Optimal. Leaf size=91 \[ -\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.179598, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^4)^(5/2)*x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.5405, size = 85, normalized size = 0.93 \[ - \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{16} - \frac{15 a b \sqrt{a + \frac{b}{x^{4}}}}{16 x^{2}} - \frac{5 b \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{8 x^{2}} + \frac{x^{2} \left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**4)**(5/2)*x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.131894, size = 94, normalized size = 1.03 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (\sqrt{a x^4+b} \left (-8 a^2 x^8+9 a b x^4+2 b^2\right )+15 a^2 \sqrt{b} x^8 \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )\right )}{16 x^6 \sqrt{a x^4+b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^4)^(5/2)*x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.027, size = 108, normalized size = 1.2 \[ -{\frac{{x}^{2}}{16} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,\sqrt{b}{a}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{8}-8\,{a}^{2}{x}^{8}\sqrt{a{x}^{4}+b}+9\,ab\sqrt{a{x}^{4}+b}{x}^{4}+2\,{b}^{2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^4)^(5/2)*x,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((a + b/x^4)^(5/2)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.253666, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x^{6} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) + 2 \,{\left (8 \, a^{2} x^{8} - 9 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{32 \, x^{6}}, -\frac{15 \, a^{2} \sqrt{-b} x^{6} \arctan \left (\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-b}}\right ) -{\left (8 \, a^{2} x^{8} - 9 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{16 \, x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^4)^(5/2)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 22.2637, size = 124, normalized size = 1.36 \[ \frac{a^{\frac{5}{2}} x^{2}}{2 \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{a^{\frac{3}{2}} b}{16 x^{2} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{11 \sqrt{a} b^{2}}{16 x^{6} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{16} - \frac{b^{3}}{8 \sqrt{a} x^{10} \sqrt{1 + \frac{b}{a x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**4)**(5/2)*x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.232008, size = 103, normalized size = 1.13 \[ \frac{1}{16} \,{\left (\frac{15 \, b \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 8 \, \sqrt{a x^{4} + b} - \frac{9 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} b - 7 \, \sqrt{a x^{4} + b} b^{2}}{a^{2} x^{8}}\right )} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^4)^(5/2)*x,x, algorithm="giac")
[Out]