3.1926 \(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^4} \, dx\)

Optimal. Leaf size=53 \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{2 b^{3/2}}-\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, 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, 321, 217, 206} \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{2 b^{3/2}}-\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x^4),x]

[Out]

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

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])

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 71, normalized size = 1.34 \[ \frac {a \left (a x^2+b\right ) \left (\frac {\tanh ^{-1}\left (\sqrt {\frac {a x^2}{b}+1}\right )}{2 \sqrt {\frac {a x^2}{b}+1}}-\frac {b}{2 a x^2}\right )}{b^2 x \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^4),x]

[Out]

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

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fricas [A]  time = 0.87, size = 128, normalized size = 2.42 \[ \left [\frac {a \sqrt {b} x \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, b \sqrt {\frac {a x^{2} + b}{x^{2}}}}{4 \, b^{2} x}, -\frac {a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + b \sqrt {\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [sign(t_nostep),sign(t_nostep+sqrt(b)/a*sign(t_nostep))]sym2poly/r2sym(const gen & e,const index_
m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.01, size = 73, normalized size = 1.38 \[ -\frac {\sqrt {a \,x^{2}+b}\, \left (-a b \,x^{2} \ln \left (\frac {2 b +2 \sqrt {a \,x^{2}+b}\, \sqrt {b}}{x}\right )+\sqrt {a \,x^{2}+b}\, b^{\frac {3}{2}}\right )}{2 \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, b^{\frac {5}{2}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^2)^(1/2)/x^4,x)

[Out]

-1/2*(a*x^2+b)^(1/2)*(-a*ln(2*(b+(a*x^2+b)^(1/2)*b^(1/2))/x)*x^2*b+(a*x^2+b)^(1/2)*b^(3/2))/((a*x^2+b)/x^2)^(1
/2)/x^3/b^(5/2)

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maxima [A]  time = 1.99, size = 76, normalized size = 1.43 \[ -\frac {\sqrt {a + \frac {b}{x^{2}}} a x}{2 \, {\left ({\left (a + \frac {b}{x^{2}}\right )} b x^{2} - b^{2}\right )}} - \frac {a \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{4 \, b^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(a + b/x^2)*a*x/((a + b/x^2)*b*x^2 - b^2) - 1/4*a*log((sqrt(a + b/x^2)*x - sqrt(b))/(sqrt(a + b/x^2)*
x + sqrt(b)))/b^(3/2)

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mupad [B]  time = 1.57, size = 60, normalized size = 1.13 \[ \left \{\begin {array}{cl} -\frac {1}{3\,\sqrt {a}\,x^3} & \text {\ if\ \ }b=0\\ \frac {a\,\ln \left (2\,\sqrt {a+\frac {b}{x^2}}+\frac {2\,\sqrt {b}}{x}\right )}{2\,b^{3/2}}-\frac {\sqrt {a+\frac {b}{x^2}}}{2\,b\,x} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x^2)^(1/2)),x)

[Out]

piecewise(b == 0, -1/(3*a^(1/2)*x^3), b ~= 0, (a*log(2*(a + b/x^2)^(1/2) + (2*b^(1/2))/x))/(2*b^(3/2)) - (a +
b/x^2)^(1/2)/(2*b*x))

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sympy [A]  time = 2.68, size = 42, normalized size = 0.79 \[ - \frac {\sqrt {a} \sqrt {1 + \frac {b}{a x^{2}}}}{2 b x} + \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{2 b^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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