Optimal. Leaf size=267 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt [8]{-a} b^{7/8}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt [8]{-a} b^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt [8]{-a} b^{7/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} \sqrt [8]{-a} b^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} \sqrt [8]{-a} b^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt [8]{-a} b^{7/8}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.466515, 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 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt [8]{-a} b^{7/8}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt [8]{-a} b^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt [8]{-a} b^{7/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} \sqrt [8]{-a} b^{7/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} \sqrt [8]{-a} b^{7/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt [8]{-a} b^{7/8}} \]
Antiderivative was successfully verified.
[In] Int[x^6/(a + b*x^8),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 91.6954, size = 246, normalized size = 0.92 \[ \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 b^{\frac{7}{8}} \sqrt [8]{- a}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 b^{\frac{7}{8}} \sqrt [8]{- a}} + \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 b^{\frac{7}{8}} \sqrt [8]{- a}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 b^{\frac{7}{8}} \sqrt [8]{- a}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 b^{\frac{7}{8}} \sqrt [8]{- a}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 b^{\frac{7}{8}} \sqrt [8]{- a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(b*x**8+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.302144, size = 324, normalized size = 1.21 \[ \frac{\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 \sqrt [8]{a} b^{7/8}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(a + b*x^8),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.016, size = 27, normalized size = 0.1 \[{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{\it \_R}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(b*x^8+a),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{b x^{8} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^8 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.240528, size = 579, normalized size = 2.17 \[ \frac{1}{16} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \arctan \left (\frac{a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}}}{x + \sqrt{-a b^{5} \left (-\frac{1}{a b^{7}}\right )^{\frac{3}{4}} + x^{2}}}\right ) + \sqrt{2} \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \log \left (a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} + x\right ) - \sqrt{2} \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \log \left (-a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} + x\right ) + 4 \, \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \arctan \left (\frac{a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}}}{a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} + \sqrt{2} x + \sqrt{2} \sqrt{\sqrt{2} a b^{6} x \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} - a b^{5} \left (-\frac{1}{a b^{7}}\right )^{\frac{3}{4}} + x^{2}}}\right ) + 4 \, \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}}}{a b^{6} \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} - \sqrt{2} x - \sqrt{2} \sqrt{-\sqrt{2} a b^{6} x \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} - a b^{5} \left (-\frac{1}{a b^{7}}\right )^{\frac{3}{4}} + x^{2}}}\right ) + \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a b^{6} x \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} - a b^{5} \left (-\frac{1}{a b^{7}}\right )^{\frac{3}{4}} + x^{2}\right ) - \left (-\frac{1}{a b^{7}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a b^{6} x \left (-\frac{1}{a b^{7}}\right )^{\frac{7}{8}} - a b^{5} \left (-\frac{1}{a b^{7}}\right )^{\frac{3}{4}} + x^{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^8 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.482933, size = 26, normalized size = 0.1 \[ \operatorname{RootSum}{\left (16777216 t^{8} a b^{7} + 1, \left ( t \mapsto t \log{\left (2097152 t^{7} a b^{6} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(b*x**8+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.25922, size = 579, normalized size = 2.17 \[ \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^8 + a),x, algorithm="giac")
[Out]