3.2067 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{16 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2} \]

[Out]

(-3*a*Sqrt[a + b/x^4])/(16*x^2) - (a + b/x^4)^(3/2)/(8*x^2) - (3*a^2*ArcTanh[Sqr
t[b]/(Sqrt[a + b/x^4]*x^2)])/(16*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.131929, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{16 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)/x^3,x]

[Out]

(-3*a*Sqrt[a + b/x^4])/(16*x^2) - (a + b/x^4)^(3/2)/(8*x^2) - (3*a^2*ArcTanh[Sqr
t[b]/(Sqrt[a + b/x^4]*x^2)])/(16*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.35633, size = 66, normalized size = 0.93 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{16 \sqrt{b}} - \frac{3 a \sqrt{a + \frac{b}{x^{4}}}}{16 x^{2}} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{8 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*atanh(sqrt(b)/(x**2*sqrt(a + b/x**4)))/(16*sqrt(b)) - 3*a*sqrt(a + b/x**
4)/(16*x**2) - (a + b/x**4)**(3/2)/(8*x**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.141109, size = 70, normalized size = 0.99 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (-\frac{3 a^2 x^8 \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{a x^4+b}}-5 a x^4-2 b\right )}{16 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*(-2*b - 5*a*x^4 - (3*a^2*x^8*ArcTanh[Sqrt[b + a*x^4]/Sqrt[b]])/
(Sqrt[b]*Sqrt[b + a*x^4])))/(16*x^6)

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 93, normalized size = 1.3 \[ -{\frac{1}{16\,{x}^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{8}+5\,a\sqrt{a{x}^{4}+b}{x}^{4}\sqrt{b}+2\,{b}^{3/2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)/x^3,x)

[Out]

-1/16*((a*x^4+b)/x^4)^(3/2)/x^2*(3*a^2*ln(2*(b^(1/2)*(a*x^4+b)^(1/2)+b)/x^2)*x^8
+5*a*(a*x^4+b)^(1/2)*x^4*b^(1/2)+2*b^(3/2)*(a*x^4+b)^(1/2))/(a*x^4+b)^(3/2)/b^(1
/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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25221, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} x^{6} \log \left (-\frac{2 \, b x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + 2 \, b\right )} \sqrt{b}}{x^{4}}\right ) - 2 \,{\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{32 \, b x^{6}}, -\frac{3 \, a^{2} \sqrt{-b} x^{6} \arctan \left (\frac{b}{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{16 \, b x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/32*(3*a^2*sqrt(b)*x^6*log(-(2*b*x^2*sqrt((a*x^4 + b)/x^4) - (a*x^4 + 2*b)*sqr
t(b))/x^4) - 2*(5*a*b*x^4 + 2*b^2)*sqrt((a*x^4 + b)/x^4))/(b*x^6), -1/16*(3*a^2*
sqrt(-b)*x^6*arctan(b/(sqrt(-b)*x^2*sqrt((a*x^4 + b)/x^4))) + (5*a*b*x^4 + 2*b^2
)*sqrt((a*x^4 + b)/x^4))/(b*x^6)]

_______________________________________________________________________________________

Sympy [A]  time = 13.076, size = 75, normalized size = 1.06 \[ - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{16 x^{2}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{4}}}}{8 x^{6}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{16 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**(3/2)*sqrt(1 + b/(a*x**4))/(16*x**2) - sqrt(a)*b*sqrt(1 + b/(a*x**4))/(8*x
**6) - 3*a**2*asinh(sqrt(b)/(sqrt(a)*x**2))/(16*sqrt(b))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23397, size = 82, normalized size = 1.15 \[ \frac{1}{16} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{4} + b} b}{a^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*a^2*(3*arctan(sqrt(a*x^4 + b)/sqrt(-b))/sqrt(-b) - (5*(a*x^4 + b)^(3/2) - 3
*sqrt(a*x^4 + b)*b)/(a^2*x^8))