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

Optimal. Leaf size=18 \[ \frac{2 \left (a+b x^3\right )^{5/2}}{15 b} \]

[Out]

(2*(a + b*x^3)^(5/2))/(15*b)

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Rubi [A]  time = 0.0108311, 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 \[ \frac{2 \left (a+b x^3\right )^{5/2}}{15 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2))/(15*b)

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Rubi in Sympy [A]  time = 2.26303, size = 14, normalized size = 0.78 \[ \frac{2 \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a + b*x**3)**(5/2)/(15*b)

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Mathematica [A]  time = 0.0144207, size = 18, normalized size = 1. \[ \frac{2 \left (a+b x^3\right )^{5/2}}{15 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x^3)^(5/2))/(15*b)

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Maple [A]  time = 0.006, size = 15, normalized size = 0.8 \[{\frac{2}{15\,b} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^3+a)^(3/2),x)

[Out]

2/15*(b*x^3+a)^(5/2)/b

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Maxima [A]  time = 1.42774, size = 19, normalized size = 1.06 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b

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Fricas [A]  time = 0.216869, size = 43, normalized size = 2.39 \[ \frac{2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \sqrt{b x^{3} + a}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b^2*x^6 + 2*a*b*x^3 + a^2)*sqrt(b*x^3 + a)/b

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Sympy [A]  time = 2.02559, size = 65, normalized size = 3.61 \[ \begin{cases} \frac{2 a^{2} \sqrt{a + b x^{3}}}{15 b} + \frac{4 a x^{3} \sqrt{a + b x^{3}}}{15} + \frac{2 b x^{6} \sqrt{a + b x^{3}}}{15} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a**2*sqrt(a + b*x**3)/(15*b) + 4*a*x**3*sqrt(a + b*x**3)/15 + 2*b*x
**6*sqrt(a + b*x**3)/15, Ne(b, 0)), (a**(3/2)*x**3/3, True))

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GIAC/XCAS [A]  time = 0.239484, size = 19, normalized size = 1.06 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b