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

Optimal. Leaf size=79 \[ \frac{a^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{1}{3} x^2 \sqrt [4]{a+b x^4} \]

[Out]

(x^2*(a + b*x^4)^(1/4))/3 + (a^(3/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqr
t[b]*x^2)/Sqrt[a]]/2, 2])/(3*Sqrt[b]*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.091143, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{a^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{1}{3} x^2 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^4)^(1/4),x]

[Out]

(x^2*(a + b*x^4)^(1/4))/3 + (a^(3/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqr
t[b]*x^2)/Sqrt[a]]/2, 2])/(3*Sqrt[b]*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 8.27723, size = 66, normalized size = 0.84 \[ \frac{a^{\frac{3}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{x^{2} \sqrt [4]{a + b x^{4}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2, 2)/(3*sq
rt(b)*(a + b*x**4)**(3/4)) + x**2*(a + b*x**4)**(1/4)/3

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Mathematica [C]  time = 0.0383573, size = 63, normalized size = 0.8 \[ \frac{x^2 \left (a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+2 \left (a+b x^4\right )\right )}{6 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(2*(a + b*x^4) + a*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -
((b*x^4)/a)]))/(6*(a + b*x^4)^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(1/4)*x, x)

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Sympy [A]  time = 2.33196, size = 29, normalized size = 0.37 \[ \frac{\sqrt [4]{a} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)*x, x)