Optimal. Leaf size=62 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}} \]
[Out]
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Rubi [A] time = 0.120745, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)^(3/2)/(d^2 - e^2*x^4),x]
[Out]
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Rubi in Sympy [A] time = 21.5816, size = 56, normalized size = 0.9 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{\sqrt{e}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{\sqrt{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**(3/2)/(-e**2*x**4+d**2),x)
[Out]
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Mathematica [A] time = 0.0421149, size = 61, normalized size = 0.98 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )-\log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{\sqrt{e}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)^(3/2)/(d^2 - e^2*x^4),x]
[Out]
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Maple [B] time = 0.066, size = 1442, normalized size = 23.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^(3/2)/(-e^2*x^4+d^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x^2 + d)^(3/2)/(e^2*x^4 - d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.307087, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \log \left (\frac{17 \, e^{2} x^{4} + 14 \, d e x^{2} + d^{2} + \frac{4 \, \sqrt{2}{\left (3 \, e^{2} x^{3} + d e x\right )} \sqrt{e x^{2} + d}}{\sqrt{e}}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 2 \, \log \left (2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{4 \, \sqrt{e}}, \frac{\sqrt{2} \sqrt{-e} \sqrt{-\frac{1}{e}} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )}}{4 \, \sqrt{e x^{2} + d} e x \sqrt{-\frac{1}{e}}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, \sqrt{-e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x^2 + d)^(3/2)/(e^2*x^4 - d^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{d + e x^{2}}}{- d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**(3/2)/(-e**2*x**4+d**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}{e^{2} x^{4} - d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x^2 + d)^(3/2)/(e^2*x^4 - d^2),x, algorithm="giac")
[Out]