3.2095 \(\int \frac{1}{(a+\frac{b}{x^4})^{3/2}} \, dx\)

Optimal. Leaf size=258 \[ -\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{4 a^{7/4} \sqrt{a+\frac{b}{x^4}}}+\frac{3 x \sqrt{a+\frac{b}{x^4}}}{2 a^2}-\frac{3 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{2 a^2 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}+\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+\frac{b}{x^4}}}-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

(-3*Sqrt[b]*Sqrt[a + b/x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(2*a*Sqrt[a + b/x^4]) + (3*Sqrt[a + b/x^4]*
x)/(2*a^2) + (3*b^(1/4)*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])/(2*a^(7/4)*Sqrt[a + b/x^4]) - (3*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)
^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(4*a^(7/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.129783, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {242, 290, 325, 305, 220, 1196} \[ \frac{3 x \sqrt{a+\frac{b}{x^4}}}{2 a^2}-\frac{3 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{2 a^2 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+\frac{b}{x^4}}}-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^4)^(-3/2),x]

[Out]

(-3*Sqrt[b]*Sqrt[a + b/x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(2*a*Sqrt[a + b/x^4]) + (3*Sqrt[a + b/x^4]*
x)/(2*a^2) + (3*b^(1/4)*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])/(2*a^(7/4)*Sqrt[a + b/x^4]) - (3*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)
^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(4*a^(7/4)*Sqrt[a + b/x^4])

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 290

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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{1}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt{a+\frac{b}{x^4}} x}{2 a^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{2 a^2}\\ &=-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt{a+\frac{b}{x^4}} x}{2 a^2}-\frac{\left (3 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{2 a^{3/2}}+\frac{\left (3 \sqrt{b}\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 )}{2 a^{3/2}}\\ &=-\frac{3 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{2 a^2 \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt{a+\frac{b}{x^4}} x}{2 a^2}+\frac{3 \sqrt [4]{b} \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 )}{2 a^{7/4} \sqrt{a+\frac{b}{x^4}}}-\frac{3 \sqrt [4]{b} \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 )}{4 a^{7/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.0176038, size = 53, normalized size = 0.21 \[ \frac{x-x \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{a x^4}{b}\right )}{a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^4)^(-3/2),x]

[Out]

(x - x*Sqrt[1 + (a*x^4)/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((a*x^4)/b)])/(a*Sqrt[a + b/x^4])

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Maple [C]  time = 0.011, size = 187, normalized size = 0.7 \begin{align*} -{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ({x}^{3}{a}^{{\frac{3}{2}}}\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}-3\,i\sqrt{b}\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}}}}}a{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +3\,i\sqrt{b}\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}}}}}a{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{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(1/(a+b/x^4)^(3/2),x)

[Out]

-1/2*(a*x^4+b)*(x^3*a^(3/2)*(I*a^(1/2)/b^(1/2))^(1/2)-3*I*b^(1/2)*(-(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)*a*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+3*I*b^(1/2)*(-(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)*a*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I))/((a
*x^4+b)/x^4)^(3/2)/x^6/a^(5/2)/(I*a^(1/2)/b^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2),x, algorithm="maxima")

[Out]

integrate((a + b/x^4)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{8} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a^{2} x^{8} + 2 \, a b x^{4} + b^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2),x, algorithm="fricas")

[Out]

integral(x^8*sqrt((a*x^4 + b)/x^4)/(a^2*x^8 + 2*a*b*x^4 + b^2), x)

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Sympy [C]  time = 1.59152, size = 41, normalized size = 0.16 \begin{align*} - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**4)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2),x, algorithm="giac")

[Out]

integrate((a + b/x^4)^(-3/2), x)