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

Optimal. Leaf size=53 \[ \frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0412784, antiderivative size = 53, 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 \[ \frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt((a + b*x**2)**2), x)

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Mathematica [A]  time = 0.0235722, size = 44, normalized size = 0.83 \[ \frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 0.6 \[{(b{x}^{2}+a)\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((b*x^2+a)^2)^(1/2),x)

[Out]

1/((b*x^2+a)^2)^(1/2)*(b*x^2+a)/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266776, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{2 \, \sqrt{-a b}}, \frac{\arctan \left (\frac{\sqrt{a b} x}{a}\right )}{\sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.363505, size = 53, normalized size = 1. \[ - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- a \sqrt{- \frac{1}{a b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (a \sqrt{- \frac{1}{a b}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a*b))*log(-a*sqrt(-1/(a*b)) + x)/2 + sqrt(-1/(a*b))*log(a*sqrt(-1/(a*b
)) + x)/2

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GIAC/XCAS [A]  time = 0.272995, size = 31, normalized size = 0.58 \[ \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(b*x/sqrt(a*b))*sign(b*x^2 + a)/sqrt(a*b)