3.2065 \(\int \left (a+\frac{b}{x^4}\right )^{3/2} x \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x^4}}}{4 x^2}-\frac{3}{4} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.139001, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x^4}}}{4 x^2}-\frac{3}{4} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.63545, size = 63, normalized size = 0.91 \[ - \frac{3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{4} - \frac{3 b \sqrt{a + \frac{b}{x^{4}}}}{4 x^{2}} + \frac{x^{2} \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.113487, size = 79, normalized size = 1.14 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (\left (b-2 a x^4\right ) \sqrt{a x^4+b}+3 a \sqrt{b} x^4 \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )\right )}{4 x^2 \sqrt{a x^4+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 85, normalized size = 1.2 \[ -{\frac{{x}^{2}}{4} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,\sqrt{b}a\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{4}-2\,a{x}^{4}\sqrt{a{x}^{4}+b}+b\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)*x,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250062, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{b} x^{2} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) + 2 \,{\left (2 \, a x^{4} - b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{8 \, x^{2}}, -\frac{3 \, a \sqrt{-b} x^{2} \arctan \left (\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-b}}\right ) -{\left (2 \, a x^{4} - b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{4 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*a*sqrt(b)*x^2*log((a*x^4 - 2*sqrt(b)*x^2*sqrt((a*x^4 + b)/x^4) + 2*b)/x^
4) + 2*(2*a*x^4 - b)*sqrt((a*x^4 + b)/x^4))/x^2, -1/4*(3*a*sqrt(-b)*x^2*arctan(x
^2*sqrt((a*x^4 + b)/x^4)/sqrt(-b)) - (2*a*x^4 - b)*sqrt((a*x^4 + b)/x^4))/x^2]

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Sympy [A]  time = 12.2408, size = 95, normalized size = 1.38 \[ \frac{a^{\frac{3}{2}} x^{2}}{2 \sqrt{1 + \frac{b}{a x^{4}}}} + \frac{\sqrt{a} b}{4 x^{2} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{4} - \frac{b^{2}}{4 \sqrt{a} x^{6} \sqrt{1 + \frac{b}{a x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224268, size = 77, normalized size = 1.12 \[ \frac{1}{4} \,{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{a x^{4} + b} - \frac{\sqrt{a x^{4} + b} b}{a x^{4}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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