Integrand size = 29, antiderivative size = 68 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {\text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(633\) vs. \(2(68)=136\).
Time = 0.24 (sec) , antiderivative size = 633, normalized size of antiderivative = 9.31, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2081, 926, 93} \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {-a} x+\sqrt {b}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}} \]
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Rule 93
Rule 926
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+a x^2\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = -\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{2 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}+\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
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Time = 1.42 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{2 b}\) | \(59\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.29 (sec) , antiderivative size = 1358, normalized size of antiderivative = 19.97 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 1.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{2} + b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 0.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 5.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (a\,x^2+b\right )} \,d x \]
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