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

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (a+b x^2\right )^{5/2}}{5 b} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2}}{5 b} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x^2)^(5/2)/(5*b)

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fricas [B]  time = 0.78, size = 32, normalized size = 1.78 \begin {gather*} \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{5 \, b} \end {gather*}

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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giac [A]  time = 0.67, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{5 \, b} \end {gather*}

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b} \end {gather*}

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 = 1.33, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{5 \, b} \end {gather*}

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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mupad [B]  time = 4.57, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b\,x^2+a\right )}^{5/2}}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^(3/2),x)

[Out]

(a + b*x^2)^(5/2)/(5*b)

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sympy [A]  time = 0.59, size = 61, normalized size = 3.39 \begin {gather*} \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 {gather*}

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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