3.407 \(\int \frac{x^2}{\sqrt{a+b x^3}} \, dx\)

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

[Out]

(2*Sqrt[a + b*x^3])/(3*b)

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Rubi [A]  time = 0.0113335, 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 \[ \frac{2 \sqrt{a+b x^3}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a + b*x^3],x]

[Out]

(2*Sqrt[a + b*x^3])/(3*b)

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Rubi in Sympy [A]  time = 2.17225, size = 14, normalized size = 0.78 \[ \frac{2 \sqrt{a + b x^{3}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a + b*x**3)/(3*b)

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Mathematica [A]  time = 0.00920303, size = 18, normalized size = 1. \[ \frac{2 \sqrt{a+b x^3}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^3])/(3*b)

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[{\frac{2}{3\,b}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43358, size = 19, normalized size = 1.06 \[ \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(b*x^3 + a)/b

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Fricas [A]  time = 0.221886, size = 19, normalized size = 1.06 \[ \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(b*x^3 + a)/b

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Sympy [A]  time = 1.59361, size = 24, normalized size = 1.33 \[ \begin{cases} \frac{2 \sqrt{a + b x^{3}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218793, size = 19, normalized size = 1.06 \[ \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(b*x^3 + a)/b