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

Optimal. Leaf size=18 \[ \frac{3 \left (a+b x^2\right )^{5/3}}{10 b} \]

[Out]

(3*(a + b*x^2)^(5/3))/(10*b)

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Rubi [A]  time = 0.0113831, 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 \[ \frac{3 \left (a+b x^2\right )^{5/3}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(5/3))/(10*b)

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Rubi in Sympy [A]  time = 2.14743, size = 14, normalized size = 0.78 \[ \frac{3 \left (a + b x^{2}\right )^{\frac{5}{3}}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(a + b*x**2)**(5/3)/(10*b)

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Mathematica [A]  time = 0.00827124, size = 18, normalized size = 1. \[ \frac{3 \left (a+b x^2\right )^{5/3}}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x^2)^(5/3))/(10*b)

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Maple [A]  time = 0.005, size = 15, normalized size = 0.8 \[{\frac{3}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^(2/3),x)

[Out]

3/10*(b*x^2+a)^(5/3)/b

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Maxima [A]  time = 1.32962, size = 19, normalized size = 1.06 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}}}{10 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(b*x^2 + a)^(5/3)/b

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Fricas [A]  time = 0.208338, size = 19, normalized size = 1.06 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}}}{10 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(b*x^2 + a)^(5/3)/b

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Sympy [A]  time = 1.06609, size = 42, normalized size = 2.33 \[ \begin{cases} \frac{3 a \left (a + b x^{2}\right )^{\frac{2}{3}}}{10 b} + \frac{3 x^{2} \left (a + b x^{2}\right )^{\frac{2}{3}}}{10} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{2}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((3*a*(a + b*x**2)**(2/3)/(10*b) + 3*x**2*(a + b*x**2)**(2/3)/10, Ne(b,
0)), (a**(2/3)*x**2/2, True))

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GIAC/XCAS [A]  time = 0.21362, size = 19, normalized size = 1.06 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}}}{10 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(b*x^2 + a)^(5/3)/b