3.2090 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2} x} \, dx\)

Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

-1/(2*a*Sqrt[a + b/x^4]) + ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]]/(2*a^(3/2))

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Rubi [A]  time = 0.0879748, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^4)^(3/2)*x),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]) + ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]]/(2*a^(3/2))

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Rubi in Sympy [A]  time = 7.12803, size = 37, normalized size = 0.8 \[ - \frac{1}{2 a \sqrt{a + \frac{b}{x^{4}}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(3/2)/x,x)

[Out]

-1/(2*a*sqrt(a + b/x**4)) + atanh(sqrt(a + b/x**4)/sqrt(a))/(2*a**(3/2))

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Mathematica [A]  time = 0.0398852, size = 70, normalized size = 1.52 \[ \frac{\sqrt{a x^4+b} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )-\sqrt{a} x^2}{2 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^4)^(3/2)*x),x]

[Out]

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

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Maple [A]  time = 0.02, size = 67, normalized size = 1.5 \[{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ( -{x}^{2}{a}^{{\frac{3}{2}}}+\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a\sqrt{a{x}^{4}+b} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(3/2)/x,x)

[Out]

1/2*(a*x^4+b)*(-x^2*a^(3/2)+ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*a*(a*x^4+b)^(1/2))/(
(a*x^4+b)/x^4)^(3/2)/x^6/a^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260137, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + b\right )} \sqrt{a} \log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{4 \,{\left (a^{3} x^{4} + a^{2} b\right )}}, -\frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} +{\left (a x^{4} + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{2 \,{\left (a^{3} x^{4} + a^{2} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.51045, size = 187, normalized size = 4.07 \[ - \frac{2 a^{3} x^{4} \sqrt{1 + \frac{b}{a x^{4}}}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} - \frac{a^{3} x^{4} \log{\left (\frac{b}{a x^{4}} \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{4} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{4}} \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(3/2)/x,x)

[Out]

-2*a**3*x**4*sqrt(1 + b/(a*x**4))/(4*a**(9/2)*x**4 + 4*a**(7/2)*b) - a**3*x**4*l
og(b/(a*x**4))/(4*a**(9/2)*x**4 + 4*a**(7/2)*b) + 2*a**3*x**4*log(sqrt(1 + b/(a*
x**4)) + 1)/(4*a**(9/2)*x**4 + 4*a**(7/2)*b) - a**2*b*log(b/(a*x**4))/(4*a**(9/2
)*x**4 + 4*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**4)) + 1)/(4*a**(9/2)*x**4
+ 4*a**(7/2)*b)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^4)^(3/2)*x), x)