3.1208 \(\int \frac{1}{x^5 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=81 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}-\frac{\left (a-b x^4\right )^{3/4}}{4 a x^4} \]

[Out]

-(a - b*x^4)^(3/4)/(4*a*x^4) + (b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(5/4))
- (b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(5/4))

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Rubi [A]  time = 0.120557, antiderivative size = 81, 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 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}-\frac{\left (a-b x^4\right )^{3/4}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a - b*x^4)^(3/4)/(4*a*x^4) + (b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(5/4))
- (b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(5/4))

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Rubi in Sympy [A]  time = 12.9221, size = 66, normalized size = 0.81 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{4 a x^{4}} + \frac{b \operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a - b*x**4)**(3/4)/(4*a*x**4) + b*atan((a - b*x**4)**(1/4)/a**(1/4))/(8*a**(5/
4)) - b*atanh((a - b*x**4)**(1/4)/a**(1/4))/(8*a**(5/4))

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Mathematica [C]  time = 0.0561116, size = 70, normalized size = 0.86 \[ \frac{-b x^4 \sqrt [4]{1-\frac{a}{b x^4}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{a}{b x^4}\right )-a+b x^4}{4 a x^4 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-b*x^4+a)^(1/4),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252146, size = 254, normalized size = 3.14 \[ -\frac{4 \, a x^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{3}{4}}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{3} + \sqrt{a^{3} b^{4} \sqrt{\frac{b^{4}}{a^{5}}} + \sqrt{-b x^{4} + a} b^{6}}}\right ) + a x^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - a x^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) + 4 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{16 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(4*a*x^4*(b^4/a^5)^(1/4)*arctan(a^4*(b^4/a^5)^(3/4)/((-b*x^4 + a)^(1/4)*b^
3 + sqrt(a^3*b^4*sqrt(b^4/a^5) + sqrt(-b*x^4 + a)*b^6))) + a*x^4*(b^4/a^5)^(1/4)
*log(a^4*(b^4/a^5)^(3/4) + (-b*x^4 + a)^(1/4)*b^3) - a*x^4*(b^4/a^5)^(1/4)*log(-
a^4*(b^4/a^5)^(3/4) + (-b*x^4 + a)^(1/4)*b^3) + 4*(-b*x^4 + a)^(3/4))/(a*x^4)

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Sympy [A]  time = 5.25813, size = 41, normalized size = 0.51 \[ \frac{e^{- \frac{5 i \pi }{4}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x^{5} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229973, size = 290, normalized size = 3.58 \[ -\frac{1}{32} \, b{\left (\frac{2 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{2 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} + \frac{8 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{a b x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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