3.794 \(\int x^2 \sqrt [4]{a-b x^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{4 a^{5/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a-b x^2\right )^{3/4}}-\frac{2 a x \sqrt [4]{a-b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt [4]{a-b x^2} \]

[Out]

(-2*a*x*(a - b*x^2)^(1/4))/(21*b) + (2*x^3*(a - b*x^2)^(1/4))/7 + (4*a^(5/2)*(1
- (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*b^(3/2)*(a -
b*x^2)^(3/4))

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Rubi [A]  time = 0.102623, antiderivative size = 101, 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 \[ \frac{4 a^{5/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a-b x^2\right )^{3/4}}-\frac{2 a x \sqrt [4]{a-b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt [4]{a-b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a - b*x^2)^(1/4),x]

[Out]

(-2*a*x*(a - b*x^2)^(1/4))/(21*b) + (2*x^3*(a - b*x^2)^(1/4))/7 + (4*a^(5/2)*(1
- (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*b^(3/2)*(a -
b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 14.465, size = 87, normalized size = 0.86 \[ \frac{4 a^{\frac{5}{2}} \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{21 b^{\frac{3}{2}} \left (a - b x^{2}\right )^{\frac{3}{4}}} - \frac{2 a x \sqrt [4]{a - b x^{2}}}{21 b} + \frac{2 x^{3} \sqrt [4]{a - b x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(5/2)*(1 - b*x**2/a)**(3/4)*elliptic_f(asin(sqrt(b)*x/sqrt(a))/2, 2)/(21*b*
*(3/2)*(a - b*x**2)**(3/4)) - 2*a*x*(a - b*x**2)**(1/4)/(21*b) + 2*x**3*(a - b*x
**2)**(1/4)/7

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Mathematica [C]  time = 0.0711751, size = 79, normalized size = 0.78 \[ \frac{2 \left (a^2 x \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )-a^2 x+4 a b x^3-3 b^2 x^5\right )}{21 b \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(a^2*x) + 4*a*b*x^3 - 3*b^2*x^5 + a^2*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometri
c2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(21*b*(a - b*x^2)^(3/4))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{x}^{2}\sqrt [4]{-b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-b*x^2+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4)*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-b*x^2 + a)^(1/4)*x^2, x)

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Sympy [A]  time = 2.54451, size = 31, normalized size = 0.31 \[ \frac{\sqrt [4]{a} x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*x**3*hyper((-1/4, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a)/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-b*x^2 + a)^(1/4)*x^2, x)