3.4.56 \(\int x \sqrt {a+b x^2} \, dx\) [356]

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

[Out]

1/3*(b*x^2+a)^(3/2)/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 = {267} \begin {gather*} \frac {\left (a+b x^2\right )^{3/2}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + b*x^2],x]

[Out]

(a + b*x^2)^(3/2)/(3*b)

Rule 267

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

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[a + b*x^2],x]

[Out]

(a + b*x^2)^(3/2)/(3*b)

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Maple [A]
time = 0.04, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}\)	\(15\)
derivativedivides	\(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}\)	\(15\)
default	\(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}\)	\(15\)
trager	\(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}\)	\(15\)
risch	\(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(b*x^2+a)^(3/2)/b

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

1/3*(b*x^2 + a)^(3/2)/b

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Fricas [A]
time = 0.85, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/3*(b*x^2 + a)^(3/2)/b

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 0.06, size = 39, normalized size = 2.17 \begin {gather*} \begin {cases} \frac {a \sqrt {a + b x^{2}}}{3 b} + \frac {x^{2} \sqrt {a + b x^{2}}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} 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)**(1/2),x)

[Out]

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

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Giac [A]
time = 1.16, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/3*(b*x^2 + a)^(3/2)/b

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Mupad [B]
time = 4.63, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b\,x^2+a\right )}^{3/2}}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^(1/2),x)

[Out]

(a + b*x^2)^(3/2)/(3*b)

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