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

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

[Out]

-2/(3*b*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.0107943, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2}{3 b \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^3)^(3/2),x]

[Out]

-2/(3*b*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 2.19983, size = 15, normalized size = 0.83 \[ - \frac{2}{3 b \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*b*sqrt(a + b*x**3))

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Mathematica [A]  time = 0.0104929, size = 18, normalized size = 1. \[ -\frac{2}{3 b \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(3*b*Sqrt[a + b*x^3])

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Maple [A]  time = 0.005, size = 15, normalized size = 0.8 \[ -{\frac{2}{3\,b}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44872, size = 19, normalized size = 1.06 \[ -\frac{2}{3 \, \sqrt{b x^{3} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(sqrt(b*x^3 + a)*b)

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Fricas [A]  time = 0.222588, size = 19, normalized size = 1.06 \[ -\frac{2}{3 \, \sqrt{b x^{3} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(sqrt(b*x^3 + a)*b)

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Sympy [A]  time = 2.04845, size = 26, normalized size = 1.44 \[ \begin{cases} - \frac{2}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209972, size = 19, normalized size = 1.06 \[ -\frac{2}{3 \, \sqrt{b x^{3} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(sqrt(b*x^3 + a)*b)