3.2103 \(\int \frac{1}{(a+\frac{b}{x^4})^{5/2}} \, dx\)

Optimal. Leaf size=277 \[ -\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{8 a^{11/4} \sqrt{a+\frac{b}{x^4}}}+\frac{7 x \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{7 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{4 a^3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}+\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{11/4} \sqrt{a+\frac{b}{x^4}}}-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

(-7*Sqrt[b]*Sqrt[a + b/x^4])/(4*a^3*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(6*a*(a + b/x^4)^(3/2)) - (7*x)/(12*a^2*Sqr
t[a + b/x^4]) + (7*Sqrt[a + b/x^4]*x)/(4*a^3) + (7*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])/(4*a^(11/4)*Sqrt[a + b/x^4]) - (7*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])
/(8*a^(11/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.159775, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{7 x \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{7 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{4 a^3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{11/4} \sqrt{a+\frac{b}{x^4}}}+\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{11/4} \sqrt{a+\frac{b}{x^4}}}-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^4)^(-5/2),x]

[Out]

(-7*Sqrt[b]*Sqrt[a + b/x^4])/(4*a^3*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(6*a*(a + b/x^4)^(3/2)) - (7*x)/(12*a^2*Sqr
t[a + b/x^4]) + (7*Sqrt[a + b/x^4]*x)/(4*a^3) + (7*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])/(4*a^(11/4)*Sqrt[a + b/x^4]) - (7*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])
/(8*a^(11/4)*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 )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{6 a}\\ &=-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^2}\\ &=-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{7 \sqrt{a+\frac{b}{x^4}} x}{4 a^3}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^3}\\ &=-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{7 \sqrt{a+\frac{b}{x^4}} x}{4 a^3}-\frac{\left (7 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{4 a^{5/2}}+\frac{\left (7 \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 )}{4 a^{5/2}}\\ &=-\frac{7 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{4 a^3 \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}+\frac{7 \sqrt{a+\frac{b}{x^4}} x}{4 a^3}+\frac{7 \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 )}{4 a^{11/4} \sqrt{a+\frac{b}{x^4}}}-\frac{7 \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 )}{8 a^{11/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.0405056, size = 81, normalized size = 0.29 \[ \frac{-7 x \left (a x^4+b\right ) \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{a x^4}{b}\right )+3 a x^5+7 b x}{3 a^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^4)^(-5/2),x]

[Out]

(7*b*x + 3*a*x^5 - 7*x*(b + a*x^4)*Sqrt[1 + (a*x^4)/b]*Hypergeometric2F1[3/4, 5/2, 7/4, -((a*x^4)/b)])/(3*a^2*
Sqrt[a + b/x^4]*(b + a*x^4))

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Maple [C]  time = 0.017, size = 503, normalized size = 1.8 \begin{align*} -{\frac{1}{12\,{x}^{10}} \left ( 9\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{9/2}{x}^{11}-21\,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}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{b}{x}^{8}{a}^{4}+21\,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}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{b}{x}^{8}{a}^{4}+16\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{7/2}{x}^{7}b-42\,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}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{3}{2}}}{x}^{4}{a}^{3}+42\,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}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{3}{2}}}{x}^{4}{a}^{3}+7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{5/2}{x}^{3}{b}^{2}-21\,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}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{5}{2}}}{a}^{2}+21\,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}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{5}{2}}}{a}^{2} \right ){a}^{-{\frac{9}{2}}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\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)^(5/2),x)

[Out]

-1/12*(9*(I*a^(1/2)/b^(1/2))^(1/2)*a^(9/2)*x^11-21*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)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(1/2)*x^8*a^4+21*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)*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(1/2)*x^8*
a^4+16*(I*a^(1/2)/b^(1/2))^(1/2)*a^(7/2)*x^7*b-42*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)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(3/2)*x^4*a^3+42*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)*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(3/2)*x^4*a
^3+7*(I*a^(1/2)/b^(1/2))^(1/2)*a^(5/2)*x^3*b^2-21*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)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(5/2)*a^2+21*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)*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(5/2)*a^2)/a^(9
/2)/((a*x^4+b)/x^4)^(5/2)/x^10/(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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(5/2),x, algorithm="maxima")

[Out]

integrate((a + b/x^4)^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{12} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a^{3} x^{12} + 3 \, a^{2} b x^{8} + 3 \, a b^{2} x^{4} + b^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(5/2),x, algorithm="fricas")

[Out]

integral(x^12*sqrt((a*x^4 + b)/x^4)/(a^3*x^12 + 3*a^2*b*x^8 + 3*a*b^2*x^4 + b^3), x)

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Sympy [C]  time = 1.62733, size = 41, normalized size = 0.15 \begin{align*} - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{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)**(5/2),x)

[Out]

-x*gamma(-1/4)*hyper((-1/4, 5/2), (3/4,), b*exp_polar(I*pi)/(a*x**4))/(4*a**(5/2)*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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(5/2),x, algorithm="giac")

[Out]

integrate((a + b/x^4)^(-5/2), x)