3.2066 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2} \]

[Out]

-(a*Sqrt[a + b/x^4])/2 - (a + b/x^4)^(3/2)/6 + (a^(3/2)*ArcTanh[Sqrt[a + b/x^4]/
Sqrt[a]])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.103374, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)/x,x]

[Out]

-(a*Sqrt[a + b/x^4])/2 - (a + b/x^4)^(3/2)/6 + (a^(3/2)*ArcTanh[Sqrt[a + b/x^4]/
Sqrt[a]])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.63705, size = 48, normalized size = 0.81 \[ \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2} - \frac{a \sqrt{a + \frac{b}{x^{4}}}}{2} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**4)**(3/2)/x,x)

[Out]

a**(3/2)*atanh(sqrt(a + b/x**4)/sqrt(a))/2 - a*sqrt(a + b/x**4)/2 - (a + b/x**4)
**(3/2)/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.0881435, size = 82, normalized size = 1.39 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (3 a^{3/2} x^6 \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )-\sqrt{a x^4+b} \left (4 a x^4+b\right )\right )}{6 x^4 \sqrt{a x^4+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(3/2)/x,x]

[Out]

(Sqrt[a + b/x^4]*(-(Sqrt[b + a*x^4]*(b + 4*a*x^4)) + 3*a^(3/2)*x^6*ArcTanh[(Sqrt
[a]*x^2)/Sqrt[b + a*x^4]]))/(6*x^4*Sqrt[b + a*x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 79, normalized size = 1.3 \[{\frac{1}{6} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{3/2}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{6}-4\,a{x}^{4}\sqrt{a{x}^{4}+b}-b\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)/x,x)

[Out]

1/6*((a*x^4+b)/x^4)^(3/2)*(3*a^(3/2)*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^6-4*a*x^4
*(a*x^4+b)^(1/2)-b*(a*x^4+b)^(1/2))/(a*x^4+b)^(3/2)

_______________________________________________________________________________________

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((a + b/x^4)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.255385, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a^{\frac{3}{2}} x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - 2 \,{\left (4 \, a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{12 \, x^{4}}, \frac{3 \, \sqrt{-a} a x^{4} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) -{\left (4 \, a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(3/2)/x,x, algorithm="fricas")

[Out]

[1/12*(3*a^(3/2)*x^4*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 2
*(4*a*x^4 + b)*sqrt((a*x^4 + b)/x^4))/x^4, 1/6*(3*sqrt(-a)*a*x^4*arctan(a/(sqrt(
-a)*sqrt((a*x^4 + b)/x^4))) - (4*a*x^4 + b)*sqrt((a*x^4 + b)/x^4))/x^4]

_______________________________________________________________________________________

Sympy [A]  time = 9.8054, size = 80, normalized size = 1.36 \[ - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{3} - \frac{a^{\frac{3}{2}} \log{\left (\frac{b}{a x^{4}} \right )}}{4} + \frac{a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{2} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{4}}}}{6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(3/2)/x,x)

[Out]

-2*a**(3/2)*sqrt(1 + b/(a*x**4))/3 - a**(3/2)*log(b/(a*x**4))/4 + a**(3/2)*log(s
qrt(1 + b/(a*x**4)) + 1)/2 - sqrt(a)*b*sqrt(1 + b/(a*x**4))/(6*x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.238974, size = 68, normalized size = 1.15 \[ -\frac{a^{2} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{6} \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(3/2)/x,x, algorithm="giac")

[Out]

-1/2*a^2*arctan(sqrt(a + b/x^4)/sqrt(-a))/sqrt(-a) - 1/6*(a + b/x^4)^(3/2) - 1/2
*sqrt(a + b/x^4)*a