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

Optimal. Leaf size=99 \[ -\frac{3 \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a-b x^2}}-\frac{3 \left (a-b x^2\right )^{3/4}}{a^2 x}+\frac{2}{a x \sqrt [4]{a-b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.101465, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a-b x^2}}-\frac{3 \left (a-b x^2\right )^{3/4}}{a^2 x}+\frac{2}{a x \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.4233, size = 82, normalized size = 0.83 \[ \frac{2}{a x \sqrt [4]{a - b x^{2}}} - \frac{3 \left (a - b x^{2}\right )^{\frac{3}{4}}}{a^{2} x} - \frac{3 \sqrt{b} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{a^{\frac{3}{2}} \sqrt [4]{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(-b*x**2+a)**(5/4),x)

[Out]

2/(a*x*(a - b*x**2)**(1/4)) - 3*(a - b*x**2)**(3/4)/(a**2*x) - 3*sqrt(b)*(1 - b*
x**2/a)**(1/4)*elliptic_e(asin(sqrt(b)*x/sqrt(a))/2, 2)/(a**(3/2)*(a - b*x**2)**
(1/4))

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Mathematica [C]  time = 0.0513109, size = 71, normalized size = 0.72 \[ \frac{-3 b x^2 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-2 a+6 b x^2}{2 a^2 x \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a + 6*b*x^2 - 3*b*x^2*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2,
(b*x^2)/a])/(2*a^2*x*(a - b*x^2)^(1/4))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-b*x^2+a)^(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (b x^{4} - a x^{2}\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.00131, size = 29, normalized size = 0.29 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac{5}{4}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(-b*x**2+a)**(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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