3.3 \(\int \frac{1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=60 \[ \frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(Sqrt[a]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))

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Rubi [A]  time = 0.0122802, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1089, 215} \[ \frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/4),x]

[Out]

(Sqrt[a]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(1/4))

Rule 1089

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p]
)/(1 + (2*c*x^2)/b)^(2*FracPart[p]), Int[(1 + (2*c*x^2)/b)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2
- 4*a*c, 0] &&  !IntegerQ[2*p]

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0149033, size = 49, normalized size = 0.82 \[ \frac{\sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt [4]{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-1/4),x]

[Out]

(Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*((a + b*x^2)^2)^(1/4))

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Maple [F]  time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [4]{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28972, size = 217, normalized size = 3.62 \begin{align*} \left [\frac{\log \left (-2 \, b x^{2} - 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} \sqrt{b} x - a\right )}{2 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} \sqrt{-b} x}{b x^{2} + a}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15798, size = 36, normalized size = 0.6 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{-b x^{2} - a}{\left | x \right |}}{\sqrt{b} x^{2}}\right )}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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