3.1084 \(\int \frac{1}{x^5 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=78 \[ -\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}-\frac{\left (a+b x^4\right )^{3/4}}{4 a x^4} \]

[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 = 0.112341, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{5/4}}-\frac{\left (a+b x^4\right )^{3/4}}{4 a x^4} \]

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 = 12.1201, size = 66, normalized size = 0.85 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{4 a x^{4}} - \frac{b \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}}} \]

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 = 0.0560111, size = 69, normalized size = 0.88 \[ \frac{b x^4 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )-a-b x^4}{4 a x^4 \sqrt [4]{a+b x^4}} \]

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 = 0.047, 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 = 0.330745, size = 247, normalized size = 3.17 \[ \frac{4 \, a x^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} + \sqrt{a^{3} b^{4} \sqrt{\frac{b^{4}}{a^{5}}} + \sqrt{b x^{4} + a} b^{6}}}\right ) + a x^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - a x^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (\frac{b^{4}}{a^{5}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - 4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{16 \, a x^{4}} \]

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)*lo
g(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 = 5.11182, size = 39, normalized size = 0.5 \[ - \frac{\Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x^{5} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226952, size = 281, normalized size = 3.6 \[ \frac{1}{32} \, b{\left (\frac{2 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{2 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{a b x^{4}}\right )} \]

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))