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

Optimal. Leaf size=65 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0654669, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.67731, size = 56, normalized size = 0.86 \[ \frac{\sqrt{a^{2} - b^{2} x^{4}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a - b x^{2}}} \right )}}{\sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a**2 - b**2*x**4)*atan(sqrt(b)*x/sqrt(a - b*x**2))/(sqrt(b)*sqrt(a - b*x**2
)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.0367116, size = 50, normalized size = 0.77 \[ \frac{i \log \left (\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(I*Log[(-2*I)*Sqrt[b]*x + (2*Sqrt[a^2 - b^2*x^4])/Sqrt[a + b*x^2]])/Sqrt[b]

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Maple [A]  time = 0.025, size = 69, normalized size = 1.1 \[{1\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{1}{b} \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }}}}} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279092, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} b x +{\left (2 \, b^{2} x^{4} + a b x^{2} - a^{2}\right )} \sqrt{-b}}{b x^{2} + a}\right )}{2 \, \sqrt{-b}}, -\frac{\arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{b}}{b^{2} x^{3} + a b x}\right )}{\sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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