3.47 \(\int x \sqrt {1+x^2} \, dx\)

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

[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 = {261} \[ \frac {1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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 \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 \[ \frac {1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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fricas [A]  time = 0.39, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]

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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giac [A]  time = 0.01, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]

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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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2+1)^(1/2),x)

[Out]

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

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maxima [A]  time = 0.44, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]

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

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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sympy [B]  time = 0.20, size = 22, normalized size = 1.69 \[ \frac {x^{2} \sqrt {x^{2} + 1}}{3} + \frac {\sqrt {x^{2} + 1}}{3} \]

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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