3.468 \(\int \frac{1}{\left (1+2 x^2\right )^{5/2}} \, dx\)

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]

x/(3*(1 + 2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 + 2*x^2])

_______________________________________________________________________________________

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]

x/(3*(1 + 2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 + 2*x^2])

_______________________________________________________________________________________

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]

2*x/(3*sqrt(2*x**2 + 1)) + x/(3*(2*x**2 + 1)**(3/2))

_______________________________________________________________________________________

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]

(x*(3 + 4*x^2))/(3*(1 + 2*x^2)^(3/2))

_______________________________________________________________________________________

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]

1/3*x*(4*x^2+3)/(2*x^2+1)^(3/2)

_______________________________________________________________________________________

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]

2/3*x/sqrt(2*x^2 + 1) + 1/3*x/(2*x^2 + 1)^(3/2)

_______________________________________________________________________________________

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]

-1/3*(12*x^5 + 17*x^3 - (4*x^5 + 11*x^3 + 6*x)*sqrt(2*x^2 + 1) + 6*x)/(4*x^6 + 1
2*x^4 + 9*x^2 - (6*x^4 + 7*x^2 + 2)*sqrt(2*x^2 + 1) + 2)

_______________________________________________________________________________________

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]

4*x**3/(6*x**2*sqrt(2*x**2 + 1) + 3*sqrt(2*x**2 + 1)) + 3*x/(6*x**2*sqrt(2*x**2
+ 1) + 3*sqrt(2*x**2 + 1))

_______________________________________________________________________________________

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]

1/3*(4*x^2 + 3)*x/(2*x^2 + 1)^(3/2)