Optimal. Leaf size=145 \[ \frac {1}{162} (-14 a-81 b) \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {1}{162} \left (14 \sqrt {3} a+81 \sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )+\frac {1}{324} (14 a+81 b) \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right )+\frac {1}{108} \left (x^3-x\right )^{2/3} \left (18 a x^4+21 a x^2+28 a\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 271, normalized size of antiderivative = 1.87, number of steps used = 13, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2053, 2011, 329, 275, 239, 2024} \begin {gather*} \frac {7}{27} a \left (x^3-x\right )^{2/3}+\frac {1}{6} a \left (x^3-x\right )^{2/3} x^4+\frac {7}{36} a \left (x^3-x\right )^{2/3} x^2-\frac {7 a \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{54 \sqrt [3]{x^3-x}}+\frac {7 a \sqrt [3]{x^2-1} \sqrt [3]{x} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{27 \sqrt {3} \sqrt [3]{x^3-x}}-\frac {3 b \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}}+\frac {\sqrt {3} b \sqrt [3]{x^2-1} \sqrt [3]{x} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 275
Rule 329
Rule 2011
Rule 2024
Rule 2053
Rubi steps
\begin {align*} \int \frac {b+a x^6}{\sqrt [3]{-x+x^3}} \, dx &=\int \left (\frac {b}{\sqrt [3]{-x+x^3}}+\frac {a x^6}{\sqrt [3]{-x+x^3}}\right ) \, dx\\ &=a \int \frac {x^6}{\sqrt [3]{-x+x^3}} \, dx+b \int \frac {1}{\sqrt [3]{-x+x^3}} \, dx\\ &=\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {1}{9} (7 a) \int \frac {x^4}{\sqrt [3]{-x+x^3}} \, dx+\frac {\left (b \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 {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {1}{27} (14 a) \int \frac {x^2}{\sqrt [3]{-x+x^3}} \, dx+\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^3}}\\ &=\frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {1}{81} (14 a) \int \frac {1}{\sqrt [3]{-x+x^3}} \, dx+\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 b \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}}+\frac {\left (14 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \, dx}{81 \sqrt [3]{-x+x^3}}\\ &=\frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 b \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}}+\frac {\left (14 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{27 \sqrt [3]{-x+x^3}}\\ &=\frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {3 b \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}}+\frac {\left (7 a \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{27 \sqrt [3]{-x+x^3}}\\ &=\frac {7}{27} a \left (-x+x^3\right )^{2/3}+\frac {7}{36} a x^2 \left (-x+x^3\right )^{2/3}+\frac {1}{6} a x^4 \left (-x+x^3\right )^{2/3}+\frac {7 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{27 \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{-x+x^3}}-\frac {7 a \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{54 \sqrt [3]{-x+x^3}}-\frac {3 b \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*}
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Mathematica [A] time = 0.15, size = 220, normalized size = 1.52 \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \left (-2 (14 a+81 b) \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )+2 \sqrt {3} (14 a+81 b) \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )+54 a \left (x^2-1\right )^{2/3} x^{14/3}+63 a \left (x^2-1\right )^{2/3} x^{8/3}+84 a \left (x^2-1\right )^{2/3} x^{2/3}+14 a \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )+81 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 )\right )}{324 \sqrt [3]{x \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 145, normalized size = 1.00 \begin {gather*} \frac {1}{108} \left (-x+x^3\right )^{2/3} \left (28 a+21 a x^2+18 a x^4\right )+\frac {1}{162} \left (14 \sqrt {3} a+81 \sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )+\frac {1}{162} (-14 a-81 b) \log \left (-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{324} (14 a+81 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 = 156.26, size = 128, normalized size = 0.88 \begin {gather*} \frac {1}{162} \, \sqrt {3} {\left (14 \, a + 81 \, b\right )} \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 ) - \frac {1}{324} \, {\left (14 \, a + 81 \, b\right )} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + \frac {1}{108} \, {\left (18 \, a x^{4} + 21 \, a x^{2} + 28 \, a\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 133, normalized size = 0.92 \begin {gather*} \frac {1}{108} \, {\left (28 \, a {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} - 77 \, a {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} + 67 \, a {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )} x^{6} - \frac {1}{162} \, \sqrt {3} {\left (14 \, a + 81 \, 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}{324} \, {\left (14 \, a + 81 \, 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}{162} \, {\left (14 \, a + 81 \, b\right )} \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.70, size = 68, normalized size = 0.47
method | result | size |
meijerg | \(\frac {3 a \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {20}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {10}{3}\right ], \left [\frac {13}{3}\right ], x^{2}\right )}{20 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {3 b \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\) | \(68\) |
risch | \(\frac {a \left (18 x^{4}+21 x^{2}+28\right ) x \left (x^{2}-1\right )}{108 \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}+\frac {3 b \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {7 a \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right )}{27 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\) | \(98\) |
trager | \(\frac {a \left (18 x^{4}+21 x^{2}+28\right ) \left (x^{3}-x \right )^{\frac {2}{3}}}{108}-\frac {\left (14 a +81 b \right ) \left (18 \ln \left (-228744 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2} x^{2}+40608 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+96390 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -124290 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) x^{2}+914976 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+1550 x^{2}+26964 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )-465\right ) \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )+\ln \left (178524 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2} x^{2}-40608 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+136998 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -106308 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) x^{2}-714096 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-1705 x^{2}+58266 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )+1085\right )-\ln \left (-228744 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2} x^{2}+40608 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+96390 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -124290 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right ) x^{2}+914976 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+1550 x^{2}+26964 \RootOf \left (324 \textit {\_Z}^{2}-18 \textit {\_Z} +1\right )-465\right )\right )}{162}\) | \(465\) |
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 {a x^{6} + b}{{\left (x^{3} - x\right )}^{\frac {1}{3}}}\,{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 {a\,x^6+b}{{\left (x^3-x\right )}^{1/3}} \,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 {a x^{6} + b}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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