3.1261 \(\int \frac{1}{x^8 (a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{4 b^{5/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{7 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac{\sqrt [4]{a-b x^4}}{7 a x^7} \]

[Out]

-(a - b*x^4)^(1/4)/(7*a*x^7) - (2*b*(a - b*x^4)^(1/4))/(7*a^2*x^3) - (4*b^(5/2)*(1 - a/(b*x^4))^(3/4)*x^3*Elli
pticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(7*a^(5/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.0486328, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {325, 237, 335, 275, 232} \[ -\frac{4 b^{5/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac{\sqrt [4]{a-b x^4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(7*a*x^7) - (2*b*(a - b*x^4)^(1/4))/(7*a^2*x^3) - (4*b^(5/2)*(1 - a/(b*x^4))^(3/4)*x^3*Elli
pticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(7*a^(5/2)*(a - b*x^4)^(3/4))

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 237

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

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 232

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(3/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{7 a x^7}+\frac{(6 b) \int \frac{1}{x^4 \left (a-b x^4\right )^{3/4}} \, dx}{7 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{7 a x^7}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}+\frac{\left (4 b^2\right ) \int \frac{1}{\left (a-b x^4\right )^{3/4}} \, dx}{7 a^2}\\ &=-\frac{\sqrt [4]{a-b x^4}}{7 a x^7}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}+\frac{\left (4 b^2 \left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1-\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{7 a^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{7 a x^7}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac{\left (4 b^2 \left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{7 a^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{7 a x^7}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac{\left (2 b^2 \left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{7 a^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{7 a x^7}-\frac{2 b \sqrt [4]{a-b x^4}}{7 a^2 x^3}-\frac{4 b^{5/2} \left (1-\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} \left (a-b x^4\right )^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0106436, size = 52, normalized size = 0.47 \[ -\frac{\left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (-\frac{7}{4},\frac{3}{4};-\frac{3}{4};\frac{b x^4}{a}\right )}{7 x^7 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a - b*x^4)^(3/4)),x]

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{8}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^8/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{b x^{12} - a x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^4 + a)^(1/4)/(b*x^12 - a*x^8), x)

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Sympy [C]  time = 2.22842, size = 31, normalized size = 0.28 \begin{align*} \frac{i e^{- \frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{10 b^{\frac{3}{4}} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(-b*x**4+a)**(3/4),x)

[Out]

I*exp(-I*pi/4)*hyper((3/4, 5/2), (7/2,), a/(b*x**4))/(10*b**(3/4)*x**10)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^8), x)