3.142 \(\int 5 x \sqrt {1+x^2} \, dx\)

Optimal. Leaf size=13 \[ \frac {5}{3} \left (x^2+1\right )^{3/2} \]

[Out]

5/3*(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 = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 261} \[ \frac {5}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[5*x*Sqrt[1 + x^2],x]

[Out]

(5*(1 + x^2)^(3/2))/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 5 x \sqrt {1+x^2} \, dx &=5 \int x \sqrt {1+x^2} \, dx\\ &=\frac {5}{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 \[ \frac {5}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[5*x*Sqrt[1 + x^2],x]

[Out]

(5*(1 + x^2)^(3/2))/3

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fricas [A]  time = 0.40, size = 9, normalized size = 0.69 \[ \frac {5}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x*(x^2+1)^(1/2),x, algorithm="fricas")

[Out]

5/3*(x^2 + 1)^(3/2)

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giac [A]  time = 0.99, size = 9, normalized size = 0.69 \[ \frac {5}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x*(x^2+1)^(1/2),x, algorithm="giac")

[Out]

5/3*(x^2 + 1)^(3/2)

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5*(x^2+1)^(1/2)*x,x)

[Out]

5/3*(x^2+1)^(3/2)

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maxima [A]  time = 0.43, size = 9, normalized size = 0.69 \[ \frac {5}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x*(x^2+1)^(1/2),x, algorithm="maxima")

[Out]

5/3*(x^2 + 1)^(3/2)

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mupad [B]  time = 0.03, size = 9, normalized size = 0.69 \[ \frac {5\,{\left (x^2+1\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5*x*(x^2 + 1)^(1/2),x)

[Out]

(5*(x^2 + 1)^(3/2))/3

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sympy [B]  time = 0.20, size = 26, normalized size = 2.00 \[ \frac {5 x^{2} \sqrt {x^{2} + 1}}{3} + \frac {5 \sqrt {x^{2} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x*(x**2+1)**(1/2),x)

[Out]

5*x**2*sqrt(x**2 + 1)/3 + 5*sqrt(x**2 + 1)/3

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