3.1.47 \(\int x \sqrt {1+x^2} \, dx\) [47]

Optimal. Leaf size=13 \[ \frac {1}{3} \left (1+x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \begin {gather*} \frac {1}{3} \left (x^2+1\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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 {1+x^2} \, dx &=\frac {1}{3} \left (1+x^2\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{3} \left (1+x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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Maple [A]
time = 0.06, size = 10, normalized size = 0.77

method	result	size
gosper	\(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
derivativedivides	\(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
default	\(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
risch	\(\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
trager	\(\left (\frac {x^{2}}{3}+\frac {1}{3}\right ) \sqrt {x^{2}+1}\)	\(16\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x^{2}+2\right ) \sqrt {x^{2}+1}}{3}}{4 \sqrt {\pi }}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.20, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^2 + 1)^(3/2)

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Fricas [A]
time = 1.65, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^2 + 1)^(3/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).
time = 0.06, size = 22, normalized size = 1.69 \begin {gather*} \frac {x^{2} \sqrt {x^{2} + 1}}{3} + \frac {\sqrt {x^{2} + 1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2+1)**(1/2),x)

[Out]

x**2*sqrt(x**2 + 1)/3 + sqrt(x**2 + 1)/3

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Giac [A]
time = 0.50, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^2 + 1)^(3/2)

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Mupad [B]
time = 0.02, size = 9, normalized size = 0.69 \begin {gather*} \frac {{\left (x^2+1\right )}^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 + 1)^(1/2),x)

[Out]

(x^2 + 1)^(3/2)/3

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