Optimal. Leaf size=33 \[ \frac{2 x}{3 \sqrt{2 x^2+1}}+\frac{x}{3 \left (2 x^2+1\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0120243, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 x}{3 \sqrt{2 x^2+1}}+\frac{x}{3 \left (2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x^2)^(-5/2),x]
[Out]
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Rubi in Sympy [A] time = 0.596979, size = 27, normalized size = 0.82 \[ \frac{2 x}{3 \sqrt{2 x^{2} + 1}} + \frac{x}{3 \left (2 x^{2} + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*x**2+1)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0127785, size = 23, normalized size = 0.7 \[ \frac{x \left (4 x^2+3\right )}{3 \left (2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x^2)^(-5/2),x]
[Out]
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Maple [A] time = 0.004, size = 20, normalized size = 0.6 \[{\frac{x \left ( 4\,{x}^{2}+3 \right ) }{3} \left ( 2\,{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*x^2+1)^(5/2),x)
[Out]
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Maxima [A] time = 1.3432, size = 34, normalized size = 1.03 \[ \frac{2 \, x}{3 \, \sqrt{2 \, x^{2} + 1}} + \frac{x}{3 \,{\left (2 \, x^{2} + 1\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 1)^(-5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202123, size = 112, normalized size = 3.39 \[ -\frac{12 \, x^{5} + 17 \, x^{3} -{\left (4 \, x^{5} + 11 \, x^{3} + 6 \, x\right )} \sqrt{2 \, x^{2} + 1} + 6 \, x}{3 \,{\left (4 \, x^{6} + 12 \, x^{4} + 9 \, x^{2} -{\left (6 \, x^{4} + 7 \, x^{2} + 2\right )} \sqrt{2 \, x^{2} + 1} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 1)^(-5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.5528, size = 61, normalized size = 1.85 \[ \frac{4 x^{3}}{6 x^{2} \sqrt{2 x^{2} + 1} + 3 \sqrt{2 x^{2} + 1}} + \frac{3 x}{6 x^{2} \sqrt{2 x^{2} + 1} + 3 \sqrt{2 x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x**2+1)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.202205, size = 26, normalized size = 0.79 \[ \frac{{\left (4 \, x^{2} + 3\right )} x}{3 \,{\left (2 \, x^{2} + 1\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 1)^(-5/2),x, algorithm="giac")
[Out]