Integrand size = 24, antiderivative size = 96 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {455, 57, 631, 210, 31} \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
[In]
[Out]
Rule 31
Rule 57
Rule 210
Rule 455
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{a+b x}} \, dx,x,x^3\right ) \\ & = \frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b)^{2/3}+\sqrt [3]{a+b} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a+b}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \\ & = \frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}\right )}{\sqrt [3]{a+b}} \\ & = -\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.90 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=\frac {2 \sqrt {-6+6 i \sqrt {3}} \arctan \left (\frac {1+\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )-i \left (-i+\sqrt {3}\right ) \left (\log \left (\left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right ) \left (2 \sqrt [3]{a+b}+\sqrt [3]{a+b x^3}-i \sqrt {3} \sqrt [3]{a+b x^3}\right )\right )-2 \log \left (2 \sqrt [3]{a+b}+\left (1+i \sqrt {3}\right ) \sqrt [3]{a+b x^3}\right )\right )}{12 \sqrt [3]{a+b}} \]
[In]
[Out]
Time = 0.91 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {\arctan \left (\frac {\left (\left (a +b \right )^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 \left (a +b \right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (a +b \right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (a +b \right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (a +b \right )^{\frac {2}{3}}\right )}{2}}{3 \left (a +b \right )^{\frac {1}{3}}}\) | \(92\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (75) = 150\).
Time = 0.26 (sec) , antiderivative size = 387, normalized size of antiderivative = 4.03 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )} - {\left (a + b\right )} {\left (-a - b\right )}^{\frac {1}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}}\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} + 3 \, a - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + b}{x^{3} - 1}\right ) + {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {2}{3}}\right ) - 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {1}{3}}\right )}{6 \, {\left (a + b\right )}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left (-a - b\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}}\right ) - {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {1}{3}}\right )}{6 \, {\left (a + b\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=- \int \frac {x^{2}}{x^{3} \sqrt [3]{a + b x^{3}} - \sqrt [3]{a + b x^{3}}}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a + b\right )}^{\frac {1}{3}}}\right )}{{\left (a + b\right )}^{\frac {1}{3}}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {2}{3}}\right )}{{\left (a + b\right )}^{\frac {1}{3}}} + \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left (a + b\right )}^{\frac {1}{3}}\right )}{{\left (a + b\right )}^{\frac {1}{3}}}}{6 \, b} \]
[In]
[Out]
none
Time = 3.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {{\left (a + b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a + b\right )}^{\frac {1}{3}}}\right )}{\sqrt {3} a + \sqrt {3} b} + \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {2}{3}}\right )}{6 \, {\left (a + b\right )}^{\frac {1}{3}}} - \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left (a + b\right )}^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a + b\right )}^{\frac {1}{3}}} \]
[In]
[Out]
Time = 0.61 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.64 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}-\frac {9\,a+9\,b}{9\,{\left (-a-b\right )}^{2/3}}\right )}{3\,{\left (-a-b\right )}^{1/3}}+\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,a+9\,b\right )}{36\,{\left (-a-b\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a-b\right )}^{1/3}}-\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,a+9\,b\right )}{36\,{\left (-a-b\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a-b\right )}^{1/3}} \]
[In]
[Out]