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

Optimal. Leaf size=192 \[ -\frac{\sqrt{x^2+2} (a-2 b) \Pi \left (1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} a b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} x}{b \sqrt{x^2+1}}+\frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}-\frac{\sqrt{2} \sqrt{x^2+2} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

[Out]

(x*Sqrt[2 + x^2])/(b*Sqrt[1 + x^2]) - (Sqrt[2]*Sqrt[2 + x^2]*EllipticE[ArcTan[x]
, 1/2])/(b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) + (Sqrt[2 + x^2]*EllipticF[A
rcTan[x], 1/2])/(Sqrt[2]*b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) - ((a - 2*b)
*Sqrt[2 + x^2]*EllipticPi[1 - b/a, ArcTan[x], 1/2])/(Sqrt[2]*a*b*Sqrt[1 + x^2]*S
qrt[(2 + x^2)/(1 + x^2)])

_______________________________________________________________________________________

Rubi [A]  time = 0.292636, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\sqrt{x^2+2} (a-2 b) \Pi \left (1-\frac{b}{a};\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} a b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}+\frac{\sqrt{x^2+2} x}{b \sqrt{x^2+1}}+\frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}}-\frac{\sqrt{2} \sqrt{x^2+2} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{b \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[2 + x^2])/(b*Sqrt[1 + x^2]) - (Sqrt[2]*Sqrt[2 + x^2]*EllipticE[ArcTan[x]
, 1/2])/(b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) + (Sqrt[2 + x^2]*EllipticF[A
rcTan[x], 1/2])/(Sqrt[2]*b*Sqrt[1 + x^2]*Sqrt[(2 + x^2)/(1 + x^2)]) - ((a - 2*b)
*Sqrt[2 + x^2]*EllipticPi[1 - b/a, ArcTan[x], 1/2])/(Sqrt[2]*a*b*Sqrt[1 + x^2]*S
qrt[(2 + x^2)/(1 + x^2)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.3078, size = 165, normalized size = 0.86 \[ \frac{x \sqrt{x^{2} + 2}}{b \sqrt{x^{2} + 1}} - \frac{\sqrt{2} \sqrt{x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{b \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} + \frac{\sqrt{2} \sqrt{x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 b \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} - \frac{\sqrt{2} \left (a - 2 b\right ) \sqrt{x^{2} + 2} \Pi \left (1 - \frac{b}{a}; \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 a b \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.183848, size = 71, normalized size = 0.37 \[ \frac{i \left ((a-b) \left (a F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )-(a-2 b) \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )\right )-2 a b E\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )\right )}{\sqrt{2} a b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.015, size = 121, normalized size = 0.6 \[{\frac{i}{a{b}^{2}} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ){a}^{2}-2\,{\it EllipticF} \left ( i/2x\sqrt{2},\sqrt{2} \right ) ba-{a}^{2}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) +3\,{\it EllipticPi} \left ( i/2x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) ba-2\,{\it EllipticPi} \left ( i/2x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ){b}^{2}-{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) ba \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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