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

Optimal. Leaf size=122 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b} \]

[Out]

(8*a^2*x)/(15*b^2*(a + b*x^2)^(1/4)) - (4*a*x*(a + b*x^2)^(3/4))/(15*b^2) + (2*x
^3*(a + b*x^2)^(3/4))/(9*b) - (8*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[
(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(15*b^(5/2)*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.121214, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(1/4),x]

[Out]

(8*a^2*x)/(15*b^2*(a + b*x^2)^(1/4)) - (4*a*x*(a + b*x^2)^(3/4))/(15*b^2) + (2*x
^3*(a + b*x^2)^(3/4))/(9*b) - (8*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[
(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(15*b^(5/2)*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 a^{3} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{15 b^{2}} + \frac{8 a^{2} x}{15 b^{2} \sqrt [4]{a + b x^{2}}} - \frac{4 a x \left (a + b x^{2}\right )^{\frac{3}{4}}}{15 b^{2}} + \frac{2 x^{3} \left (a + b x^{2}\right )^{\frac{3}{4}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**3*Integral((a + b*x**2)**(-5/4), x)/(15*b**2) + 8*a**2*x/(15*b**2*(a + b*x
**2)**(1/4)) - 4*a*x*(a + b*x**2)**(3/4)/(15*b**2) + 2*x**3*(a + b*x**2)**(3/4)/
(9*b)

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Mathematica [C]  time = 0.056046, size = 79, normalized size = 0.65 \[ \frac{2 \left (6 a^2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-6 a^2 x-a b x^3+5 b^2 x^5\right )}{45 b^2 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-6*a^2*x - a*b*x^3 + 5*b^2*x^5 + 6*a^2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometri
c2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(45*b^2*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{{x}^{4}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^2 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(b*x^2 + a)^(1/4), x)

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Sympy [A]  time = 2.57385, size = 27, normalized size = 0.22 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*hyper((1/4, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^2 + a)^(1/4), x)