3.1248 \(\int \frac{1}{x^{11} (a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=133 \[ \frac{3 b^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}} \]

[Out]

-(a - b*x^4)^(1/4)/(10*a*x^10) - (3*b*(a - b*x^4)^(1/4))/(20*a^2*x^6) - (3*b^2*(a - b*x^4)^(1/4))/(8*a^3*x^2)
+ (3*b^(5/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(8*a^(5/2)*(a - b*x^4)^(3/4)
)

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Rubi [A]  time = 0.0859719, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {275, 325, 233, 232} \[ -\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac{3 b^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^11*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(10*a*x^10) - (3*b*(a - b*x^4)^(1/4))/(20*a^2*x^6) - (3*b^2*(a - b*x^4)^(1/4))/(8*a^3*x^2)
+ (3*b^(5/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(8*a^(5/2)*(a - b*x^4)^(3/4)
)

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 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 233

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

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^{11} \left (a-b x^4\right )^{3/4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^6 \left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}}+\frac{(9 b) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{20 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{8 a^2}\\ &=-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{16 a^3}\\ &=-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac{\left (3 b^3 \left (1-\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{16 a^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac{3 b^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0110971, size = 52, normalized size = 0.39 \[ -\frac{\left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (-\frac{5}{2},\frac{3}{4};-\frac{3}{2};\frac{b x^4}{a}\right )}{10 x^{10} \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^11*(a - b*x^4)^(3/4)),x]

[Out]

-((1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, (b*x^4)/a])/(10*x^10*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{11}} \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^11/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^11/(-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^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^11), 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^{15} - a x^{11}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^4 + a)^(1/4)/(b*x^15 - a*x^11), x)

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Sympy [C]  time = 3.92006, size = 34, normalized size = 0.26 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{10 a^{\frac{3}{4}} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**11/(-b*x**4+a)**(3/4),x)

[Out]

-hyper((-5/2, 3/4), (-3/2,), b*x**4*exp_polar(2*I*pi)/a)/(10*a**(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^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^11), x)