3.1898 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x^4} \, dx\)

Optimal. Leaf size=74 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 b^{3/2}}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3} \]

[Out]

-Sqrt[a + b/x^2]/(4*x^3) - (a*Sqrt[a + b/x^2])/(8*b*x) + (a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(8*b^(3/2)
)

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Rubi [A]  time = 0.0365519, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {335, 279, 321, 217, 206} \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 b^{3/2}}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^2]/x^4,x]

[Out]

-Sqrt[a + b/x^2]/(4*x^3) - (a*Sqrt[a + b/x^2])/(8*b*x) + (a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(8*b^(3/2)
)

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \sqrt{a+b x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )}{8 b}\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^2}} x}\right )}{8 b}\\ &=-\frac{\sqrt{a+\frac{b}{x^2}}}{4 x^3}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 b x}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^2}} x}\right )}{8 b^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0098692, size = 47, normalized size = 0.64 \[ -\frac{a^2 x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right ) \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{a x^2}{b}+1\right )}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]/x^4,x]

[Out]

-(a^2*Sqrt[a + b/x^2]*x*(b + a*x^2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (a*x^2)/b])/(3*b^3)

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Maple [A]  time = 0.008, size = 106, normalized size = 1.4 \begin{align*}{\frac{1}{8\,{b}^{2}{x}^{3}}\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}^{4}{a}^{2}-\sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}+ \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{2}a-2\, \left ( a{x}^{2}+b \right ) ^{3/2}b \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^4,x)

[Out]

1/8*((a*x^2+b)/x^2)^(1/2)/x^3*(b^(1/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^4*a^2-(a*x^2+b)^(1/2)*x^4*a^2+(a*
x^2+b)^(3/2)*x^2*a-2*(a*x^2+b)^(3/2)*b)/(a*x^2+b)^(1/2)/b^2

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53698, size = 367, normalized size = 4.96 \begin{align*} \left [\frac{a^{2} \sqrt{b} x^{3} \log \left (-\frac{a x^{2} + 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \,{\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, b^{2} x^{3}}, -\frac{a^{2} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, b^{2} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/16*(a^2*sqrt(b)*x^3*log(-(a*x^2 + 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b)/x^2) - 2*(a*b*x^2 + 2*b^2)*sqrt(
(a*x^2 + b)/x^2))/(b^2*x^3), -1/8*(a^2*sqrt(-b)*x^3*arctan(sqrt(-b)*x*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) + (a*
b*x^2 + 2*b^2)*sqrt((a*x^2 + b)/x^2))/(b^2*x^3)]

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Sympy [A]  time = 3.63857, size = 92, normalized size = 1.24 \begin{align*} - \frac{a^{\frac{3}{2}}}{8 b x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{3 \sqrt{a}}{8 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8 b^{\frac{3}{2}}} - \frac{b}{4 \sqrt{a} x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**(1/2)/x**4,x)

[Out]

-a**(3/2)/(8*b*x*sqrt(1 + b/(a*x**2))) - 3*sqrt(a)/(8*x**3*sqrt(1 + b/(a*x**2))) + a**2*asinh(sqrt(b)/(sqrt(a)
*x))/(8*b**(3/2)) - b/(4*sqrt(a)*x**5*sqrt(1 + b/(a*x**2)))

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Giac [A]  time = 1.2719, size = 86, normalized size = 1.16 \begin{align*} -\frac{1}{8} \, a^{2}{\left (\frac{\arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (a x^{2} + b\right )}^{\frac{3}{2}} + \sqrt{a x^{2} + b} b}{a^{2} b x^{4}}\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^4,x, algorithm="giac")

[Out]

-1/8*a^2*(arctan(sqrt(a*x^2 + b)/sqrt(-b))/(sqrt(-b)*b) + ((a*x^2 + b)^(3/2) + sqrt(a*x^2 + b)*b)/(a^2*b*x^4))
*sgn(x)