3.2.42 \(\int 5 x \sqrt {1+x^2} \, dx\) [142]

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

[Out]

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

________________________________________________________________________________________

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, 267} \begin {gather*} \frac {5}{3} \left (x^2+1\right )^{3/2} \end {gather*}

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 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 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 = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {5}{3} \left (1+x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5*x*Sqrt[1 + x^2],x]

[Out]

(5*(1 + x^2)^(3/2))/3

________________________________________________________________________________________

Mathics [A]
time = 1.83, size = 9, normalized size = 0.69 \begin {gather*} \frac {5 {\left (1+x^2\right )}^{\frac {3}{2}}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[5*x*Sqrt[1 + x^2],x]')

[Out]

5 (1 + x ^ 2) ^ (3 / 2) / 3

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 10, normalized size = 0.77

method	result	size
gosper	\(\frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
derivativedivides	\(\frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
default	\(\frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
risch	\(\frac {5 \left (x^{2}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
trager	\(5 \left (\frac {x^{2}}{3}+\frac {1}{3}\right ) \sqrt {x^{2}+1}\)	\(17\)
meijerg	\(-\frac {5 \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x^{2}+2\right ) \sqrt {x^{2}+1}}{3}\right )}{4 \sqrt {\pi }}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 9, normalized size = 0.69 \begin {gather*} \frac {5}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.32, size = 9, normalized size = 0.69 \begin {gather*} \frac {5}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\)
time = 0.08, size = 26, normalized size = 2.00 \begin {gather*} \frac {5 x^{2} \sqrt {x^{2} + 1}}{3} + \frac {5 \sqrt {x^{2} + 1}}{3} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 18, normalized size = 1.38 \begin {gather*} \frac {5}{3} \sqrt {x^{2}+1} \left (x^{2}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/3*(x^2 + 1)^(3/2)

________________________________________________________________________________________

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

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

________________________________________________________________________________________