Optimal. Leaf size=234 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.08368, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^6)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**6+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.300508, size = 193, normalized size = 0.82 \[ \frac{\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac{2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{5/6}}+\frac{12 \sqrt [6]{a} x^5}{a+b x^6}}{72 a^{7/6}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^6)^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.354, size = 349, normalized size = 1.5 \[{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{1}{18\,ab}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{72\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{72\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^6+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^6 + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.244848, size = 620, normalized size = 2.65 \[ \frac{12 \, x^{5} + 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}}}{a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + 2 \, x + 2 \, \sqrt{a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}}}\right ) + 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}}}{a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - 2 \, x - 2 \, \sqrt{-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}}}\right ) +{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) -{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) + 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right ) - 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right )}{72 \,{\left (a b x^{6} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^6 + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.13771, size = 46, normalized size = 0.2 \[ \frac{x^{5}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{7} b^{5} + 1, \left ( t \mapsto t \log{\left (60466176 t^{5} a^{6} b^{4} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**6+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^6 + a)^2,x, algorithm="giac")
[Out]