3.1896 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 \sqrt{b}} \]

[Out]

-Sqrt[a + b/x^2]/(2*x) - (a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(2*Sqrt[b])

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Rubi [A]  time = 0.0204868, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {335, 195, 217, 206} \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^2]/x^2,x]

[Out]

-Sqrt[a + b/x^2]/(2*x) - (a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(2*Sqrt[b])

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^2}}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^2}} x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^2}} x}\right )}{2 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0293788, size = 68, normalized size = 1.36 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (a x^2 \sqrt{\frac{a x^2}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{a x^2}{b}+1}\right )+a x^2+b\right )}{2 x \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]/x^2,x]

[Out]

-(Sqrt[a + b/x^2]*(b + a*x^2 + a*x^2*Sqrt[1 + (a*x^2)/b]*ArcTanh[Sqrt[1 + (a*x^2)/b]]))/(2*x*(b + a*x^2))

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Maple [B]  time = 0.005, size = 85, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,bx}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}a-\sqrt{a{x}^{2}+b}{x}^{2}a+ \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^(1/2)/x^2,x)

[Out]

-1/2*((a*x^2+b)/x^2)^(1/2)/x*(b^(1/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^2*a-(a*x^2+b)^(1/2)*x^2*a+(a*x^2+b
)^(3/2))/(a*x^2+b)^(1/2)/b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56459, size = 300, normalized size = 6. \begin{align*} \left [\frac{a \sqrt{b} x \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \, b x}, \frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \, b x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(a*sqrt(b)*x*log(-(a*x^2 - 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b)/x^2) - 2*b*sqrt((a*x^2 + b)/x^2))/(b*
x), 1/2*(a*sqrt(-b)*x*arctan(sqrt(-b)*x*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) - b*sqrt((a*x^2 + b)/x^2))/(b*x)]

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Sympy [A]  time = 1.93967, size = 42, normalized size = 0.84 \begin{align*} - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{2}}}}{2 x} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*sqrt(1 + b/(a*x**2))/(2*x) - a*asinh(sqrt(b)/(sqrt(a)*x))/(2*sqrt(b))

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Giac [A]  time = 1.23113, size = 61, normalized size = 1.22 \begin{align*} \frac{1}{2} \, a{\left (\frac{\arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{a x^{2} + b}}{a x^{2}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*(arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) - sqrt(a*x^2 + b)/(a*x^2))*sgn(x)