Integrand size = 13, antiderivative size = 134 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=-\frac {x}{3 a \left (b+a x^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{4/3} b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{4/3} b^{2/3}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {269, 294, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3}}-\frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{4/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{4/3} b^{2/3}}-\frac {x}{3 a \left (a x^3+b\right )} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 269
Rule 294
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{\left (b+a x^3\right )^2} \, dx \\ & = -\frac {x}{3 a \left (b+a x^3\right )}+\frac {\int \frac {1}{b+a x^3} \, dx}{3 a} \\ & = -\frac {x}{3 a \left (b+a x^3\right )}+\frac {\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 a b^{2/3}}+\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 a b^{2/3}} \\ & = -\frac {x}{3 a \left (b+a x^3\right )}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{4/3} b^{2/3}}-\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{18 a^{4/3} b^{2/3}}+\frac {\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a \sqrt [3]{b}} \\ & = -\frac {x}{3 a \left (b+a x^3\right )}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{4/3} b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 a^{4/3} b^{2/3}} \\ & = -\frac {x}{3 a \left (b+a x^3\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{4/3} b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{4/3} b^{2/3}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=\frac {-\frac {6 \sqrt [3]{a} x}{b+a x^3}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{b^{2/3}}}{18 a^{4/3}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.32
method | result | size |
risch | \(-\frac {x}{3 a \left (a \,x^{3}+b \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 a^{2}}\) | \(43\) |
default | \(-\frac {x}{3 a \left (a \,x^{3}+b \right )}+\frac {\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}}{3 a}\) | \(112\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.92 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=\left [-\frac {6 \, a b^{2} x - 3 \, \sqrt {\frac {1}{3}} {\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b x - b^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a x^{3} + b}\right ) + {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) - 2 \, {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3}\right )}}, -\frac {6 \, a b^{2} x - 6 \, \sqrt {\frac {1}{3}} {\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) + {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) - 2 \, {\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3}\right )}}\right ] \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=- \frac {x}{3 a^{2} x^{3} + 3 a b} + \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{2} - 1, \left ( t \mapsto t \log {\left (9 t a b + x \right )} \right )\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=-\frac {x}{3 \, {\left (a^{2} x^{3} + a b\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=-\frac {\left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b} - \frac {x}{3 \, {\left (a x^{3} + b\right )} a} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b} + \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b} \]
[In]
[Out]
Time = 5.99 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^3} \, dx=\frac {\ln \left (a^{1/3}\,x+b^{1/3}\right )}{9\,a^{4/3}\,b^{2/3}}-\frac {x}{3\,a\,\left (a\,x^3+b\right )}+\frac {\ln \left (a\,x+\frac {a^{2/3}\,b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}}-\frac {\ln \left (a\,x-\frac {a^{2/3}\,b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}} \]
[In]
[Out]