3.2081 \(\int \frac{(a+\frac{b}{x^4})^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=278 \[ -\frac{4 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^3 \sqrt{a+\frac{b}{x^4}}}{39 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3} \]

[Out]

(-4*a^2*Sqrt[a + b/x^4])/(39*x^3) - (10*a*(a + b/x^4)^(3/2))/(117*x^3) - (a + b/x^4)^(5/2)/(13*x^3) - (8*a^3*S
qrt[a + b/x^4])/(39*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2)*x) + (8*a^(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4]) - (4*a^
(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.163778, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{4 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^3 \sqrt{a+\frac{b}{x^4}}}{39 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^4)^(5/2)/x^4,x]

[Out]

(-4*a^2*Sqrt[a + b/x^4])/(39*x^3) - (10*a*(a + b/x^4)^(3/2))/(117*x^3) - (a + b/x^4)^(5/2)/(13*x^3) - (8*a^3*S
qrt[a + b/x^4])/(39*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2)*x) + (8*a^(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4]) - (4*a^
(13/4)*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])/(39*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{\left (a+\frac{b}{x^4}\right )^{5/2}}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \left (a+b x^4\right )^{5/2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3}-\frac{1}{13} (10 a) \operatorname{Subst}\left (\int x^2 \left (a+b x^4\right )^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3}-\frac{1}{39} \left (20 a^2\right ) \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3}-\frac{1}{39} \left (8 a^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3}-\frac{\left (8 a^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{39 \sqrt{b}}+\frac{\left (8 a^{7/2}\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 )}{39 \sqrt{b}}\\ &=-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3}-\frac{8 a^3 \sqrt{a+\frac{b}{x^4}}}{39 \sqrt{b} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}+\frac{8 a^{13/4} \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 )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{4 a^{13/4} \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 )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.0148682, size = 54, normalized size = 0.19 \[ -\frac{b^2 \sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{13}{4},-\frac{5}{2};-\frac{9}{4};-\frac{a x^4}{b}\right )}{13 x^{11} \sqrt{\frac{a x^4}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^4)^(5/2)/x^4,x]

[Out]

-(b^2*Sqrt[a + b/x^4]*Hypergeometric2F1[-13/4, -5/2, -9/4, -((a*x^4)/b)])/(13*x^11*Sqrt[1 + (a*x^4)/b])

________________________________________________________________________________________

Maple [C]  time = 0.02, size = 279, normalized size = 1. \begin{align*}{\frac{1}{117\,{x}^{3} \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 24\,i{a}^{{\frac{7}{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}^{13}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) -24\,i{a}^{{\frac{7}{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}^{13}b{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) -24\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}\sqrt{b}{x}^{16}{a}^{4}-55\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3/2}{x}^{12}{a}^{3}-59\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{5/2}{x}^{8}{a}^{2}-37\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{7/2}{x}^{4}a-9\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{9/2} \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)^(5/2)/x^4,x)

[Out]

1/117*((a*x^4+b)/x^4)^(5/2)*(24*I*a^(7/2)*(-(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^13*b*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)-24*I*a^(7/2)*(-(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^13*b*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)-24*(I*a^(1/2)/b^(
1/2))^(1/2)*b^(1/2)*x^16*a^4-55*(I*a^(1/2)/b^(1/2))^(1/2)*b^(3/2)*x^12*a^3-59*(I*a^(1/2)/b^(1/2))^(1/2)*b^(5/2
)*x^8*a^2-37*(I*a^(1/2)/b^(1/2))^(1/2)*b^(7/2)*x^4*a-9*(I*a^(1/2)/b^(1/2))^(1/2)*b^(9/2))/x^3/(a*x^4+b)^3/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{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x^4)^(5/2)/x^4, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 2.25858, size = 41, normalized size = 0.15 \begin{align*} - \frac{a^{\frac{5}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{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)**(5/2)/x**4,x)

[Out]

-a**(5/2)*gamma(3/4)*hyper((-5/2, 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{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x^4)^(5/2)/x^4, x)