3.935 \(\int \frac{1}{\sqrt{(a^2+x^2)^3}} \, dx\)

Optimal. Leaf size=25 \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]

[Out]

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

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Rubi [A]  time = 0.0164506, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {6720, 191} \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a^2 + x^2)^3],x]

[Out]

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

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 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}{\sqrt{\left (a^2+x^2\right )^3}} \, dx &=\frac{\left (a^2+x^2\right )^{3/2} \int \frac{1}{\left (a^2+x^2\right )^{3/2}} \, dx}{\sqrt{\left (a^2+x^2\right )^3}}\\ &=\frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}}\\ \end{align*}

Mathematica [A]  time = 0.0193384, size = 25, normalized size = 1. \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(a^2 + x^2)^3],x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 1. \begin{align*}{\frac{x \left ({a}^{2}+{x}^{2} \right ) }{{a}^{2}}{\frac{1}{\sqrt{ \left ({a}^{2}+{x}^{2} \right ) ^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a^2+x^2)^3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.07124, size = 19, normalized size = 0.76 \begin{align*} \frac{x}{\sqrt{a^{2} + x^{2}} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a^2+x^2)^3)^(1/2),x, algorithm="maxima")

[Out]

x/(sqrt(a^2 + x^2)*a^2)

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Fricas [B]  time = 1.65692, size = 131, normalized size = 5.24 \begin{align*} \frac{a^{4} + 2 \, a^{2} x^{2} + x^{4} + \sqrt{a^{6} + 3 \, a^{4} x^{2} + 3 \, a^{2} x^{4} + x^{6}} x}{a^{6} + 2 \, a^{4} x^{2} + a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a^2+x^2)^3)^(1/2),x, algorithm="fricas")

[Out]

(a^4 + 2*a^2*x^2 + x^4 + sqrt(a^6 + 3*a^4*x^2 + 3*a^2*x^4 + x^6)*x)/(a^6 + 2*a^4*x^2 + a^2*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (a^{2} + x^{2}\right )^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a**2+x**2)**3)**(1/2),x)

[Out]

Integral(1/sqrt((a**2 + x**2)**3), x)

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Giac [A]  time = 1.11909, size = 19, normalized size = 0.76 \begin{align*} \frac{x}{\sqrt{a^{2} + x^{2}} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a^2+x^2)^3)^(1/2),x, algorithm="giac")

[Out]

x/(sqrt(a^2 + x^2)*a^2)