3.856 \(\int \frac{1}{x^6 (a-b x^2)^{5/4}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{2}{a x^5 \sqrt [4]{a-b x^2}} \]

[Out]

2/(a*x^5*(a - b*x^2)^(1/4)) - (11*(a - b*x^2)^(3/4))/(5*a^2*x^5) - (77*b*(a - b*x^2)^(3/4))/(30*a^3*x^3) - (77
*b^2*(a - b*x^2)^(3/4))/(20*a^4*x) - (77*b^(5/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2
, 2])/(20*a^(7/2)*(a - b*x^2)^(1/4))

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Rubi [A]  time = 0.0610959, antiderivative size = 151, 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 = {290, 325, 229, 228} \[ -\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{2}{a x^5 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a - b*x^2)^(5/4)),x]

[Out]

2/(a*x^5*(a - b*x^2)^(1/4)) - (11*(a - b*x^2)^(3/4))/(5*a^2*x^5) - (77*b*(a - b*x^2)^(3/4))/(30*a^3*x^3) - (77
*b^2*(a - b*x^2)^(3/4))/(20*a^4*x) - (77*b^(5/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2
, 2])/(20*a^(7/2)*(a - b*x^2)^(1/4))

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 229

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

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 \left (a-b x^2\right )^{5/4}} \, dx &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}+\frac{11 \int \frac{1}{x^6 \sqrt [4]{a-b x^2}} \, dx}{a}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{(77 b) \int \frac{1}{x^4 \sqrt [4]{a-b x^2}} \, dx}{10 a^2}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}+\frac{\left (77 b^2\right ) \int \frac{1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a^3}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{\left (77 b^3\right ) \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^4}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{\left (77 b^3 \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{40 a^4 \sqrt [4]{a-b x^2}}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0102303, size = 55, normalized size = 0.36 \[ -\frac{\sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (-\frac{5}{2},\frac{5}{4};-\frac{3}{2};\frac{b x^2}{a}\right )}{5 a x^5 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a - b*x^2)^(5/4)),x]

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(-b*x^2+a)^(5/4),x)

[Out]

int(1/x^6/(-b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^2+a)^(5/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(5/4)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{b^{2} x^{10} - 2 \, a b x^{8} + a^{2} x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^2+a)^(5/4),x, algorithm="fricas")

[Out]

integral((-b*x^2 + a)^(3/4)/(b^2*x^10 - 2*a*b*x^8 + a^2*x^6), x)

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Sympy [C]  time = 1.78238, size = 34, normalized size = 0.23 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{5}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac{5}{4}} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(-b*x**2+a)**(5/4),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-b*x^2+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(5/4)*x^6), x)