Integrand size = 13, antiderivative size = 56 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=-\frac {2 b x^3}{3 a^3}+\frac {x^6}{6 a^2}+\frac {b^3}{3 a^4 \left (b+a x^3\right )}+\frac {b^2 \log \left (b+a x^3\right )}{a^4} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45} \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {b^3}{3 a^4 \left (a x^3+b\right )}+\frac {b^2 \log \left (a x^3+b\right )}{a^4}-\frac {2 b x^3}{3 a^3}+\frac {x^6}{6 a^2} \]
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Rule 45
Rule 269
Rule 272
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{11}}{\left (b+a x^3\right )^2} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {x^3}{(b+a x)^2} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {2 b}{a^3}+\frac {x}{a^2}-\frac {b^3}{a^3 (b+a x)^2}+\frac {3 b^2}{a^3 (b+a x)}\right ) \, dx,x,x^3\right ) \\ & = -\frac {2 b x^3}{3 a^3}+\frac {x^6}{6 a^2}+\frac {b^3}{3 a^4 \left (b+a x^3\right )}+\frac {b^2 \log \left (b+a x^3\right )}{a^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {-4 a b x^3+a^2 x^6+\frac {2 b^3}{b+a x^3}+6 b^2 \log \left (b+a x^3\right )}{6 a^4} \]
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Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93
method | result | size |
norman | \(\frac {\frac {b^{3}}{a^{4}}+\frac {x^{9}}{6 a}-\frac {b \,x^{6}}{2 a^{2}}}{a \,x^{3}+b}+\frac {b^{2} \ln \left (a \,x^{3}+b \right )}{a^{4}}\) | \(52\) |
default | \(\frac {\left (a \,x^{3}-2 b \right )^{2}}{6 a^{4}}+\frac {b^{2} \left (\frac {b}{a \left (a \,x^{3}+b \right )}+\frac {3 \ln \left (a \,x^{3}+b \right )}{a}\right )}{3 a^{3}}\) | \(54\) |
risch | \(\frac {x^{6}}{6 a^{2}}-\frac {2 b \,x^{3}}{3 a^{3}}+\frac {2 b^{2}}{3 a^{4}}+\frac {b^{3}}{3 a^{4} \left (a \,x^{3}+b \right )}+\frac {b^{2} \ln \left (a \,x^{3}+b \right )}{a^{4}}\) | \(59\) |
parallelrisch | \(\frac {a^{3} x^{9}-3 a^{2} b \,x^{6}+6 \ln \left (a \,x^{3}+b \right ) x^{3} a \,b^{2}+6 \ln \left (a \,x^{3}+b \right ) b^{3}+6 b^{3}}{6 a^{4} \left (a \,x^{3}+b \right )}\) | \(67\) |
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Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.25 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {a^{3} x^{9} - 3 \, a^{2} b x^{6} - 4 \, a b^{2} x^{3} + 2 \, b^{3} + 6 \, {\left (a b^{2} x^{3} + b^{3}\right )} \log \left (a x^{3} + b\right )}{6 \, {\left (a^{5} x^{3} + a^{4} b\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {b^{3}}{3 a^{5} x^{3} + 3 a^{4} b} + \frac {x^{6}}{6 a^{2}} - \frac {2 b x^{3}}{3 a^{3}} + \frac {b^{2} \log {\left (a x^{3} + b \right )}}{a^{4}} \]
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none
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {b^{3}}{3 \, {\left (a^{5} x^{3} + a^{4} b\right )}} + \frac {b^{2} \log \left (a x^{3} + b\right )}{a^{4}} + \frac {a x^{6} - 4 \, b x^{3}}{6 \, a^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {b^{2} \log \left ({\left | a x^{3} + b \right |}\right )}{a^{4}} + \frac {b^{3}}{3 \, {\left (a x^{3} + b\right )} a^{4}} + \frac {a^{2} x^{6} - 4 \, a b x^{3}}{6 \, a^{4}} \]
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Time = 5.60 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {x^6}{6\,a^2}+\frac {b^3}{3\,a\,\left (a^4\,x^3+b\,a^3\right )}-\frac {2\,b\,x^3}{3\,a^3}+\frac {b^2\,\ln \left (a\,x^3+b\right )}{a^4} \]
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