Optimal. Leaf size=267 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{5/8} b^{3/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{5/8} b^{3/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{5/8} b^{3/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{5/8} b^{3/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}} \]
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Rubi [A] time = 0.19443, antiderivative size = 267, normalized size of antiderivative = 1., 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 = {300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{5/8} b^{3/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{5/8} b^{3/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{5/8} b^{3/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{5/8} b^{3/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{5/8} b^{3/8}} \]
Antiderivative was successfully verified.
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Rule 300
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 298
Rule 205
Rule 208
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{\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)^{5/8} b^{3/8}}+\frac{\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)^{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*}
Mathematica [A] time = 0.081286, size = 324, normalized size = 1.21 \[ -\frac{-\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 a^{5/8} b^{3/8}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 27, normalized size = 0.1 \begin{align*}{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b x^{8} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42879, size = 1226, normalized size = 4.59 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.186023, size = 27, normalized size = 0.1 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28046, size = 579, normalized size = 2.17 \begin{align*} -\frac{\sqrt{-\sqrt{2} + 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 )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 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 )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 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 )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 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 )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \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 )}{16 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \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 )}{16 \, a} - \frac{\sqrt{\sqrt{2} + 2} \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 )}{16 \, a} + \frac{\sqrt{\sqrt{2} + 2} \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 )}{16 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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