3.2065 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x^4} \, dx\)

Optimal. Leaf size=236 \[ -\frac{a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{2 a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{5 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3} \]

[Out]

-Sqrt[a + b/x^4]/(5*x^3) - (2*a*Sqrt[a + b/x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2)*x) + (2*a^(5/4)*Sqrt[(a +
b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(5*b^
(3/4)*Sqrt[a + b/x^4]) - (a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b/x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.115039, antiderivative size = 236, normalized size of antiderivative = 1., 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 = {335, 279, 305, 220, 1196} \[ -\frac{a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{2 a^{5/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{5 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^4]/x^4,x]

[Out]

-Sqrt[a + b/x^4]/(5*x^3) - (2*a*Sqrt[a + b/x^4])/(5*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2)*x) + (2*a^(5/4)*Sqrt[(a +
b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(5*b^
(3/4)*Sqrt[a + b/x^4]) - (a^(5/4)*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])/(5*b^(3/4)*Sqrt[a + b/x^4])

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]

Rule 279

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

Rule 305

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

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 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^4}}}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \sqrt{a+b x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{1}{5} (2 a) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{\left (2 a^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{5 \sqrt{b}}+\frac{\left (2 a^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{5 \sqrt{b}}\\ &=-\frac{\sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{5 \sqrt{b} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}+\frac{2 a^{5/4} \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 ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{a^{5/4} \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 )}{5 b^{3/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.0098849, size = 51, normalized size = 0.22 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{a x^4}{b}\right )}{5 x^3 \sqrt{\frac{a x^4}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^4]/x^4,x]

[Out]

-(Sqrt[a + b/x^4]*Hypergeometric2F1[-5/4, -1/2, -1/4, -((a*x^4)/b)])/(5*x^3*Sqrt[1 + (a*x^4)/b])

________________________________________________________________________________________

Maple [C]  time = 0.015, size = 234, normalized size = 1. \begin{align*} -{\frac{1}{5\,{x}^{3} \left ( a{x}^{4}+b \right ) }\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( -2\,i{a}^{{\frac{3}{2}}}\sqrt{-{ \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{ \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{5}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +2\,i{a}^{{\frac{3}{2}}}\sqrt{-{ \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{ \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{5}b{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +2\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}\sqrt{b}{x}^{8}{a}^{2}+3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3/2}{x}^{4}a+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{{\frac{5}{2}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*((a*x^4+b)/x^4)^(1/2)*(-2*I*a^(3/2)*(-(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)*x^5*b*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+2*I*a^(3/2)*(-(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)*x^5*b*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+2*(I*a^(1/2)/b^(1/2))
^(1/2)*b^(1/2)*x^8*a^2+3*(I*a^(1/2)/b^(1/2))^(1/2)*b^(3/2)*x^4*a+(I*a^(1/2)/b^(1/2))^(1/2)*b^(5/2))/x^3/(a*x^4
+b)/b^(3/2)/(I*a^(1/2)/b^(1/2))^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a + b/x^4)/x^4, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 1.69351, size = 41, normalized size = 0.17 \begin{align*} - \frac{\sqrt{a} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*exp_polar(I*pi)/(a*x**4))/(4*x**3*gamma(7/4))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a + b/x^4)/x^4, x)