3.2089 \(\int \frac{x}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{a^2}-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

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

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Rubi [A]  time = 0.0508975, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{a^2}-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.34434, size = 32, normalized size = 0.8 \[ - \frac{x^{2}}{2 a \sqrt{a + \frac{b}{x^{4}}}} + \frac{x^{2} \sqrt{a + \frac{b}{x^{4}}}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0278523, size = 30, normalized size = 0.75 \[ \frac{a x^4+2 b}{2 a^2 x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 38, normalized size = 1. \[{\frac{ \left ( a{x}^{4}+b \right ) \left ( a{x}^{4}+2\,b \right ) }{2\,{a}^{2}{x}^{6}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^4)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.41906, size = 49, normalized size = 1.22 \[ \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a^{2}} + \frac{b}{2 \, \sqrt{a + \frac{b}{x^{4}}} a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239978, size = 57, normalized size = 1.42 \[ \frac{{\left (a x^{6} + 2 \, b x^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \,{\left (a^{3} x^{4} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.87987, size = 42, normalized size = 1.05 \[ \frac{x^{4}}{2 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} + \frac{\sqrt{b}}{a^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(a + b/x^4)^(3/2), x)