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

Optimal. Leaf size=42 \[ x \sqrt {\frac {a}{x^2}+b}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b}}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {1972, 242, 277, 217, 206} \begin {gather*} x \sqrt {\frac {a}{x^2}+b}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b}}\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]*ArcTanh[Sqrt[a]/(Sqrt[b + a/x^2]*x)]

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 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} \tanh ^{-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 = 62, normalized size = 1.48 \begin {gather*} x \sqrt {\frac {a}{x^2}+b}-\frac {\sqrt {a} x \sqrt {\frac {a}{x^2}+b} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a+b x^2}} \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*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a + b*x^2]

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IntegrateAlgebraic [A]  time = 3.84, size = 61, normalized size = 1.45 \begin {gather*} \frac {x \sqrt {\frac {a}{x^2}+b} \left (\sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )}{\sqrt {a+b x^2}} \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]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/Sqrt[a + b*x^2]

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fricas [A]  time = 0.43, size = 108, normalized size = 2.57 \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 [B]  time = 0.17, size = 69, normalized size = 1.64 \begin {gather*} \frac {a \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} + \sqrt {b x^{2} + a} \mathrm {sgn}\relax (x) - \frac {{\left (a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\relax (x)}{\sqrt {-a}} \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]

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

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maple [A]  time = 0.08, size = 61, normalized size = 1.45 \begin {gather*} \frac {\sqrt {\frac {b \,x^{2}+a}{x^{2}}}\, \left (-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\sqrt {b \,x^{2}+a}\right ) x}{\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/(b*x^2+a)^(1/2)*((b*x^2+a)^(1/2)-a^(1/2)*ln(2*(a+(b*x^2+a)^(1/2)*a^(1/2))/x))

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maxima [A]  time = 2.94, size = 53, normalized size = 1.26 \begin {gather*} \sqrt {b + \frac {a}{x^{2}}} x + \frac {1}{2} \, \sqrt {a} \log \left (\frac {\sqrt {b + \frac {a}{x^{2}}} x - \sqrt {a}}{\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 + 1/2*sqrt(a)*log((sqrt(b + a/x^2)*x - sqrt(a))/(sqrt(b + a/x^2)*x + sqrt(a)))

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mupad [B]  time = 5.57, size = 55, normalized size = 1.31 \begin {gather*} x\,\sqrt {b+\frac {a}{x^2}}+\frac {\sqrt {a}\,\mathrm {asin}\left (\frac {\sqrt {a}\,1{}\mathrm {i}}{\sqrt {b}\,x}\right )\,\sqrt {b+\frac {a}{x^2}}\,1{}\mathrm {i}}{\sqrt {b}\,\sqrt {\frac {a}{b\,x^2}+1}} \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)*1i)/(b^(1/2)*x))*(b + a/x^2)^(1/2)*1i)/(b^(1/2)*(a/(b*x^2) + 1)^(
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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