Optimal. Leaf size=47 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.173782, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.26988, size = 44, normalized size = 0.94 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{b x^{2} + \sqrt{a + b^{2} x^{4}}}} \right )}}{2 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+(b**2*x**4+a)**(1/2))**(1/2)/(b**2*x**4+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0830897, size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^2+\sqrt{a+b^2 x^4}}}{\sqrt{a+b^2 x^4}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt{b{x}^{2}+\sqrt{{b}^{2}{x}^{4}+a}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}}}{\sqrt{b^{2} x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 1.52291, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \log \left (4 \, b^{2} x^{4} + 4 \, \sqrt{b^{2} x^{4} + a} b x^{2} + 2 \,{\left (\sqrt{2} b^{\frac{3}{2}} x^{3} + \sqrt{2} \sqrt{b^{2} x^{4} + a} \sqrt{b} x\right )} \sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}} + a\right )}{4 \, \sqrt{b}}, \frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{b}} \arctan \left (\frac{\sqrt{2} \sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}}}{2 \, b x \sqrt{-\frac{1}{b}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + \sqrt{a + b^{2} x^{4}}}}{\sqrt{a + b^{2} x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+(b**2*x**4+a)**(1/2))**(1/2)/(b**2*x**4+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + \sqrt{b^{2} x^{4} + a}}}{\sqrt{b^{2} x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a),x, algorithm="giac")
[Out]