3.769 \(\int \frac{\sqrt{b x^2+\sqrt{a+b^2 x^4}}}{\sqrt{a+b^2 x^4}} \, dx\)

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]

ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

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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]

ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

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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]

sqrt(2)*atanh(sqrt(2)*sqrt(b)*x/sqrt(b*x**2 + sqrt(a + b**2*x**4)))/(2*sqrt(b))

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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]

Integrate[Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4], x]

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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]

int((b*x^2+(b^2*x^4+a)^(1/2))^(1/2)/(b^2*x^4+a)^(1/2),x)

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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]

integrate(sqrt(b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)

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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]

[1/4*sqrt(2)*log(4*b^2*x^4 + 4*sqrt(b^2*x^4 + a)*b*x^2 + 2*(sqrt(2)*b^(3/2)*x^3
+ sqrt(2)*sqrt(b^2*x^4 + a)*sqrt(b)*x)*sqrt(b*x^2 + sqrt(b^2*x^4 + a)) + a)/sqrt
(b), 1/2*sqrt(2)*sqrt(-1/b)*arctan(1/2*sqrt(2)*sqrt(b*x^2 + sqrt(b^2*x^4 + a))/(
b*x*sqrt(-1/b)))]

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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]

Integral(sqrt(b*x**2 + sqrt(a + b**2*x**4))/sqrt(a + b**2*x**4), x)

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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]

integrate(sqrt(b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)