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

Optimal. Leaf size=81 \[ \frac{4 a^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a-b x^2\right )^{3/4}}-\frac{2 x \sqrt [4]{a-b x^2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0700971, antiderivative size = 81, 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 \[ \frac{4 a^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a-b x^2\right )^{3/4}}-\frac{2 x \sqrt [4]{a-b x^2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a - b*x^2)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.6095, size = 68, normalized size = 0.84 \[ \frac{4 a^{\frac{3}{2}} \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 b^{\frac{3}{2}} \left (a - b x^{2}\right )^{\frac{3}{4}}} - \frac{2 x \sqrt [4]{a - b x^{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0510693, size = 64, normalized size = 0.79 \[ \frac{2 x \left (a \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )-a+b x^2\right )}{3 b \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^2+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(-b*x^2 + a)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(-b*x^2 + a)^(3/4), x)

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Sympy [A]  time = 2.45568, size = 29, normalized size = 0.36 \[ \frac{x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(-b*x^2 + a)^(3/4), x)