3.598 \(\int \frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac {x \log (x) \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {435, 23, 29} \[ \frac {x \log (x) \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]

[Out]

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 435

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(x^(n*FracPart[q])*(c +
d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x], x] /; FreeQ[{a, b,
c, d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \[ \frac {x \log (x) \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 51, normalized size = 1.82 \[ -\arctan \left (\frac {\sqrt {-b x^{2} + a} {\left (x^{3} + x\right )} \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b x^{4} - {\left (a + b\right )} x^{2} + a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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((b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),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(x)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(t_nostep)]Sign error %%%{b,2%%%}Limit: Max order reached or unable to make series expansion Error:
Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.01, size = 42, normalized size = 1.50 \[ -\frac {\sqrt {\frac {b \,x^{2}-a}{x^{2}}}\, \sqrt {-b \,x^{2}+a}\, x \ln \relax (x )}{b \,x^{2}-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 1.04, size = 4, normalized size = 0.14 \[ -i \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {a}{x^{2}} + b}}{\sqrt {a - b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-a/x**2 + b)/sqrt(a - b*x**2), x)

________________________________________________________________________________________