3.1001 \(\int x^2 \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=77 \[ -\frac{a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a+b x^4} \]

[Out]

(x^3*(a + b*x^4)^(1/4))/4 - (a*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(3/4)
) + (a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(3/4))

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Rubi [A]  time = 0.0759054, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^4)^(1/4),x]

[Out]

(x^3*(a + b*x^4)^(1/4))/4 - (a*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(3/4)
) + (a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(3/4))

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Rubi in Sympy [A]  time = 10.5406, size = 68, normalized size = 0.88 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{3}{4}}} + \frac{a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{3}{4}}} + \frac{x^{3} \sqrt [4]{a + b x^{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(8*b**(3/4)) + a*atanh(b**(1/4)*x/(a + b
*x**4)**(1/4))/(8*b**(3/4)) + x**3*(a + b*x**4)**(1/4)/4

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Mathematica [C]  time = 0.0434783, size = 63, normalized size = 0.82 \[ \frac{x^3 \left (a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+3 \left (a+b x^4\right )\right )}{12 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(3*(a + b*x^4) + a*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -
((b*x^4)/a)]))/(12*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{x}^{2}\sqrt [4]{b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^4+a)^(1/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283194, size = 234, normalized size = 3.04 \[ \frac{1}{4} \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3} - \frac{1}{4} \, \left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x}{x \sqrt{\frac{\sqrt{\frac{a^{4}}{b^{3}}} b^{2} x^{2} + \sqrt{b x^{4} + a} a^{2}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}\right ) + \frac{1}{16} \, \left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{\left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) - \frac{1}{16} \, \left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*x^4 + a)^(1/4)*x^3 - 1/4*(a^4/b^3)^(1/4)*arctan((a^4/b^3)^(1/4)*b*x/(x*sq
rt((sqrt(a^4/b^3)*b^2*x^2 + sqrt(b*x^4 + a)*a^2)/x^2) + (b*x^4 + a)^(1/4)*a)) +
1/16*(a^4/b^3)^(1/4)*log(((a^4/b^3)^(1/4)*b*x + (b*x^4 + a)^(1/4)*a)/x) - 1/16*(
a^4/b^3)^(1/4)*log(-((a^4/b^3)^(1/4)*b*x - (b*x^4 + a)^(1/4)*a)/x)

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Sympy [A]  time = 4.18308, size = 39, normalized size = 0.51 \[ \frac{\sqrt [4]{a} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/4)*x**3*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*x**4*exp_polar(I*pi)/a)/(4
*gamma(7/4))

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GIAC/XCAS [A]  time = 0.235306, size = 304, normalized size = 3.95 \[ \frac{1}{32} \,{\left (\frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}{a} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} - \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(8*(b*x^4 + a)^(1/4)*x^3/a + 2*sqrt(2)*(-b)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(
2)*(-b)^(1/4) + 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b + 2*sqrt(2)*(-b)^(1/4)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) - 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b + sqr
t(2)*(-b)^(1/4)*ln(sqrt(-b) + sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^
4 + a)/x^2)/b - sqrt(2)*(-b)^(1/4)*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^
(1/4)/x + sqrt(b*x^4 + a)/x^2)/b)*a