-\frac {5}{4489 (-2+5 x)}+\frac {25}{4489 (-77+25 x)}\right ) \, dx+625 \int \left (\frac {1}{625}-\frac {8}{8375 (-2+5 x)^2}-\frac {844}{561125 (-2+5 x)}+\frac {456533}{2805625 (-77+25 x)}\right ) \, dx+1765 \int \left (-\frac {2}{335 (-2+5 x)^2}-\frac {77}{22445 (-2+5 x)}+\frac {77}{4489 (-77+25 x)}\right ) \, dx-2425 \int \left (-\frac {4}{1675 (-2+5 x)^2}-\frac {288}{112225 (-2+5 x)}+\frac {5929}{112225 (-77+25 x)}\right ) \, dx \\ & = x+\frac {5}{67} \log (77-25 x)-\frac {\log (77-25 x)}{2-5 x}-\frac {5}{67} \log (2-5 x)+\frac {25}{67} \int \frac {1}{-2+5 x} \, dx+\frac {125}{67} \int \frac {1}{77-25 x} \, dx \\ & = x-\frac {\log (77-25 x)}{2-5 x} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=x+\frac {10}{67} \text {arctanh}\left (\frac {1}{67} (-87+50 x)\right )+\left (\frac {5}{67}+\frac {1}{-2+5 x}\right ) \log (77-25 x)-\frac {5}{67} \log (2-5 x) \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {\ln \left (-25 x +77\right )}{5 x -2}+x\) | \(17\) |
norman | \(\frac {\ln \left (-25 x +77\right )+5 x^{2}-\frac {4}{5}}{5 x -2}\) | \(22\) |
parallelrisch | \(\frac {-41000+15625 x^{2}+96250 x +3125 \ln \left (-25 x +77\right )}{15625 x -6250}\) | \(28\) |
derivativedivides | \(x -\frac {77}{25}+\frac {5 \ln \left (-25 x +77\right )}{67}-\frac {5 \ln \left (-25 x +77\right ) \left (-25 x +77\right )}{67 \left (-25 x +10\right )}\) | \(32\) |
default | \(x -\frac {77}{25}+\frac {5 \ln \left (-25 x +77\right )}{67}-\frac {5 \ln \left (-25 x +77\right ) \left (-25 x +77\right )}{67 \left (-25 x +10\right )}\) | \(32\) |
parts | \(x +\frac {5 \ln \left (25 x -77\right )}{67}-\frac {5 \ln \left (5 x -2\right )}{67}+\frac {5 \ln \left (-25 x +10\right )}{67}-\frac {5 \ln \left (-25 x +77\right ) \left (-25 x +77\right )}{67 \left (-25 x +10\right )}\) | \(47\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=\frac {5 \, x^{2} - 2 \, x + \log \left (-25 \, x + 77\right )}{5 \, x - 2} \]
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Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.44 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=x + \frac {\log {\left (77 - 25 x \right )}}{5 x - 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=\frac {112225 \, x^{2} + 5 \, {\left (1790 \, x + 3773\right )} \log \left (-25 \, x + 77\right ) - 44890 \, x + 23986}{22445 \, {\left (5 \, x - 2\right )}} - \frac {358}{335 \, {\left (5 \, x - 2\right )}} - \frac {358}{4489} \, \log \left (25 \, x - 77\right ) \]
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=x + \frac {\log \left (-25 \, x + 77\right )}{5 \, x - 2} \]
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Time = 12.63 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {-358+1765 x-2425 x^2+625 x^3+(385-125 x) \log (77-25 x)}{-308+1640 x-2425 x^2+625 x^3} \, dx=x+\frac {\ln \left (77-25\,x\right )}{5\,x-2} \]
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