Optimal. Leaf size=277 \[ -\frac {b^{5/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}+\frac {b^{5/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {1}{5 a x^5} \]
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Rubi [A] time = 0.23, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {325, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac {b^{5/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}+\frac {b^{5/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {1}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 297
Rule 298
Rule 300
Rule 325
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^8\right )} \, dx &=-\frac {1}{5 a x^5}-\frac {b \int \frac {x^2}{a+b x^8} \, dx}{a}\\ &=-\frac {1}{5 a x^5}-\frac {b \int \frac {x^2}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 (-a)^{3/2}}-\frac {b \int \frac {x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 (-a)^{3/2}}\\ &=-\frac {1}{5 a x^5}-\frac {b^{3/4} \int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 (-a)^{3/2}}+\frac {b^{3/4} \int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 (-a)^{3/2}}+\frac {b^{3/4} \int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 (-a)^{3/2}}-\frac {b^{3/4} \int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 (-a)^{3/2}}\\ &=-\frac {1}{5 a x^5}+\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {b^{5/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {\sqrt {b} \int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 (-a)^{3/2}}-\frac {\sqrt {b} \int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 (-a)^{3/2}}-\frac {b^{5/8} \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)^{13/8}}-\frac {b^{5/8} \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)^{13/8}}\\ &=-\frac {1}{5 a x^5}+\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {b^{5/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {b^{5/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \operatorname {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)^{13/8}}+\frac {b^{5/8} \operatorname {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)^{13/8}}\\ &=-\frac {1}{5 a x^5}+\frac {b^{5/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}+\frac {b^{5/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{13/8}}-\frac {b^{5/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{13/8}}-\frac {b^{5/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}+\frac {b^{5/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{13/8}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 395, normalized size = 1.43 \[ \frac {-8 a^{5/8}+10 b^{5/8} x^5 \sin \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )+10 b^{5/8} x^5 \sin \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )+10 b^{5/8} x^5 \cos \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )-10 b^{5/8} x^5 \cos \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac {\pi }{8}\right )\right )-5 b^{5/8} x^5 \cos \left (\frac {\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+5 b^{5/8} x^5 \cos \left (\frac {\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+5 b^{5/8} x^5 \sin \left (\frac {\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-5 b^{5/8} x^5 \sin \left (\frac {\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{40 a^{13/8} x^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 524, normalized size = 1.89 \[ \frac {20 \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{8} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {5}{8}} - \sqrt {2} a^{8} \sqrt {\frac {a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} + \sqrt {2} a^{5} b^{2} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{4} x^{2}}{b^{4}}} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {5}{8}} - b^{3}}{b^{3}}\right ) + 20 \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{8} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {5}{8}} - \sqrt {2} a^{8} \sqrt {\frac {a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} - \sqrt {2} a^{5} b^{2} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{4} x^{2}}{b^{4}}} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {5}{8}} + b^{3}}{b^{3}}\right ) - 5 \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} + \sqrt {2} a^{5} b^{2} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{4} x^{2}\right ) + 5 \, \sqrt {2} a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} - \sqrt {2} a^{5} b^{2} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{4} x^{2}\right ) - 40 \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a^{8} x \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {5}{8}} - a^{8} \sqrt {\frac {a^{10} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{4}} + b^{4} x^{2}}{b^{4}}} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {5}{8}}}{b^{3}}\right ) + 10 \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) - 10 \, a x^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {1}{8}} \log \left (-a^{5} \left (-\frac {b^{5}}{a^{13}}\right )^{\frac {3}{8}} + b^{2} x\right ) - 16}{80 \, a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 453, normalized size = 1.64 \[ \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {1}{5 \, a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 36, normalized size = 0.13 \[ -\frac {\ln \left (-\RootOf \left (b \,\textit {\_Z}^{8}+a \right )+x \right )}{8 a \RootOf \left (b \,\textit {\_Z}^{8}+a \right )^{5}}-\frac {1}{5 a \,x^{5}} \]
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 \[ -\frac {-\frac {1}{8} \, b {\left (\frac {2 \, \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 )}{a \sqrt {2 \, \sqrt {2} + 4}} + \frac {2 \, \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 )}{a \sqrt {2 \, \sqrt {2} + 4}} - \frac {2 \, \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 )}{a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {2 \, \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 )}{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 )}{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 )}{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 )}{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 )}{a \sqrt {-2 \, \sqrt {2} + 4}}\right )}}{a} - \frac {1}{5 \, a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 118, normalized size = 0.43 \[ -\frac {1}{5\,a\,x^5}-\frac {{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{13/8}}-\frac {{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{13/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{13/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{5/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{13/8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 36, normalized size = 0.13 \[ \operatorname {RootSum} {\left (16777216 t^{8} a^{13} + b^{5}, \left (t \mapsto t \log {\left (\frac {512 t^{3} a^{5}}{b^{2}} + x \right )} \right )\right )} - \frac {1}{5 a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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