Integrand size = 13, antiderivative size = 46 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {x^3}{3 a^2}-\frac {b^2}{3 a^3 \left (b+a x^3\right )}-\frac {2 b \log \left (b+a x^3\right )}{3 a^3} \]
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Time = 0.02 (sec) , antiderivative size = 46, 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} \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=-\frac {b^2}{3 a^3 \left (a x^3+b\right )}-\frac {2 b \log \left (a x^3+b\right )}{3 a^3}+\frac {x^3}{3 a^2} \]
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Rule 45
Rule 269
Rule 272
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^8}{\left (b+a x^3\right )^2} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(b+a x)^2} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{a^2}+\frac {b^2}{a^2 (b+a x)^2}-\frac {2 b}{a^2 (b+a x)}\right ) \, dx,x,x^3\right ) \\ & = \frac {x^3}{3 a^2}-\frac {b^2}{3 a^3 \left (b+a x^3\right )}-\frac {2 b \log \left (b+a x^3\right )}{3 a^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {a x^3-\frac {b^2}{b+a x^3}-2 b \log \left (b+a x^3\right )}{3 a^3} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {x^{3}}{3 a^{2}}-\frac {b^{2}}{3 a^{3} \left (a \,x^{3}+b \right )}-\frac {2 b \ln \left (a \,x^{3}+b \right )}{3 a^{3}}\) | \(41\) |
norman | \(\frac {\frac {x^{6}}{3 a}-\frac {2 b^{2}}{3 a^{3}}}{a \,x^{3}+b}-\frac {2 b \ln \left (a \,x^{3}+b \right )}{3 a^{3}}\) | \(43\) |
default | \(\frac {x^{3}}{3 a^{2}}-\frac {b \left (\frac {b}{a \left (a \,x^{3}+b \right )}+\frac {2 \ln \left (a \,x^{3}+b \right )}{a}\right )}{3 a^{2}}\) | \(44\) |
parallelrisch | \(-\frac {-a^{2} x^{6}+2 \ln \left (a \,x^{3}+b \right ) x^{3} a b +2 b^{2} \ln \left (a \,x^{3}+b \right )+2 b^{2}}{3 a^{3} \left (a \,x^{3}+b \right )}\) | \(57\) |
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {a^{2} x^{6} + a b x^{3} - b^{2} - 2 \, {\left (a b x^{3} + b^{2}\right )} \log \left (a x^{3} + b\right )}{3 \, {\left (a^{4} x^{3} + a^{3} b\right )}} \]
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Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=- \frac {b^{2}}{3 a^{4} x^{3} + 3 a^{3} b} + \frac {x^{3}}{3 a^{2}} - \frac {2 b \log {\left (a x^{3} + b \right )}}{3 a^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=-\frac {b^{2}}{3 \, {\left (a^{4} x^{3} + a^{3} b\right )}} + \frac {x^{3}}{3 \, a^{2}} - \frac {2 \, b \log \left (a x^{3} + b\right )}{3 \, a^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {x^{3}}{3 \, a^{2}} - \frac {2 \, b \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{3}} - \frac {b^{2}}{3 \, {\left (a x^{3} + b\right )} a^{3}} \]
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Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\left (a+\frac {b}{x^3}\right )^2} \, dx=\frac {x^3}{3\,a^2}-\frac {b^2}{3\,\left (a^4\,x^3+b\,a^3\right )}-\frac {2\,b\,\ln \left (a\,x^3+b\right )}{3\,a^3} \]
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