Integrand size = 22, antiderivative size = 129 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=\frac {\left (3 b+a x^2\right ) \sqrt [3]{x+x^3}}{2 x}+\frac {1}{6} \left (-\sqrt {3} a+3 \sqrt {3} b\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )+\frac {1}{6} (-a+3 b) \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{12} (a-3 b) \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2063, 2029, 2057, 335, 281, 337} \[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=-\frac {x^{2/3} \left (x^2+1\right )^{2/3} (a-3 b) \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x^3+x\right )^{2/3}}+\frac {1}{2} x \sqrt [3]{x^3+x} (a-3 b)-\frac {x^{2/3} \left (x^2+1\right )^{2/3} (a-3 b) \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \left (x^3+x\right )^{2/3}}+\frac {3 b \left (x^3+x\right )^{4/3}}{2 x^2} \]
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Rule 281
Rule 335
Rule 337
Rule 2029
Rule 2057
Rule 2063
Rubi steps \begin{align*} \text {integral}& = \frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-(-a+3 b) \int \sqrt [3]{x+x^3} \, dx \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {1}{3} (-a+3 b) \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{3 \left (x+x^3\right )^{2/3}} \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x+x^3\right )^{2/3}} \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (x+x^3\right )^{2/3}} \\ & = \frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x+x^3\right )^{2/3}}-\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 \left (x+x^3\right )^{2/3}} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.62 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=\frac {\left (3 b+a x^2\right ) \sqrt [3]{x+x^3}}{2 x}-\frac {(a-3 b) \sqrt [3]{x+x^3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )}{2 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {(-a+3 b) \sqrt [3]{x+x^3} \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {(a-3 b) \sqrt [3]{x+x^3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )}{12 \sqrt [3]{x} \sqrt [3]{1+x^2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(\frac {3 a \,x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}+\frac {3 b \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{2}\right )}{2 x^{\frac {2}{3}}}\) | \(36\) |
pseudoelliptic | \(\frac {3 \left (-a \,x^{2}-3 b \right ) {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+\left (a -3 b \right ) \left (-\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right ) x}{6 \left (-{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}\right )}\) | \(150\) |
trager | \(\frac {\left (a \,x^{2}+3 b \right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{2 x}+\frac {\left (a -3 b \right ) \left (3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-4 x^{2}+48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3\right )-3 \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right )\right )}{6}\) | \(447\) |
risch | \(\text {Expression too large to display}\) | \(745\) |
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Time = 40.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=-\frac {2 \, \sqrt {3} {\left (a - 3 \, b\right )} x \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + {\left (a - 3 \, b\right )} x \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) - 6 \, {\left (a x^{2} + 3 \, b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{12 \, x} \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{2} - b\right )}{x^{2}}\, dx \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=\int { \frac {{\left (a x^{2} - b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{x^{2}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=\frac {1}{2} \, a x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{12} \, {\left (a - 3 \, b\right )} \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, {\left (a - 3 \, b\right )} \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) + \frac {3}{2} \, b {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \]
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Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx=-\int \frac {\left (b-a\,x^2\right )\,{\left (x^3+x\right )}^{1/3}}{x^2} \,d x \]
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