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

Optimal. Leaf size=39 \[ \frac{2 x}{3 a^2 \sqrt{a+c x^2}}+\frac{x}{3 a \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.020612, antiderivative size = 39, 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 \[ \frac{2 x}{3 a^2 \sqrt{a+c x^2}}+\frac{x}{3 a \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(-5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.01375, size = 32, normalized size = 0.82 \[ \frac{x}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{2 x}{3 a^{2} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+a)**(5/2),x)

[Out]

x/(3*a*(a + c*x**2)**(3/2)) + 2*x/(3*a**2*sqrt(a + c*x**2))

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Mathematica [A]  time = 0.0210235, size = 29, normalized size = 0.74 \[ \frac{x \left (3 a+2 c x^2\right )}{3 a^2 \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(-5/2),x]

[Out]

(x*(3*a + 2*c*x^2))/(3*a^2*(a + c*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 26, normalized size = 0.7 \[{\frac{x \left ( 2\,c{x}^{2}+3\,a \right ) }{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+a)^(5/2),x)

[Out]

1/3*x*(2*c*x^2+3*a)/(c*x^2+a)^(3/2)/a^2

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Maxima [A]  time = 0.702105, size = 42, normalized size = 1.08 \[ \frac{2 \, x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(-5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225107, size = 63, normalized size = 1.62 \[ \frac{{\left (2 \, c x^{3} + 3 \, a x\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(-5/2),x, algorithm="fricas")

[Out]

1/3*(2*c*x^3 + 3*a*x)*sqrt(c*x^2 + a)/(a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)

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Sympy [A]  time = 2.66581, size = 95, normalized size = 2.44 \[ \frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+a)**(5/2),x)

[Out]

3*a*x/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a)) + 2
*c*x**3/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.21492, size = 36, normalized size = 0.92 \[ \frac{x{\left (\frac{2 \, c x^{2}}{a^{2}} + \frac{3}{a}\right )}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(-5/2),x, algorithm="giac")

[Out]

1/3*x*(2*c*x^2/a^2 + 3/a)/(c*x^2 + a)^(3/2)