3.275 \(\int x \sqrt{4+x^2} \log (x) \, dx\)

Optimal. Leaf size=60 \[ -\frac{1}{9} \left (x^2+4\right )^{3/2}-\frac{4 \sqrt{x^2+4}}{3}+\frac{1}{3} \left (x^2+4\right )^{3/2} \log (x)+\frac{8}{3} \tanh ^{-1}\left (\frac{\sqrt{x^2+4}}{2}\right ) \]

[Out]

(-4*Sqrt[4 + x^2])/3 - (4 + x^2)^(3/2)/9 + (8*ArcTanh[Sqrt[4 + x^2]/2])/3 + ((4 + x^2)^(3/2)*Log[x])/3

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Rubi [A]  time = 0.0454512, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2338, 266, 50, 63, 207} \[ -\frac{1}{9} \left (x^2+4\right )^{3/2}-\frac{4 \sqrt{x^2+4}}{3}+\frac{1}{3} \left (x^2+4\right )^{3/2} \log (x)+\frac{8}{3} \tanh ^{-1}\left (\frac{\sqrt{x^2+4}}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[4 + x^2]*Log[x],x]

[Out]

(-4*Sqrt[4 + x^2])/3 - (4 + x^2)^(3/2)/9 + (8*ArcTanh[Sqrt[4 + x^2]/2])/3 + ((4 + x^2)^(3/2)*Log[x])/3

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[(f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*r*(q + 1)), x] - Dist[(b*f^m*n*p)/(e*r*(q + 1)), Int[
((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x \sqrt{4+x^2} \log (x) \, dx &=\frac{1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac{1}{3} \int \frac{\left (4+x^2\right )^{3/2}}{x} \, dx\\ &=\frac{1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac{1}{6} \operatorname{Subst}\left (\int \frac{(4+x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=-\frac{1}{9} \left (4+x^2\right )^{3/2}+\frac{1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sqrt{4+x}}{x} \, dx,x,x^2\right )\\ &=-\frac{4}{3} \sqrt{4+x^2}-\frac{1}{9} \left (4+x^2\right )^{3/2}+\frac{1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac{8}{3} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{4+x}} \, dx,x,x^2\right )\\ &=-\frac{4}{3} \sqrt{4+x^2}-\frac{1}{9} \left (4+x^2\right )^{3/2}+\frac{1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac{16}{3} \operatorname{Subst}\left (\int \frac{1}{-4+x^2} \, dx,x,\sqrt{4+x^2}\right )\\ &=-\frac{4}{3} \sqrt{4+x^2}-\frac{1}{9} \left (4+x^2\right )^{3/2}+\frac{8}{3} \tanh ^{-1}\left (\frac{\sqrt{4+x^2}}{2}\right )+\frac{1}{3} \left (4+x^2\right )^{3/2} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0436634, size = 53, normalized size = 0.88 \[ \frac{1}{3} \left (-\frac{1}{3} \left (x^2+16\right ) \sqrt{x^2+4}+\left (x^2+4\right )^{3/2} \log (x)+8 \log \left (\sqrt{x^2+4}+2\right )-8 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[4 + x^2]*Log[x],x]

[Out]

(-(Sqrt[4 + x^2]*(16 + x^2))/3 - 8*Log[x] + (4 + x^2)^(3/2)*Log[x] + 8*Log[2 + Sqrt[4 + x^2]])/3

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Maple [A]  time = 0.306, size = 75, normalized size = 1.3 \begin{align*} \left ( -{\frac{2}{9}\sqrt{1+{\frac{{x}^{2}}{4}}}}+{\frac{2\,\ln \left ( x \right ) }{3}\sqrt{1+{\frac{{x}^{2}}{4}}}} \right ){x}^{2}+{\frac{32}{9}}-{\frac{32}{9}\sqrt{1+{\frac{{x}^{2}}{4}}}}+\ln \left ( x \right ) \left ( -{\frac{8}{3}}+{\frac{8}{3}\sqrt{1+{\frac{{x}^{2}}{4}}}} \right ) +{\frac{8}{3}\ln \left ({\frac{1}{2}}+{\frac{1}{2}\sqrt{1+{\frac{{x}^{2}}{4}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(x)*(x^2+4)^(1/2),x)

