3.92 \(\int \frac{\sqrt{2+x^2}}{\left (1+x^2\right )^{3/2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=121 \[ \frac{\sqrt{2} \sqrt{x^2+2} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}-\frac{2 b \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)} \]

[Out]

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

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Rubi [A]  time = 0.195056, antiderivative size = 121, 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 \[ \frac{\sqrt{2} \sqrt{x^2+2} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}} (a-b)}-\frac{2 b \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2} (a-b)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + x^2]/((1 + x^2)^(3/2)*(a + b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.4648, size = 109, normalized size = 0.9 \[ \frac{\sqrt{2} \sqrt{x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \left (a - b\right ) \sqrt{x^{2} + 1}} - \frac{2 \sqrt{2} b \sqrt{x^{2} + 1} \Pi \left (1 - \frac{2 b}{a}; \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{a \sqrt{\frac{2 x^{2} + 2}{x^{2} + 2}} \left (a - b\right ) \sqrt{x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+2)**(1/2)/(x**2+1)**(3/2)/(b*x**2+a),x)

[Out]

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

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Mathematica [C]  time = 0.434484, size = 122, normalized size = 1.01 \[ \frac{\frac{2 i \sqrt{2} b \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )}{a}-i \sqrt{2} \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )+\frac{2 \sqrt{x^2+2} x}{\sqrt{x^2+1}}-i \sqrt{2} F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )+2 i \sqrt{2} E\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )}{2 a-2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 + x^2]/((1 + x^2)^(3/2)*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.05, size = 147, normalized size = 1.2 \[{\frac{1}{a \left ({x}^{4}+3\,{x}^{2}+2 \right ) \left ( a-b \right ) } \left ( i{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) a\sqrt{{x}^{2}+1}\sqrt{{x}^{2}+2}-i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) a\sqrt{{x}^{2}+1}\sqrt{{x}^{2}+2}+2\,i{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) b\sqrt{{x}^{2}+1}\sqrt{{x}^{2}+2}+a{x}^{3}+2\,ax \right ) \sqrt{{x}^{2}+1}\sqrt{{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2)^(1/2)/(x^2+1)^(3/2)/(b*x^2+a),x)

[Out]

(I*EllipticE(1/2*I*x*2^(1/2),2^(1/2))*a*(x^2+1)^(1/2)*(x^2+2)^(1/2)-I*EllipticPi
(1/2*I*x*2^(1/2),2*b/a,2^(1/2))*a*(x^2+1)^(1/2)*(x^2+2)^(1/2)+2*I*EllipticPi(1/2
*I*x*2^(1/2),2*b/a,2^(1/2))*b*(x^2+1)^(1/2)*(x^2+2)^(1/2)+a*x^3+2*a*x)*(x^2+1)^(
1/2)*(x^2+2)^(1/2)/a/(x^4+3*x^2+2)/(a-b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )}{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{2} + 2}}{{\left (b x^{4} +{\left (a + b\right )} x^{2} + a\right )} \sqrt{x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+2)**(1/2)/(x**2+1)**(3/2)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )}{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate(sqrt(x^2 + 2)/((b*x^2 + a)*(x^2 + 1)^(3/2)), x)