Optimal. Leaf size=122 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]
[Out]
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Rubi [A] time = 0.344605, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)*(c + d*x^2)*Sqrt[e + f*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 41.8518, size = 104, normalized size = 0.85 \[ \frac{d \operatorname{atanh}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{c} \sqrt{e + f x^{2}}} \right )}}{\sqrt{c} \left (a d - b c\right ) \sqrt{c f - d e}} - \frac{b \operatorname{atanh}{\left (\frac{x \sqrt{a f - b e}}{\sqrt{a} \sqrt{e + f x^{2}}} \right )}}{\sqrt{a} \left (a d - b c\right ) \sqrt{a f - b e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)/(d*x**2+c)/(f*x**2+e)**(1/2),x)
[Out]
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Mathematica [A] time = 0.276808, size = 113, normalized size = 0.93 \[ \frac{\frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}}{b c-a d} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)*(c + d*x^2)*Sqrt[e + f*x^2]),x]
[Out]
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Maple [B] time = 0.056, size = 782, normalized size = 6.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)/(d*x^2+c)/(f*x^2+e)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 92.5344, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \sqrt{e + f x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)/(d*x**2+c)/(f*x**2+e)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.267234, size = 234, normalized size = 1.92 \[ -f^{\frac{3}{2}}{\left (\frac{b \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt{-a^{2} f^{2} + a b f e}}\right )}{\sqrt{-a^{2} f^{2} + a b f e}{\left (b c f - a d f\right )}} - \frac{d \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{\sqrt{-c^{2} f^{2} + c d f e}{\left (b c f - a d f\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="giac")
[Out]