Integrand size = 13, antiderivative size = 36 \[ \int \frac {x^{14}}{3+b x^5} \, dx=-\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b}+\frac {9 \log \left (3+b x^5\right )}{5 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 36, 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} \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {9 \log \left (b x^5+3\right )}{5 b^3}-\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {x^2}{3+b x} \, dx,x,x^5\right ) \\ & = \frac {1}{5} \text {Subst}\left (\int \left (-\frac {3}{b^2}+\frac {x}{b}+\frac {9}{b^2 (3+b x)}\right ) \, dx,x,x^5\right ) \\ & = -\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b}+\frac {9 \log \left (3+b x^5\right )}{5 b^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {x^{14}}{3+b x^5} \, dx=-\frac {3 x^5}{5 b^2}+\frac {x^{10}}{10 b}+\frac {9 \log \left (3+b x^5\right )}{5 b^3} \]
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Time = 4.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83
method | result | size |
meijerg | \(\frac {-\frac {b \,x^{5} \left (-b \,x^{5}+6\right )}{10}+\frac {9 \ln \left (1+\frac {b \,x^{5}}{3}\right )}{5}}{b^{3}}\) | \(30\) |
parallelrisch | \(\frac {b^{2} x^{10}-6 b \,x^{5}+18 \ln \left (b \,x^{5}+3\right )}{10 b^{3}}\) | \(30\) |
norman | \(-\frac {3 x^{5}}{5 b^{2}}+\frac {x^{10}}{10 b}+\frac {9 \ln \left (b \,x^{5}+3\right )}{5 b^{3}}\) | \(31\) |
default | \(\frac {\frac {1}{2} b \,x^{10}-3 x^{5}}{5 b^{2}}+\frac {9 \ln \left (b \,x^{5}+3\right )}{5 b^{3}}\) | \(32\) |
risch | \(\frac {x^{10}}{10 b}-\frac {3 x^{5}}{5 b^{2}}+\frac {9}{10 b^{3}}+\frac {9 \ln \left (b \,x^{5}+3\right )}{5 b^{3}}\) | \(36\) |
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {b^{2} x^{10} - 6 \, b x^{5} + 18 \, \log \left (b x^{5} + 3\right )}{10 \, b^{3}} \]
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Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {x^{10}}{10 b} - \frac {3 x^{5}}{5 b^{2}} + \frac {9 \log {\left (b x^{5} + 3 \right )}}{5 b^{3}} \]
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Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {b x^{10} - 6 \, x^{5}}{10 \, b^{2}} + \frac {9 \, \log \left (b x^{5} + 3\right )}{5 \, b^{3}} \]
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Time = 0.48 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {b x^{10} - 6 \, x^{5}}{10 \, b^{2}} + \frac {9 \, \log \left ({\left | b x^{5} + 3 \right |}\right )}{5 \, b^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {x^{14}}{3+b x^5} \, dx=\frac {18\,\ln \left (b\,x^5+3\right )-6\,b\,x^5+b^2\,x^{10}}{10\,b^3} \]
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