3.1009 \(\int x^4 \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=102 \[ \frac{a^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{a x \sqrt [4]{a+b x^4}}{12 b}+\frac{1}{6} x^5 \sqrt [4]{a+b x^4} \]

[Out]

(a*x*(a + b*x^4)^(1/4))/(12*b) + (x^5*(a + b*x^4)^(1/4))/6 + (a^(3/2)*(1 + a/(b*
x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*Sqrt[b]*(a +
b*x^4)^(3/4))

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Rubi [A]  time = 0.127131, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{a^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{a x \sqrt [4]{a+b x^4}}{12 b}+\frac{1}{6} x^5 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^4)^(1/4),x]

[Out]

(a*x*(a + b*x^4)^(1/4))/(12*b) + (x^5*(a + b*x^4)^(1/4))/6 + (a^(3/2)*(1 + a/(b*
x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*Sqrt[b]*(a +
b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 15.1155, size = 87, normalized size = 0.85 \[ \frac{a^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 \sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{a x \sqrt [4]{a + b x^{4}}}{12 b} + \frac{x^{5} \sqrt [4]{a + b x^{4}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2,
2)/(12*sqrt(b)*(a + b*x**4)**(3/4)) + a*x*(a + b*x**4)**(1/4)/(12*b) + x**5*(a
+ b*x**4)**(1/4)/6

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Mathematica [C]  time = 0.0496351, size = 76, normalized size = 0.75 \[ \frac{x \left (-a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+a^2+3 a b x^4+2 b^2 x^8\right )}{12 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a^2 + 3*a*b*x^4 + 2*b^2*x^8 - a^2*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/
4, 3/4, 5/4, -((b*x^4)/a)]))/(12*b*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{x}^{4}\sqrt [4]{b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)*x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(1/4)*x^4, x)

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Sympy [A]  time = 2.91306, size = 39, normalized size = 0.38 \[ \frac{\sqrt [4]{a} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*x**5*gamma(5/4)*hyper((-1/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4
*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(1/4)*x^4, x)