Optimal. Leaf size=49 \[ \frac{\Pi \left (\frac{2 b}{a d};\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{-d}} \]
[Out]
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Rubi [A] time = 0.146693, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ \frac{\Pi \left (\frac{2 b}{a d};\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{-d}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 41.6285, size = 177, normalized size = 3.61 \[ - \frac{\sqrt{3} \sqrt{f} \sqrt{d x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \left (- a f + 3 b\right ) \sqrt{f x^{2} + 3}} + \frac{3 \sqrt{3} b \sqrt{d x^{2} + 2} \Pi \left (1 - \frac{3 b}{a f}; \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 a \sqrt{f} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \left (- a f + 3 b\right ) \sqrt{f x^{2} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)/(d*x**2+2)**(1/2)/(f*x**2+3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0988031, size = 52, normalized size = 1.06 \[ -\frac{i \Pi \left (\frac{2 b}{a d};i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]
[Out]
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Maple [A] time = 0.032, size = 53, normalized size = 1.1 \[{\frac{\sqrt{2}}{2\,a}{\it EllipticPi} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},3\,{\frac{b}{af}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{-d}{\frac{1}{\sqrt{-f}}}} \right ){\frac{1}{\sqrt{-f}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right ) \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)/(d*x**2+2)**(1/2)/(f*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)),x, algorithm="giac")
[Out]