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

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 = 0.0056857, 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, Rules used = {192, 191} \[ \frac{2 x}{3 a^2 \sqrt{a+c x^2}}+\frac{x}{3 a \left (a+c x^2\right )^{3/2}} \]

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])

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+c x^2\right )^{5/2}} \, dx &=\frac{x}{3 a \left (a+c x^2\right )^{3/2}}+\frac{2 \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac{x}{3 a \left (a+c x^2\right )^{3/2}}+\frac{2 x}{3 a^2 \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0085884, 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 = 0.049, size = 26, normalized size = 0.7 \begin{align*}{\frac{x \left ( 2\,c{x}^{2}+3\,a \right ) }{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 1.13467, size = 42, normalized size = 1.08 \begin{align*} \frac{2 \, x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 = 2.26115, size = 99, normalized size = 2.54 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B]  time = 1.18182, size = 95, normalized size = 2.44 \begin{align*} \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}}} \end{align*}

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 [A]  time = 1.23899, size = 36, normalized size = 0.92 \begin{align*} \frac{x{\left (\frac{2 \, c x^{2}}{a^{2}} + \frac{3}{a}\right )}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)