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

Optimal. Leaf size=43 \[ x \sqrt {b-\frac {a}{x^2}}+\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {b-\frac {a}{x^2}}}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1972, 242, 277, 217, 203} \begin {gather*} x \sqrt {b-\frac {a}{x^2}}+\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {b-\frac {a}{x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-a + b*x^2)/x^2],x]

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 242

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

Rule 277

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

Rule 1972

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] &&  !BinomialMatchQ[
u, x]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 68, normalized size = 1.58 \begin {gather*} x \sqrt {b-\frac {a}{x^2}}-\frac {\sqrt {a} x \sqrt {b-\frac {a}{x^2}} \tan ^{-1}\left (\frac {\sqrt {b x^2-a}}{\sqrt {a}}\right )}{\sqrt {b x^2-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-a + b*x^2)/x^2],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 3.95, size = 68, normalized size = 1.58 \begin {gather*} \frac {x \sqrt {b-\frac {a}{x^2}} \left (\sqrt {b x^2-a}-\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b x^2-a}}{\sqrt {a}}\right )\right )}{\sqrt {b x^2-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 118, normalized size = 2.74 \begin {gather*} \left [x \sqrt {\frac {b x^{2} - a}{x^{2}}} + \frac {1}{2} \, \sqrt {-a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {-a} x \sqrt {\frac {b x^{2} - a}{x^{2}}} - 2 \, a}{x^{2}}\right ), x \sqrt {\frac {b x^{2} - a}{x^{2}}} + \sqrt {a} \arctan \left (\frac {\sqrt {a} x \sqrt {\frac {b x^{2} - a}{x^{2}}}}{b x^{2} - a}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.24, size = 63, normalized size = 1.47 \begin {gather*} -\sqrt {a} \arctan \left (\frac {\sqrt {b x^{2} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\relax (x) + {\left (\sqrt {a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {a}}\right ) - \sqrt {-a}\right )} \mathrm {sgn}\relax (x) + \sqrt {b x^{2} - a} \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 81, normalized size = 1.88 \begin {gather*} \frac {\sqrt {\frac {b \,x^{2}-a}{x^{2}}}\, \left (a \ln \left (\frac {-2 a +2 \sqrt {-a}\, \sqrt {b \,x^{2}-a}}{x}\right )+\sqrt {-a}\, \sqrt {b \,x^{2}-a}\right ) x}{\sqrt {-a}\, \sqrt {b \,x^{2}-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.99, size = 34, normalized size = 0.79 \begin {gather*} \sqrt {b - \frac {a}{x^{2}}} x - \sqrt {a} \arctan \left (\frac {\sqrt {b - \frac {a}{x^{2}}} x}{\sqrt {a}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.38, size = 54, normalized size = 1.26 \begin {gather*} x\,\sqrt {b-\frac {a}{x^2}}+\frac {\sqrt {a}\,\mathrm {asin}\left (\frac {\sqrt {a}}{\sqrt {b}\,x}\right )\,\sqrt {b-\frac {a}{x^2}}}{\sqrt {b}\,\sqrt {1-\frac {a}{b\,x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(a - b*x^2)/x^2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((-a + b*x**2)/x**2), x)

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