Integrand size = 15, antiderivative size = 43 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=-\frac {1}{22 (3+5 x)^2}-\frac {2}{121 (3+5 x)}-\frac {4 \log (1-2 x)}{1331}+\frac {4 \log (3+5 x)}{1331} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46} \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=-\frac {2}{121 (5 x+3)}-\frac {1}{22 (5 x+3)^2}-\frac {4 \log (1-2 x)}{1331}+\frac {4 \log (5 x+3)}{1331} \]
[In]
[Out]
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8}{1331 (-1+2 x)}+\frac {5}{11 (3+5 x)^3}+\frac {10}{121 (3+5 x)^2}+\frac {20}{1331 (3+5 x)}\right ) \, dx \\ & = -\frac {1}{22 (3+5 x)^2}-\frac {2}{121 (3+5 x)}-\frac {4 \log (1-2 x)}{1331}+\frac {4 \log (3+5 x)}{1331} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=\frac {-\frac {11 (23+20 x)}{(3+5 x)^2}-8 \log (5-10 x)+8 \log (3+5 x)}{2662} \]
[In]
[Out]
Time = 2.59 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {-\frac {10 x}{121}-\frac {23}{242}}{\left (3+5 x \right )^{2}}-\frac {4 \ln \left (-1+2 x \right )}{1331}+\frac {4 \ln \left (3+5 x \right )}{1331}\) | \(32\) |
norman | \(\frac {\frac {85}{363} x +\frac {575}{2178} x^{2}}{\left (3+5 x \right )^{2}}-\frac {4 \ln \left (-1+2 x \right )}{1331}+\frac {4 \ln \left (3+5 x \right )}{1331}\) | \(35\) |
default | \(-\frac {1}{22 \left (3+5 x \right )^{2}}-\frac {2}{121 \left (3+5 x \right )}+\frac {4 \ln \left (3+5 x \right )}{1331}-\frac {4 \ln \left (-1+2 x \right )}{1331}\) | \(36\) |
parallelrisch | \(\frac {1800 \ln \left (x +\frac {3}{5}\right ) x^{2}-1800 \ln \left (x -\frac {1}{2}\right ) x^{2}+2160 \ln \left (x +\frac {3}{5}\right ) x -2160 \ln \left (x -\frac {1}{2}\right ) x +6325 x^{2}+648 \ln \left (x +\frac {3}{5}\right )-648 \ln \left (x -\frac {1}{2}\right )+5610 x}{23958 \left (3+5 x \right )^{2}}\) | \(63\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=\frac {8 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 8 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 220 \, x - 253}{2662 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=- \frac {20 x + 23}{6050 x^{2} + 7260 x + 2178} - \frac {4 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {4 \log {\left (x + \frac {3}{5} \right )}}{1331} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=-\frac {20 \, x + 23}{242 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {4}{1331} \, \log \left (5 \, x + 3\right ) - \frac {4}{1331} \, \log \left (2 \, x - 1\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=-\frac {20 \, x + 23}{242 \, {\left (5 \, x + 3\right )}^{2}} + \frac {4}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {4}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(1-2 x) (3+5 x)^3} \, dx=\frac {8\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{1331}-\frac {\frac {2\,x}{605}+\frac {23}{6050}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \]
[In]
[Out]