Optimal. Leaf size=267 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}} \]
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Rubi [A]
time = 0.13, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {306, 303,
1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}+\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}+\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 303
Rule 304
Rule 306
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^2}{a+b x^8} \, dx &=-\frac {\int \frac {x^2}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 \sqrt {-a}}\\ &=-\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 \sqrt {-a} \sqrt [4]{b}}+\frac {\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 \sqrt {-a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 \sqrt {-a} \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 \sqrt {-a} \sqrt [4]{b}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}-\frac {\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt {-a} \sqrt {b}}-\frac {\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt {-a} \sqrt {b}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{b}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{b}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{5/8} b^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}-\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}+\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{5/8} b^{3/8}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 324, normalized size = 1.21 \begin {gather*} -\frac {2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \cos \left (\frac {\pi }{8}\right )-2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \cos \left (\frac {\pi }{8}\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+2 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 a^{5/8} b^{3/8}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.17, size = 27, normalized size = 0.10
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{8 b}\) | \(27\) |
risch | \(\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{8 b}\) | \(27\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 453 vs.
\(2 (182) = 364\).
time = 0.39, size = 453, normalized size = 1.70 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \arctan \left (-\sqrt {2} a^{3} b^{2} x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {5}{8}} + \sqrt {2} \sqrt {a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} + \sqrt {2} a^{2} b x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{8}} + x^{2}} a^{3} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {5}{8}} + 1\right ) - \frac {1}{4} \, \sqrt {2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \arctan \left (-\sqrt {2} a^{3} b^{2} x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {5}{8}} + \sqrt {2} \sqrt {a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} - \sqrt {2} a^{2} b x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{8}} + x^{2}} a^{3} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {5}{8}} - 1\right ) + \frac {1}{16} \, \sqrt {2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \log \left (a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} + \sqrt {2} a^{2} b x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{8}} + x^{2}\right ) - \frac {1}{16} \, \sqrt {2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \log \left (a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} - \sqrt {2} a^{2} b x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{8}} + x^{2}\right ) + \frac {1}{2} \, \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \arctan \left (-a^{3} b^{2} x \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {5}{8}} + \sqrt {a^{4} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{4}} + x^{2}} a^{3} b^{2} \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {5}{8}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \log \left (a^{2} b \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{8}} + x\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {1}{8}} \log \left (-a^{2} b \left (-\frac {1}{a^{5} b^{3}}\right )^{\frac {3}{8}} + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 27, normalized size = 0.10 \begin {gather*} \operatorname {RootSum} {\left (16777216 t^{8} a^{5} b^{3} + 1, \left ( t \mapsto t \log {\left (- 512 t^{3} a^{2} b + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs.
\(2 (182) = 364\).
time = 1.45, size = 437, normalized size = 1.64 \begin {gather*} -\frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {3}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 110, normalized size = 0.41 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b^{1/8}\,x}{{\left (-a\right )}^{1/8}}\right )}{4\,{\left (-a\right )}^{5/8}\,b^{3/8}}+\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{5/8}\,b^{3/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,b^{3/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,b^{3/8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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