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

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

[Out]

(1 + x^2)^(3/2)/3

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Rubi [A]  time = 0.0016729, antiderivative size = 13, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0019712, size = 13, normalized size = 1. \[ \frac{1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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Maple [A]  time = 0.002, size = 10, normalized size = 0.8 \begin{align*}{\frac{1}{3} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

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 = 0.925371, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

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 = 1.56769, size = 28, normalized size = 2.15 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

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]  time = 0.190332, size = 22, normalized size = 1.69 \begin{align*} \frac{x^{2} \sqrt{x^{2} + 1}}{3} + \frac{\sqrt{x^{2} + 1}}{3} \end{align*}

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 = 1.0848, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

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)