Optimal. Leaf size=138 \[ \frac {1}{6} (a-3 b) \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {1}{6} \left (\sqrt {3} a-3 \sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )+\frac {\sqrt [3]{x^3-x} \left (a x^2-3 b\right )}{2 x}+\frac {1}{12} (3 b-a) \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 225, normalized size of antiderivative = 1.63, number of steps used = 12, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2038, 2004, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \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 (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{6 \left (x^3-x\right )^{2/3}}-\frac {x^{2/3} \left (x^2-1\right )^{2/3} (a-3 b) \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{12 \left (x^3-x\right )^{2/3}}+\frac {x^{2/3} \left (x^2-1\right )^{2/3} (a-3 b) \tan ^{-1}\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 {3 b \left (x^3-x\right )^{4/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2004
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (b+a x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\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 {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}-\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \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} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}+\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (-x+x^3\right )^{2/3}}-\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \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} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}-\frac {(a-3 b) x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (-x+x^3\right )^{2/3}}+\frac {\left ((-a+3 b) x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\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} \tan ^{-1}\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 (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}-\frac {(a-3 b) x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (-x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 66, normalized size = 0.48 \begin {gather*} \frac {3 \sqrt [3]{x \left (x^2-1\right )} \left (x^2 (a-3 b) \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^2\right )-2 b \left (1-x^2\right )^{4/3}\right )}{4 x \sqrt [3]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 138, normalized size = 1.00 \begin {gather*} \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 ) \tan ^{-1}\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 ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 78.70, size = 123, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {3} {\left (a - 3 \, b\right )} x \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 ) + {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 105, normalized size = 0.76 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.97, size = 68, normalized size = 0.49
method | result | size |
meijerg | \(\frac {3 a \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {4}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}}}-\frac {3 b \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{2}\right )}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}}}\) | \(68\) |
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 (12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (22320 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}+27072 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+27072 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +28932 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-89280 \RootOf \left (144 \textit {\_Z}^{2}+12 \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}-46908 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-4942\right )-12 \ln \left (22320 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-27072 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -25212 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-89280 \RootOf \left (144 \textit {\_Z}^{2}+12 \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}+32028 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-1653\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-\ln \left (22320 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-27072 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -25212 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-89280 \RootOf \left (144 \textit {\_Z}^{2}+12 \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}+32028 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-1653\right )\right )}{6}\) | \(475\) |
risch | \(\frac {\left (a \,x^{2}-3 b \right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}{2 x}+\frac {\left (a -3 b \right ) \left (\RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (\frac {59 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-3750 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-1746 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-295 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+12600 x^{4}+5850 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+5652 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+24624 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+1746 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+236 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-16380 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-1902 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+3780}{\left (-1+x \right ) \left (1+x \right )}\right )+6 \ln \left (-\frac {-35 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-1956 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-4104 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+175 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+23364 x^{4}+5850 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+2010 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+10476 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+4104 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-140 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-38232 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-54 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+14868}{\left (-1+x \right ) \left (1+x \right )}\right )\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{36 x \left (x^{2}-1\right )}\) | \(552\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + b\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-x\right )}^{1/3}\,\left (a\,x^2+b\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (a x^{2} + b\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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