[Out]

(-2/9*(1+1/4*x^2)^(1/2)+2/3*ln(x)*(1+1/4*x^2)^(1/2))*x^2+32/9-32/9*(1+1/4*x^2)^(1/2)+ln(x)*(-8/3+8/3*(1+1/4*x^
2)^(1/2))+8/3*ln(1/2+1/2*(1+1/4*x^2)^(1/2))

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Maxima [A]  time = 1.73041, size = 53, normalized size = 0.88 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 4\right )}^{\frac{3}{2}} \log \left (x\right ) - \frac{1}{9} \,{\left (x^{2} + 4\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{x^{2} + 4} + \frac{8}{3} \, \operatorname{arsinh}\left (\frac{2}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x)*(x^2+4)^(1/2),x, algorithm="maxima")

[Out]

1/3*(x^2 + 4)^(3/2)*log(x) - 1/9*(x^2 + 4)^(3/2) - 4/3*sqrt(x^2 + 4) + 8/3*arcsinh(2/abs(x))

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Fricas [A]  time = 1.33906, size = 162, normalized size = 2.7 \begin{align*} -\frac{1}{9} \,{\left (x^{2} - 3 \,{\left (x^{2} + 4\right )} \log \left (x\right ) + 16\right )} \sqrt{x^{2} + 4} + \frac{8}{3} \, \log \left (-x + \sqrt{x^{2} + 4} + 2\right ) - \frac{8}{3} \, \log \left (-x + \sqrt{x^{2} + 4} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x)*(x^2+4)^(1/2),x, algorithm="fricas")

[Out]

-1/9*(x^2 - 3*(x^2 + 4)*log(x) + 16)*sqrt(x^2 + 4) + 8/3*log(-x + sqrt(x^2 + 4) + 2) - 8/3*log(-x + sqrt(x^2 +
4) - 2)

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Sympy [A]  time = 24.8481, size = 65, normalized size = 1.08 \begin{align*} \frac{\left (x^{2} + 4\right )^{\frac{3}{2}} \log{\left (x \right )}}{3} - \frac{\left (x^{2} + 4\right )^{\frac{3}{2}}}{9} - \frac{4 \sqrt{x^{2} + 4}}{3} - \frac{4 \log{\left (\sqrt{x^{2} + 4} - 2 \right )}}{3} + \frac{4 \log{\left (\sqrt{x^{2} + 4} + 2 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(x)*(x**2+4)**(1/2),x)

[Out]

(x**2 + 4)**(3/2)*log(x)/3 - (x**2 + 4)**(3/2)/9 - 4*sqrt(x**2 + 4)/3 - 4*log(sqrt(x**2 + 4) - 2)/3 + 4*log(sq
rt(x**2 + 4) + 2)/3

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Giac [A]  time = 1.29545, size = 73, normalized size = 1.22 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 4\right )}^{\frac{3}{2}} \log \left (x\right ) - \frac{1}{9} \,{\left (x^{2} + 4\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{x^{2} + 4} + \frac{4}{3} \, \log \left (\sqrt{x^{2} + 4} + 2\right ) - \frac{4}{3} \, \log \left (\sqrt{x^{2} + 4} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x)*(x^2+4)^(1/2),x, algorithm="giac")

[Out]

1/3*(x^2 + 4)^(3/2)*log(x) - 1/9*(x^2 + 4)^(3/2) - 4/3*sqrt(x^2 + 4) + 4/3*log(sqrt(x^2 + 4) + 2) - 4/3*log(sq
rt(x^2 + 4) - 2)