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

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)

________________________________________________________________________________________

Rubi [A]  time = 0.0046283, 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, Rules used = {261} \[ \frac{2 \left (a+b x^3\right )^{5/2}}{15 b} \]

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)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^2 \left (a+b x^3\right )^{3/2} \, dx &=\frac{2 \left (a+b x^3\right )^{5/2}}{15 b}\\ \end{align*}

Mathematica [A]  time = 0.0055945, 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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Maxima [A]  time = 0.951958, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b

________________________________________________________________________________________

Fricas [B]  time = 1.41843, size = 70, normalized size = 3.89 \begin{align*} \frac{2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \sqrt{b x^{3} + a}}{15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.939794, size = 65, normalized size = 3.61 \begin{align*} \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} \end{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.11496, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b