3.1002 \(\int \frac{\sqrt [4]{a+b x^4}}{x^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\sqrt [4]{a+b x^4}}{x}-\frac{1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ) \]

[Out]

-((a + b*x^4)^(1/4)/x) - (b^(1/4)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/2 + (b^
(1/4)*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/2

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Rubi [A]  time = 0.0725216, antiderivative size = 73, 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 \[ -\frac{\sqrt [4]{a+b x^4}}{x}-\frac{1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^2,x]

[Out]

-((a + b*x^4)^(1/4)/x) - (b^(1/4)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/2 + (b^
(1/4)*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/2

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Rubi in Sympy [A]  time = 10.4698, size = 61, normalized size = 0.84 \[ - \frac{\sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2} + \frac{\sqrt [4]{b} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2} - \frac{\sqrt [4]{a + b x^{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(1/4)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/2 + b**(1/4)*atanh(b**(1/4)*x/(a
+ b*x**4)**(1/4))/2 - (a + b*x**4)**(1/4)/x

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Mathematica [C]  time = 0.0401937, size = 66, normalized size = 0.9 \[ \frac{b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-3 \left (a+b x^4\right )}{3 x \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^2,x)

[Out]

int((b*x^4+a)^(1/4)/x^2,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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 3.84295, size = 41, normalized size = 0.56 \[ \frac{\sqrt [4]{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232325, size = 281, normalized size = 3.85 \[ \frac{1}{4} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + \frac{1}{4} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + \frac{1}{8} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - \frac{1}{8} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*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)) + 1/4*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)) + 1/8*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) - 1/8*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*x^4 + a)^(1/4)/x