3.2064 \(\int \frac {\sqrt {a+\frac {b}{x^4}}}{x^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac {a^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}-\frac {\sqrt {a+\frac {b}{x^4}}}{3 x} \]

[Out]

-1/3*(a+b/x^4)^(1/2)/x-1/3*a^(3/4)*(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)^(1/2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))
*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),1/2*2^(1/2))*(a^(1/2)+b^(1/2)/x^2)*((a+b/x^4)/(a^(1/2)+b^(1/2)/x^2
)^2)^(1/2)/b^(1/4)/(a+b/x^4)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {335, 195, 220} \[ -\frac {a^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}-\frac {\sqrt {a+\frac {b}{x^4}}}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^4]/x^2,x]

[Out]

-Sqrt[a + b/x^4]/(3*x) - (a^(3/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Elliptic
F[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*b^(1/4)*Sqrt[a + b/x^4])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a+\frac {b}{x^4}}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \sqrt {a+b x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {a+\frac {b}{x^4}}}{3 x}-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {a+\frac {b}{x^4}}}{3 x}-\frac {a^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 51, normalized size = 0.48 \[ -\frac {\sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{4};-\frac {a x^4}{b}\right )}{3 x \sqrt {\frac {a x^4}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^4]/x^2,x]

[Out]

-1/3*(Sqrt[a + b/x^4]*Hypergeometric2F1[-3/4, -1/2, 1/4, -((a*x^4)/b)])/(x*Sqrt[1 + (a*x^4)/b])

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fricas [F]  time = 1.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {a x^{4} + b}{x^{4}}}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + \frac {b}{x^{4}}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a + b/x^4)/x^2, x)

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maple [C]  time = 0.02, size = 132, normalized size = 1.23 \[ -\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,x^{4}-2 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a \,x^{3} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b \right )}{3 \left (a \,x^{4}+b \right ) \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((a*x^4+b)/x^4)^(1/2)*(-2*a*(-(I*a^(1/2)*x^2-b^(1/2))/b^(1/2))^(1/2)*((I*a^(1/2)*x^2+b^(1/2))/b^(1/2))^(1
/2)*EllipticF((I*a^(1/2)/b^(1/2))^(1/2)*x,I)*x^3+(I*a^(1/2)/b^(1/2))^(1/2)*a*x^4+(I*a^(1/2)/b^(1/2))^(1/2)*b)/
x/(a*x^4+b)/(I*a^(1/2)/b^(1/2))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + \frac {b}{x^{4}}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a + b/x^4)/x^2, x)

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mupad [B]  time = 1.34, size = 39, normalized size = 0.36 \[ -\frac {\sqrt {a\,x^4+b}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{4};\ \frac {5}{4};\ -\frac {b}{a\,x^4}\right )}{x\,\sqrt {\frac {b}{a}+x^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^4)^(1/2)/x^2,x)

[Out]

-((b + a*x^4)^(1/2)*hypergeom([-1/2, 1/4], 5/4, -b/(a*x^4)))/(x*(b/a + x^4)^(1/2))

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sympy [C]  time = 1.21, size = 39, normalized size = 0.36 \[ - \frac {\sqrt {a} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*x*gamma(5/4))

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