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

Optimal. Leaf size=48 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}-b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

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Rubi [A]  time = 0.108093, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2132, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}-b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]

[Out]

ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

Rule 2132

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

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])

Rubi steps

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

Mathematica [A]  time = 0.0194982, size = 48, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}-b x^2}}\right )}{\sqrt{2} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]

[Out]

ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{-b{x}^{2}+\sqrt{{b}^{2}{x}^{4}+a}}{\frac{1}{\sqrt{{b}^{2}{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-b x^{2} + \sqrt{b^{2} x^{4} + a}}}{\sqrt{b^{2} x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)

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Fricas [A]  time = 13.0062, size = 366, normalized size = 7.62 \begin{align*} \left [\frac{1}{4} \, \sqrt{2} \sqrt{-\frac{1}{b}} \log \left (4 \, b^{2} x^{4} - 4 \, \sqrt{b^{2} x^{4} + a} b x^{2} + 2 \,{\left (\sqrt{2} b^{2} x^{3} \sqrt{-\frac{1}{b}} - \sqrt{2} \sqrt{b^{2} x^{4} + a} b x \sqrt{-\frac{1}{b}}\right )} \sqrt{-b x^{2} + \sqrt{b^{2} x^{4} + a}} + a\right ), -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-b x^{2} + \sqrt{b^{2} x^{4} + a}}}{2 \, \sqrt{b} x}\right )}{2 \, \sqrt{b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(2)*sqrt(-1/b)*log(4*b^2*x^4 - 4*sqrt(b^2*x^4 + a)*b*x^2 + 2*(sqrt(2)*b^2*x^3*sqrt(-1/b) - sqrt(2)*sq
rt(b^2*x^4 + a)*b*x*sqrt(-1/b))*sqrt(-b*x^2 + sqrt(b^2*x^4 + a)) + a), -1/2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-b
*x^2 + sqrt(b^2*x^4 + a))/(sqrt(b)*x))/sqrt(b)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- b x^{2} + \sqrt{a + b^{2} x^{4}}}}{\sqrt{a + b^{2} x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-b*x**2 + sqrt(a + b**2*x**4))/sqrt(a + b**2*x**4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-b x^{2} + \sqrt{b^{2} x^{4} + a}}}{\sqrt{b^{2} x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-b*x^2 + sqrt(b^2*x^4 + a))/sqrt(b^2*x^4 + a), x)