3.1189 \(\int \frac{\sqrt [4]{a-b x^4}}{x^2} \, dx\)

Optimal. Leaf size=226 \[ -\frac{\sqrt [4]{a-b x^4}}{x}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt{2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{2 \sqrt{2}}-\frac{\sqrt [4]{b} \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.251496, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\sqrt [4]{a-b x^4}}{x}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt{2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{2 \sqrt{2}}-\frac{\sqrt [4]{b} \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2}} \]

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

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Rubi in Sympy [A]  time = 35.0631, size = 197, normalized size = 0.87 \[ - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{8} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{8} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{4} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{4} - \frac{\sqrt [4]{a - b x^{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*b**(1/4)*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqr
t(a - b*x**4) + 1)/8 + sqrt(2)*b**(1/4)*log(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/
4) + sqrt(b)*x**2/sqrt(a - b*x**4) + 1)/8 - sqrt(2)*b**(1/4)*atan(sqrt(2)*b**(1/
4)*x/(a - b*x**4)**(1/4) - 1)/4 - sqrt(2)*b**(1/4)*atan(sqrt(2)*b**(1/4)*x/(a -
b*x**4)**(1/4) + 1)/4 - (a - b*x**4)**(1/4)/x

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Mathematica [C]  time = 0.0430172, size = 68, normalized size = 0.3 \[ \frac{-b x^4 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-3 a+3 b x^4}{3 x \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a + 3*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 = 0.033, 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 = 4.16674, size = 42, normalized size = 0.19 \[ \frac{\sqrt [4]{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

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(2*I*pi)/a)/(4*
x*gamma(3/4))

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GIAC/XCAS [A]  time = 0.260656, size = 258, normalized size = 1.14 \[ \frac{1}{4} \, \sqrt{2} b^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right ) + \frac{1}{4} \, \sqrt{2} b^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right ) + \frac{1}{8} \, \sqrt{2} b^{\frac{1}{4}}{\rm ln}\left (\sqrt{b} + \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right ) - \frac{1}{8} \, \sqrt{2} b^{\frac{1}{4}}{\rm ln}\left (\sqrt{b} - \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right ) - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x} \]

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