3.1179 \(\int \frac{\sqrt [4]{a-b x^4}}{x^5} \, dx\)

Optimal. Leaf size=78 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.107347, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5307, size = 65, normalized size = 0.83 \[ - \frac{\sqrt [4]{a - b x^{4}}}{4 x^{4}} + \frac{b \operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(1/4)/x**5,x)

[Out]

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

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Mathematica [C]  time = 0.040962, size = 67, normalized size = 0.86 \[ \frac{b x^4 \left (1-\frac{a}{b x^4}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{a}{b x^4}\right )-3 a+3 b x^4}{12 x^4 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(1/4)/x^5,x]

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}\sqrt [4]{-b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x^5,x)

[Out]

int((-b*x^4+a)^(1/4)/x^5,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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263484, size = 225, normalized size = 2.88 \[ -\frac{4 \, \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b + \sqrt{\sqrt{-b x^{4} + a} b^{2} + a^{2} \sqrt{\frac{b^{4}}{a^{3}}}}}\right ) - \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} b + a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) + \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} b - a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) + 4 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.28545, size = 42, normalized size = 0.54 \[ \frac{\sqrt [4]{b} e^{- \frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**4+a)**(1/4)/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22668, size = 286, normalized size = 3.67 \[ \frac{1}{32} \, b{\left (\frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a} - \frac{8 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{b x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)/x^5,x, algorithm="giac")

[Out]

1/32*b*(2*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x^4
+ a)^(1/4))/(-a)^(1/4))/a + 2*sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-
a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a + sqrt(2)*(-a)^(1/4)*ln(sqrt(2)*(
-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a - sqrt(2)*(-a)^(1/
4)*ln(-sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a -
8*(-b*x^4 + a)^(1/4)/(b*x^4))