Optimal. Leaf size=182 \[ \frac{x \sqrt{b x^2+2}}{\sqrt{d x^2+3}}+\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]
[Out]
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Rubi [A] time = 0.246113, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{x \sqrt{b x^2+2}}{\sqrt{d x^2+3}}+\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]
[Out]
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Rubi in Sympy [A] time = 35.2931, size = 175, normalized size = 0.96 \[ - \frac{\sqrt{2} \sqrt{b} \sqrt{d x^{2} + 3} E\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{b} x}{2} \right )}\middle | 1 - \frac{2 d}{3 b}\right )}{d \sqrt{\frac{2 d x^{2} + 6}{3 b x^{2} + 6}} \sqrt{b x^{2} + 2}} + \frac{b x \sqrt{d x^{2} + 3}}{d \sqrt{b x^{2} + 2}} + \frac{\sqrt{3} \sqrt{b x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{d} x}{3} \right )}\middle | - \frac{3 b}{2 d} + 1\right )}{\sqrt{d} \sqrt{\frac{3 b x^{2} + 6}{2 d x^{2} + 6}} \sqrt{d x^{2} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+2)**(1/2)/(d*x**2+3)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0391646, size = 37, normalized size = 0.2 \[ \frac{\sqrt{2} E\left (\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{3}}\right )|\frac{3 b}{2 d}\right )}{\sqrt{-d}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]
[Out]
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Maple [A] time = 0.023, size = 37, normalized size = 0.2 \[{\sqrt{2}{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}\sqrt{-d}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{b}{d}}}} \right ){\frac{1}{\sqrt{-d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + 2}}{\sqrt{d x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + 2)/sqrt(d*x^2 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{2} + 2}}{\sqrt{d x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + 2)/sqrt(d*x^2 + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + 2}}{\sqrt{d x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+2)**(1/2)/(d*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + 2}}{\sqrt{d x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + 2)/sqrt(d*x^2 + 3),x, algorithm="giac")
[Out]