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

Optimal. Leaf size=18 \[ -\frac{3}{2 b \sqrt [3]{a+b x^2}} \]

[Out]

-3/(2*b*(a + b*x^2)^(1/3))

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Rubi [A]  time = 0.0118733, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{3}{2 b \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^2)^(4/3),x]

[Out]

-3/(2*b*(a + b*x^2)^(1/3))

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Rubi in Sympy [A]  time = 2.13948, size = 15, normalized size = 0.83 \[ - \frac{3}{2 b \sqrt [3]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/(2*b*(a + b*x**2)**(1/3))

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Mathematica [A]  time = 0.00543971, size = 18, normalized size = 1. \[ -\frac{3}{2 b \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-3/(2*b*(a + b*x^2)^(1/3))

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[ -{\frac{3}{2\,b}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^2+a)^(4/3),x)

[Out]

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

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Maxima [A]  time = 1.35662, size = 19, normalized size = 1.06 \[ -\frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2/((b*x^2 + a)^(1/3)*b)

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Fricas [A]  time = 0.212868, size = 19, normalized size = 1.06 \[ -\frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2/((b*x^2 + a)^(1/3)*b)

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Sympy [A]  time = 2.33188, size = 26, normalized size = 1.44 \[ \begin{cases} - \frac{3}{2 b \sqrt [3]{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{4}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3/(2*b*(a + b*x**2)**(1/3)), Ne(b, 0)), (x**2/(2*a**(4/3)), True))

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GIAC/XCAS [A]  time = 0.212069, size = 19, normalized size = 1.06 \[ -\frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2/((b*x^2 + a)^(1/3)*b)