Integrand size = 24, antiderivative size = 131 \[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=-\frac {3 b \left (-x+x^3\right )^{2/3}}{4 x^2}+\frac {1}{4} \left (a-i \sqrt {3} a\right ) \log \left (-i x+\sqrt {3} x-2 i \sqrt [3]{-x+x^3}\right )+\frac {1}{4} \left (a+i \sqrt {3} a\right ) \log \left (i x+\sqrt {3} x+2 i \sqrt [3]{-x+x^3}\right )-\frac {1}{2} a \log \left (-x+\sqrt [3]{-x+x^3}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2063, 2036, 335, 281, 245} \[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{x^2-1} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-x}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}}-\frac {3 b \left (x^3-x\right )^{2/3}}{4 x^2} \]
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Rule 245
Rule 281
Rule 335
Rule 2036
Rule 2063
Rubi steps \begin{align*} \text {integral}& = -\frac {3 b \left (-x+x^3\right )^{2/3}}{4 x^2}+a \int \frac {1}{\sqrt [3]{-x+x^3}} \, dx \\ & = -\frac {3 b \left (-x+x^3\right )^{2/3}}{4 x^2}+\frac {\left (a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x+x^3}} \\ & = -\frac {3 b \left (-x+x^3\right )^{2/3}}{4 x^2}+\frac {\left (3 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}} \\ & = -\frac {3 b \left (-x+x^3\right )^{2/3}}{4 x^2}+\frac {\left (3 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}} \\ & = -\frac {3 b \left (-x+x^3\right )^{2/3}}{4 x^2}+\frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{-1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \sqrt [3]{-x+x^3}} \\ \end{align*}
Time = 1.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.24 \[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3 b-3 b x^2+2 \sqrt {3} a x^{4/3} \sqrt [3]{-1+x^2} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )-2 a x^{4/3} \sqrt [3]{-1+x^2} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )+a x^{4/3} \sqrt [3]{-1+x^2} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )}{4 x \sqrt [3]{x \left (-1+x^2\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.98 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {3 b \left (x^{2}-1\right )}{4 x {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}+\frac {3 a {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\) | \(55\) |
meijerg | \(\frac {3 a {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {3 b {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {2}{3}}}{4 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {4}{3}}}\) | \(68\) |
pseudoelliptic | \(\frac {-3 b \left (x^{3}-x \right )^{\frac {2}{3}}-2 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right ) x^{2} a}{4 x^{2}}\) | \(101\) |
trager | \(-\frac {3 b \left (x^{3}-x \right )^{\frac {2}{3}}}{4 x^{2}}+\frac {a \left (72 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \ln \left (803520 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}+162432 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+162432 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +151272 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}-3214080 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}-192168 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-1653\right )-72 \ln \left (803520 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-162432 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-162432 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -173592 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}-3214080 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+7766 x^{2}+281448 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-4942\right ) \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )+\ln \left (803520 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2} x^{2}-162432 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-162432 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -173592 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right ) x^{2}-3214080 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+7766 x^{2}+281448 \operatorname {RootOf}\left (5184 \textit {\_Z}^{2}-72 \textit {\_Z} +1\right )-4942\right )\right )}{2}\) | \(461\) |
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Time = 0.88 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {2 \, \sqrt {3} a x^{2} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) - a x^{2} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} b}{4 \, x^{2}} \]
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\[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=\int \frac {a x^{2} - b}{x^{2} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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\[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=\int { \frac {a x^{2} - b}{{\left (x^{3} - x\right )}^{\frac {1}{3}} x^{2}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=-\frac {1}{2} \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, a \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, a \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) - \frac {3}{4} \, b {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \]
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Time = 5.73 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.35 \[ \int \frac {-b+a x^2}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3\,a\,x\,{\left (1-x^2\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x^2\right )}{2\,{\left (x^3-x\right )}^{1/3}}-\frac {3\,b\,{\left (x^3-x\right )}^{2/3}}{4\,x^2} \]
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