3.3 \(\int \frac{1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=60 \[ \frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(Sqrt[a]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(a^2 + 2*a*b
*x^2 + b^2*x^4)^(1/4))

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Rubi [A]  time = 0.0354016, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/4),x]

[Out]

(Sqrt[a]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(a^2 + 2*a*b
*x^2 + b^2*x^4)^(1/4))

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Rubi in Sympy [A]  time = 19.9581, size = 54, normalized size = 0.9 \[ \frac{b \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{b x}{\sqrt{a b + b^{2} x^{2}}} \right )}}{\left (a b + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(1/4),x)

[Out]

b*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/4)*atanh(b*x/sqrt(a*b + b**2*x**2))/(a*b +
b**2*x**2)**(3/2)

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Mathematica [A]  time = 0.0258873, size = 49, normalized size = 0.82 \[ \frac{\sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt [4]{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/4),x]

[Out]

(Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*((a + b*x^2)^2)^
(1/4))

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Maple [F]  time = 0.011, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [4]{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284997, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(1/4),x)

[Out]

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(-1/4), x)

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GIAC/XCAS [A]  time = 0.281421, size = 36, normalized size = 0.6 \[ -\frac{\arctan \left (\frac{\sqrt{-b x^{2} - a}{\left | x \right |}}{\sqrt{b} x^{2}}\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-1/4),x, algorithm="giac")

[Out]

-arctan(sqrt(-b*x^2 - a)*abs(x)/(sqrt(b)*x^2))/sqrt(b)