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

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

[Out]

(a + b*x^2)^(5/2)/(5*b)

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Rubi [A]  time = 0.0037953, 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, Rules used = {261} \[ \frac{\left (a+b x^2\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0030838, size = 18, normalized size = 1. \[ \frac{\left (a+b x^2\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^(3/2),x]

[Out]

(a + b*x^2)^(5/2)/(5*b)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^(3/2),x)

[Out]

1/5*(b*x^2+a)^(5/2)/b

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Maxima [A]  time = 2.1725, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^2 + a)^(5/2)/b

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Fricas [B]  time = 1.51756, size = 69, normalized size = 3.83 \begin{align*} \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)/b

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Sympy [A]  time = 0.574784, size = 61, normalized size = 3.39 \begin{align*} \begin{cases} \frac{a^{2} \sqrt{a + b x^{2}}}{5 b} + \frac{2 a x^{2} \sqrt{a + b x^{2}}}{5} + \frac{b x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*sqrt(a + b*x**2)/(5*b) + 2*a*x**2*sqrt(a + b*x**2)/5 + b*x**4*sqrt(a + b*x**2)/5, Ne(b, 0)), (
a**(3/2)*x**2/2, True))

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Giac [A]  time = 3.16103, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^2 + a)^(5/2)/b