Integrand size = 13, antiderivative size = 39 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=-\frac {9}{80 \left (3+2 x^5\right )^2}+\frac {3}{20 \left (3+2 x^5\right )}+\frac {1}{40} \log \left (3+2 x^5\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=\frac {3}{20 \left (2 x^5+3\right )}-\frac {9}{80 \left (2 x^5+3\right )^2}+\frac {1}{40} \log \left (2 x^5+3\right ) \]
[In]
[Out]
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {x^2}{(3+2 x)^3} \, dx,x,x^5\right ) \\ & = \frac {1}{5} \text {Subst}\left (\int \left (\frac {9}{4 (3+2 x)^3}-\frac {3}{2 (3+2 x)^2}+\frac {1}{4 (3+2 x)}\right ) \, dx,x,x^5\right ) \\ & = -\frac {9}{80 \left (3+2 x^5\right )^2}+\frac {3}{20 \left (3+2 x^5\right )}+\frac {1}{40} \log \left (3+2 x^5\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=\frac {1}{80} \left (\frac {3 \left (9+8 x^5\right )}{\left (3+2 x^5\right )^2}+2 \log \left (3+2 x^5\right )\right ) \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74
method | result | size |
norman | \(\frac {\frac {3 x^{5}}{10}+\frac {27}{80}}{\left (2 x^{5}+3\right )^{2}}+\frac {\ln \left (2 x^{5}+3\right )}{40}\) | \(29\) |
risch | \(\frac {\frac {3 x^{5}}{10}+\frac {27}{80}}{\left (2 x^{5}+3\right )^{2}}+\frac {\ln \left (2 x^{5}+3\right )}{40}\) | \(30\) |
meijerg | \(-\frac {x^{5} \left (6 x^{5}+6\right )}{360 \left (1+\frac {2 x^{5}}{3}\right )^{2}}+\frac {\ln \left (1+\frac {2 x^{5}}{3}\right )}{40}\) | \(33\) |
default | \(-\frac {9}{80 \left (2 x^{5}+3\right )^{2}}+\frac {3}{20 \left (2 x^{5}+3\right )}+\frac {\ln \left (2 x^{5}+3\right )}{40}\) | \(34\) |
parallelrisch | \(\frac {8 \ln \left (x^{5}+\frac {3}{2}\right ) x^{10}+27+24 \ln \left (x^{5}+\frac {3}{2}\right ) x^{5}+24 x^{5}+18 \ln \left (x^{5}+\frac {3}{2}\right )}{80 \left (2 x^{5}+3\right )^{2}}\) | \(49\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=\frac {24 \, x^{5} + 2 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )} \log \left (2 \, x^{5} + 3\right ) + 27}{80 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=\frac {24 x^{5} + 27}{320 x^{10} + 960 x^{5} + 720} + \frac {\log {\left (2 x^{5} + 3 \right )}}{40} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=\frac {3 \, {\left (8 \, x^{5} + 9\right )}}{80 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} + \frac {1}{40} \, \log \left (2 \, x^{5} + 3\right ) \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=-\frac {3 \, {\left (x^{10} + x^{5}\right )}}{20 \, {\left (2 \, x^{5} + 3\right )}^{2}} + \frac {1}{40} \, \log \left ({\left | 2 \, x^{5} + 3 \right |}\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int \frac {x^{14}}{\left (3+2 x^5\right )^3} \, dx=\frac {\ln \left (x^5+\frac {3}{2}\right )}{40}+\frac {\frac {3\,x^5}{40}+\frac {27}{320}}{x^{10}+3\,x^5+\frac {9}{4}} \]
[In]
[Out]