3.2097 \(\int \frac{1}{(a+\frac{b}{x^4})^{3/2} x^4} \, dx\)

Optimal. Leaf size=241 \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{4 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{2 a \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{1}{2 a x^3 \sqrt{a+\frac{b}{x^4}}} \]

[Out]

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

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Rubi [A]  time = 0.109225, antiderivative size = 241, 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, 290, 305, 220, 1196} \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{2 a \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{1}{2 a x^3 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^4)^(3/2)*x^4),x]

[Out]

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

Mathematica [C]  time = 0.0121228, size = 52, normalized size = 0.22 \[ \frac{x \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{a x^4}{b}\right )}{3 b \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.013, size = 187, normalized size = 0.8 \begin{align*}{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ({x}^{3}\sqrt{b}\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}\sqrt{a}-i\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}}}}}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +i\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}}}}}b{\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}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^4)^(3/2)/x^4,x)

[Out]

1/2*(a*x^4+b)*(x^3*b^(1/2)*(I*a^(1/2)/b^(1/2))^(1/2)*a^(1/2)-I*(-(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)*b*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+I*(-(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)*b*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I))/((a*x^4+b)/x^4)^
(3/2)/x^6/b^(3/2)/(I*a^(1/2)/b^(1/2))^(1/2)/a^(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}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4} \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^4,x, algorithm="fricas")

[Out]

integral(x^4*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 = 2.63988, size = 39, normalized size = 0.16 \begin{align*} - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} x^{3} \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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