3.817 \(\int \frac{x^2}{\sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=98 \[ \frac{4 a^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^2}}-\frac{4 a x}{5 b \sqrt [4]{a+b x^2}}+\frac{2 x \left (a+b x^2\right )^{3/4}}{5 b} \]

[Out]

(-4*a*x)/(5*b*(a + b*x^2)^(1/4)) + (2*x*(a + b*x^2)^(3/4))/(5*b) + (4*a^(3/2)*(1
+ (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*b^(3/2)*(a +
b*x^2)^(1/4))

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Rubi [A]  time = 0.078983, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{4 a^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^2}}-\frac{4 a x}{5 b \sqrt [4]{a+b x^2}}+\frac{2 x \left (a+b x^2\right )^{3/4}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^2)^(1/4),x]

[Out]

(-4*a*x)/(5*b*(a + b*x^2)^(1/4)) + (2*x*(a + b*x^2)^(3/4))/(5*b) + (4*a^(3/2)*(1
+ (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*b^(3/2)*(a +
b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a \int \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{5 b} + \frac{2 x \left (a + b x^{2}\right )^{\frac{3}{4}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*Integral((a + b*x**2)**(-1/4), x)/(5*b) + 2*x*(a + b*x**2)**(3/4)/(5*b)

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Mathematica [C]  time = 0.0456888, size = 62, normalized size = 0.63 \[ \frac{2 x \left (-a \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+a+b x^2\right )}{5 b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*
x^2)/a)]))/(5*b*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

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. \[ \int \frac{x^{2}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(b*x^2 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(b*x^2 + a)^(1/4), x)

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Sympy [A]  time = 2.37673, size = 27, normalized size = 0.28 \[ \frac{x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(b*x^2 + a)^(1/4), x)