Integrand size = 37, antiderivative size = 204 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}-\frac {1}{4} \text {RootSum}\left [a^6+2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(874\) vs. \(2(204)=408\).
Time = 0.56 (sec) , antiderivative size = 874, normalized size of antiderivative = 4.28, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2081, 6857, 61, 926, 93} \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log (x)}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {b}-\sqrt {2} \sqrt {-a} x\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {2} \sqrt {-a} x+\sqrt {b}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{a \sqrt [3]{x}}-1\right )}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}} \]
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Rule 61
Rule 93
Rule 926
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {b+a x^2}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+2 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 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {b}{2 x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+2 a x^2\right )}\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} \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (b 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+2 a x^2\right )} \, dx}{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} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (b 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 {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{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} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \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} a \sqrt [3]{x}}\right )}{2 a \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 {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a^3-\sqrt {2} \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 {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a^3+\sqrt {2} \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 (x)}{4 a \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 {2} \sqrt {-a} x\right )}{8 \sqrt [3]{a^3+\sqrt {2} \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 {2} \sqrt {-a} x\right )}{8 \sqrt [3]{a^3-\sqrt {2} \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 {2} \sqrt {-a} b^{3/2}}}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \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 {2} \sqrt {-a} b^{3/2}}}\right )}{8 \sqrt [3]{a^3+\sqrt {2} \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 (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.11 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )-2 \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )\right )+\log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )-a \text {RootSum}\left [a^6+2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 a \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
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Time = 0.00 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+2 a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) a -2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 a}\) | \(175\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 1462, normalized size of antiderivative = 7.17 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 2.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.15 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a x^{2} + b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{2} + b\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.18 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} + b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 1.80 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.18 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} + b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.18 \[ \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a\,x^2+b}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^2+b\right )} \,d x \]
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