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

Optimal. Leaf size=78 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.102868, antiderivative size = 78, 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 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(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[2]*Sqrt[b]*x)/Sqrt[a - b*x^2]])/(S
qrt[2]*a*Sqrt[b]*Sqrt[a^2 - b^2*x^4])

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Rubi in Sympy [A]  time = 15.4851, size = 70, normalized size = 0.9 \[ \frac{\sqrt{2} \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a - b x^{2}}} \right )}}{2 a \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0500079, size = 78, normalized size = 1. \[ \frac{\sqrt{a^2-b^2 x^4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a-b x^2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.058, size = 248, normalized size = 3.2 \[ -{\frac{1}{2}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( b\sqrt{a}\sqrt{2}\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{-b{x}^{2}+a}-\sqrt{-ab}x+a \right ) }{bx-\sqrt{-ab}}} \right ) -b\sqrt{a}\sqrt{2}\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{-b{x}^{2}+a}+\sqrt{-ab}x+a \right ) }{bx+\sqrt{-ab}}} \right ) -2\,\sqrt{b}\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{ \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }{b}}}}}} \right ) \sqrt{-ab}+2\,\sqrt{b}\arctan \left ({\frac{x\sqrt{b}}{\sqrt{-b{x}^{2}+a}}} \right ) \sqrt{-ab} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{-ab}}} \left ( \sqrt{ab}+\sqrt{-ab} \right ) ^{-1} \left ( \sqrt{ab}-\sqrt{-ab} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281267, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2} \log \left (-\frac{4 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} b x + \sqrt{2}{\left (3 \, b^{2} x^{4} + 2 \, a b x^{2} - a^{2}\right )} \sqrt{-b}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, a \sqrt{-b}}, -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{b}}{2 \,{\left (b^{2} x^{3} + a b x\right )}}\right )}{2 \, a \sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-b^2*x^4 + a^2)*sqrt(b*x^2 + a)), x)