3.1231 \(\int \frac{1}{x^6 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2 b^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0370405, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {325, 313, 335, 275, 228} \[ -\frac{2 b^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a - b*x^4)^(1/4)),x]

[Out]

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

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 313

Int[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> Dist[(x*(1 + a/(b*x^4))^(1/4))/(a + b*x^4)^(1/4), Int
[1/(x^3*(1 + a/(b*x^4))^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[b/a]

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 228

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{x^6 \sqrt [4]{a-b x^4}} \, dx &=-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5}+\frac{(2 b) \int \frac{1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{5 a}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5}+\frac{\left (2 b \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^4}} x^3} \, dx}{5 a \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5}-\frac{\left (2 b \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1-\frac{a x^4}{b}}} \, dx,x,\frac{1}{x}\right )}{5 a \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5}-\frac{\left (b \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x^2}\right )}{5 a \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{5 a x^5}-\frac{2 b^{3/2} \sqrt [4]{1-\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0097345, size = 52, normalized size = 0.6 \[ -\frac{\sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{5}{4},\frac{1}{4};-\frac{1}{4};\frac{b x^4}{a}\right )}{5 x^5 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a - b*x^4)^(1/4)),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^6/(-b*x^4+a)^(1/4),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^6), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{b x^{10} - a x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^4 + a)^(3/4)/(b*x^10 - a*x^6), x)

________________________________________________________________________________________

Sympy [C]  time = 1.51678, size = 34, normalized size = 0.4 \begin{align*} - \frac{i e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{6 \sqrt [4]{b} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(-b*x**4+a)**(1/4),x)

[Out]

-I*exp(-3*I*pi/4)*hyper((1/4, 3/2), (5/2,), a/(b*x**4))/(6*b**(1/4)*x**6)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^6), x)