### 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*(1 + x^2)^(3/2))/3

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Rubi [A]  time = 0.0024651, antiderivative size = 13, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0017622, size = 13, normalized size = 1. $\frac{5}{3} \left (x^2+1\right )^{3/2}$

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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Maple [A]  time = 0.001, size = 10, normalized size = 0.8 \begin{align*}{\frac{5}{3} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.925844, size = 12, normalized size = 0.92 \begin{align*} \frac{5}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

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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Fricas [A]  time = 2.06477, size = 28, normalized size = 2.15 \begin{align*} \frac{5}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

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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Sympy [B]  time = 0.174944, size = 26, normalized size = 2. \begin{align*} \frac{5 x^{2} \sqrt{x^{2} + 1}}{3} + \frac{5 \sqrt{x^{2} + 1}}{3} \end{align*}

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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Giac [A]  time = 1.04555, size = 12, normalized size = 0.92 \begin{align*} \frac{5}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

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)