3.91 \(\int \frac{\sqrt{2+x^2}}{\sqrt{1+x^2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=58 \[ \frac{2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}} \]

[Out]

(2*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*Sqrt[(1 + x^
2)/(2 + x^2)]*Sqrt[2 + x^2])

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Rubi [A]  time = 0.0756475, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*Sqrt[(1 + x^
2)/(2 + x^2)]*Sqrt[2 + x^2])

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Rubi in Sympy [A]  time = 13.3641, size = 58, normalized size = 1. \[ \frac{2 \sqrt{2} \sqrt{x^{2} + 1} \Pi \left (1 - \frac{2 b}{a}; \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{a \sqrt{\frac{2 x^{2} + 2}{x^{2} + 2}} \sqrt{x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.101301, size = 50, normalized size = 0.86 \[ -\frac{i \left (a F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )-(a-2 b) \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )\right )}{\sqrt{2} a b} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(a*EllipticF[I*ArcSinh[x], 1/2] - (a - 2*b)*EllipticPi[b/a, I*ArcSinh[x],
1/2]))/(Sqrt[2]*a*b)

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Maple [A]  time = 0.021, size = 64, normalized size = 1.1 \[{\frac{i}{ab} \left ( a{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) -2\,b{\it EllipticPi} \left ( i/2x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) -a{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*(a*EllipticPi(1/2*I*x*2^(1/2),2*b/a,2^(1/2))-2*b*EllipticPi(1/2*I*x*2^(1/2),2*
b/a,2^(1/2))-a*EllipticF(1/2*I*x*2^(1/2),2^(1/2)))/a/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt{x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(x^2 + 2)/((b*x^2 + a)*sqrt(x^2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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