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

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

[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 = 0.181405, 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 \[ -\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{2}{a x^5 \sqrt [4]{a-b x^2}} \]

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))

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Rubi in Sympy [A]  time = 24.2878, size = 133, normalized size = 0.88 \[ \frac{2}{a x^{5} \sqrt [4]{a - b x^{2}}} - \frac{11 \left (a - b x^{2}\right )^{\frac{3}{4}}}{5 a^{2} x^{5}} - \frac{77 b \left (a - b x^{2}\right )^{\frac{3}{4}}}{30 a^{3} x^{3}} - \frac{77 b^{2} \left (a - b x^{2}\right )^{\frac{3}{4}}}{20 a^{4} x} - \frac{77 b^{\frac{5}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{20 a^{\frac{7}{2}} \sqrt [4]{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(-b*x**2+a)**(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)*elliptic_e(asin(sqrt(b)*x/sqrt(a))/2, 2)/(20*a**(7
/2)*(a - b*x**2)**(1/4))

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Mathematica [C]  time = 0.0704395, size = 95, normalized size = 0.63 \[ \frac{-24 a^3-44 a^2 b x^2-231 b^3 x^6 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-154 a b^2 x^4+462 b^3 x^6}{120 a^4 x^5 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-24*a^3 - 44*a^2*b*x^2 - 154*a*b^2*x^4 + 462*b^3*x^6 - 231*b^3*x^6*(1 - (b*x^2)
/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a])/(120*a^4*x^5*(a - b*x^2)^
(1/4))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

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. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(5/4)*x^6),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. \[{\rm integral}\left (-\frac{1}{{\left (b x^{8} - a x^{6}\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((b*x^8 - a*x^6)*(-b*x^2 + a)^(1/4)), x)

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Sympy [A]  time = 7.19762, size = 34, normalized size = 0.23 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{5}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac{5}{4}} x^{5}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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