\(\int x^2 \sqrt {\log (a x^n)} \, dx\) [115]

Optimal result

Integrand size = 14, antiderivative size = 72 \[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=-\frac {1}{6} \sqrt {n} \sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{3} x^3 \sqrt {\log \left (a x^n\right )} \]

[Out]

-1/18*x^3*erfi(3^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*n^(1/2)*3^(1/2)*Pi^(1/2)/((a*x^n)^(3/n))+1/3*x^3*ln(a*x^n)^(1/
2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2342, 2347, 2211, 2235} \[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\frac {1}{3} x^3 \sqrt {\log \left (a x^n\right )}-\frac {1}{6} \sqrt {\frac {\pi }{3}} \sqrt {n} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right ) \]

[In]

Int[x^2*Sqrt[Log[a*x^n]],x]

[Out]

-1/6*(Sqrt[n]*Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(3/n) + (x^3*Sqrt[Log[a*x^n]])/
3

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \sqrt {\log \left (a x^n\right )}-\frac {1}{6} n \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx \\ & = \frac {1}{3} x^3 \sqrt {\log \left (a x^n\right )}-\frac {1}{6} \left (x^3 \left (a x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right ) \\ & = \frac {1}{3} x^3 \sqrt {\log \left (a x^n\right )}-\frac {1}{3} \left (x^3 \left (a x^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{\frac {3 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right ) \\ & = -\frac {1}{6} \sqrt {n} \sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{3} x^3 \sqrt {\log \left (a x^n\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.93 \[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\frac {1}{18} x^3 \left (-\sqrt {n} \sqrt {3 \pi } \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+6 \sqrt {\log \left (a x^n\right )}\right ) \]

[In]

Integrate[x^2*Sqrt[Log[a*x^n]],x]

[Out]

(x^3*(-((Sqrt[n]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(3/n)) + 6*Sqrt[Log[a*x^n]]))/18

Maple [F]

\[\int x^{2} \sqrt {\ln \left (a \,x^{n}\right )}d x\]

[In]

int(x^2*ln(a*x^n)^(1/2),x)

[Out]

int(x^2*ln(a*x^n)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*log(a*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\int x^{2} \sqrt {\log {\left (a x^{n} \right )}}\, dx \]

[In]

integrate(x**2*ln(a*x**n)**(1/2),x)

[Out]

Integral(x**2*sqrt(log(a*x**n)), x)

Maxima [F]

\[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\int { x^{2} \sqrt {\log \left (a x^{n}\right )} \,d x } \]

[In]

integrate(x^2*log(a*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2*sqrt(log(a*x^n)), x)

Giac [F]

\[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\int { x^{2} \sqrt {\log \left (a x^{n}\right )} \,d x } \]

[In]

integrate(x^2*log(a*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(x^2*sqrt(log(a*x^n)), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {\log \left (a x^n\right )} \, dx=\int x^2\,\sqrt {\ln \left (a\,x^n\right )} \,d x \]

[In]

int(x^2*log(a*x^n)^(1/2),x)

[Out]

int(x^2*log(a*x^n)^(1/2), x)