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