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

Optimal. Leaf size=103 \[ \frac{b^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.100093, antiderivative size = 103, 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 \[ \frac{b^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(3/4)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8268, size = 82, normalized size = 0.8 \[ - \frac{\left (a - b x^{2}\right )^{\frac{3}{4}}}{3 x^{3}} + \frac{b \left (a - b x^{2}\right )^{\frac{3}{4}}}{2 a x} + \frac{b^{\frac{3}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \sqrt [4]{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0481872, size = 84, normalized size = 0.82 \[ \frac{-4 a^2+3 b^2 x^4 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+10 a b x^2-6 b^2 x^4}{12 a x^3 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2 + 10*a*b*x^2 - 6*b^2*x^4 + 3*b^2*x^4*(1 - (b*x^2)/a)^(1/4)*Hypergeometri
c2F1[1/4, 1/2, 3/2, (b*x^2)/a])/(12*a*x^3*(a - b*x^2)^(1/4))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(3/4)/x^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(3/4)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^2 + a)^(3/4)/x^4, x)

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Sympy [A]  time = 4.04885, size = 36, normalized size = 0.35 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^4,x, algorithm="giac")

[Out]

Exception raised: TypeError