3.6.98 \(\int \frac {\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \, dx\) [598]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {446, 23, 29} \begin {gather*} \frac {x \log (x) \sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b x^2}} \end {gather*}

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 446

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*}

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Mathematica [A]
time = 0.05, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b-\frac {a}{x^2}} x \log (x)}{\sqrt {a-b x^2}} \end {gather*}

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]

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Maple [A]
time = 0.33, size = 30, normalized size = 1.07

method	result	size
default	\(\frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, x \ln \left (x \right )}{\sqrt {-b \,x^{2}+a}}\)	\(30\)
risch	\(\frac {i \sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, \left (b \,x^{2}-a \right ) x \sqrt {\frac {-b \,x^{2}+a}{b \,x^{2}-a}}\, \ln \left (x \right )}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(63\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 4, normalized size = 0.14 \begin {gather*} -i \, \log \left (x\right ) \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
time = 0.35, size = 51, normalized size = 1.82 \begin {gather*} -\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 ) \end {gather*}

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))

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

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)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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

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)

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