3.808 \(\int \frac{\left (a-b x^2\right )^{3/4}}{x^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0669219, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(3/4)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.75461, size = 63, normalized size = 0.83 \[ - \frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{\sqrt [4]{a - b x^{2}}} - \frac{\left (a - b x^{2}\right )^{\frac{3}{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(3/4)/x**2,x)

[Out]

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

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Mathematica [C]  time = 0.0409985, size = 70, normalized size = 0.92 \[ -\frac{3 b x \sqrt [4]{\frac{a-b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )}{2 \sqrt [4]{a-b x^2}}-\frac{\left (a-b x^2\right )^{3/4}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(3/4)/x^2,x]

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(3/4)/x^2,x)

[Out]

int((-b*x^2+a)^(3/4)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(3/4)/x^2,x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(3/4)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(3/4)/x^2,x, algorithm="fricas")

[Out]

integral((-b*x^2 + a)^(3/4)/x^2, x)

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Sympy [A]  time = 3.55102, size = 31, normalized size = 0.41 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(3/4)/x**2,x)

[Out]

-a**(3/4)*hyper((-3/4, -1/2), (1/2,), b*x**2*exp_polar(2*I*pi)/a)/x

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(3/4)/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError