3.10.35 \(\int \frac {1}{\sqrt {(a^2+x^2)^3}} \, dx\) [935]

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

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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {1973, 197} \begin {gather*} \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \end {gather*}

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 197

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]

Rule 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 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*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.00 \begin {gather*} \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \end {gather*}

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 = 0.22, size = 24, normalized size = 0.96

method	result	size
gosper	\(\frac {x \left (a^{2}+x^{2}\right )}{a^{2} \sqrt {\left (a^{2}+x^{2}\right )^{3}}}\)	\(24\)
default	\(\frac {x \left (a^{2}+x^{2}\right )}{a^{2} \sqrt {\left (a^{2}+x^{2}\right )^{3}}}\)	\(24\)
risch	\(\frac {x \left (a^{2}+x^{2}\right )}{a^{2} \sqrt {\left (a^{2}+x^{2}\right )^{3}}}\)	\(24\)
trager	\(\frac {x \sqrt {a^{6}+3 a^{4} x^{2}+3 a^{2} x^{4}+x^{6}}}{a^{2} \left (a^{2}+x^{2}\right )^{2}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 14, normalized size = 0.56 \begin {gather*} \frac {x}{\sqrt {a^{2} + x^{2}} a^{2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
time = 0.42, size = 64, normalized size = 2.56 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (a^{2} + x^{2}\right )^{3}}}\, dx \end {gather*}

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 = 3.46, size = 14, normalized size = 0.56 \begin {gather*} \frac {x}{\sqrt {a^{2} + x^{2}} a^{2}} \end {gather*}

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)

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Mupad [B]
time = 3.51, size = 25, normalized size = 1.00 \begin {gather*} \frac {x\,\sqrt {{\left (a^2+x^2\right )}^3}}{a^2\,{\left (a^2+x^2\right )}^2} \end {gather*}

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)^3)^(1/2))/(a^2*(a^2 + x^2)^2)

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