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 = 0.232816, antiderivative size = 277, normalized size of antiderivative = 1., 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 325
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{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*}
Mathematica [A] time = 0.0946008, size = 395, normalized size = 1.43 \[ \frac{-8 a^{5/8}-5 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+5 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+5 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-5 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+10 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+10 b^{5/8} x^5 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+10 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-10 b^{5/8} x^5 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{40 a^{13/8} x^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.004, size = 36, normalized size = 0.1 \begin{align*} -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}}}}-{\frac{1}{5\,a{x}^{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*} -\frac{-\frac{1}{16} \, b{\left (\frac{2 \, \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 )}{a} + \frac{2 \, \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 )}{a} - \frac{2 \, \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 )}{a} - \frac{2 \, \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 )}{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 )}{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 )}{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 )}{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 )}{a}\right )}}{a} - \frac{1}{5 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49398, size = 1241, normalized size = 4.48 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0872, size = 36, normalized size = 0.13 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2022, size = 601, normalized size = 2.17 \begin{align*} \frac{b \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^{2}} + \frac{b \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^{2}} - \frac{b \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^{2}} - \frac{b \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^{2}} - \frac{b \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^{2}} + \frac{b \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^{2}} + \frac{b \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^{2}} - \frac{b \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^{2}} - \frac{1}{5 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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