\(\int x^m \sqrt {\log (a x^n)} \, dx\) [163]

Optimal result

Integrand size = 14, antiderivative size = 86 \[ \int x^m \sqrt {\log \left (a x^n\right )} \, dx=-\frac {\sqrt {n} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (1+m)^{3/2}}+\frac {x^{1+m} \sqrt {\log \left (a x^n\right )}}{1+m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, 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} \[ \int x^m \sqrt {\log \left (a x^n\right )} \, dx=\frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {\sqrt {\pi } \sqrt {n} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (m+1)^{3/2}} \]

[In]

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

[Out]

-1/2*(Sqrt[n]*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((1 + m)^(3/2)*(a*x^n)^((1 + m)
/n)) + (x^(1 + m)*Sqrt[Log[a*x^n]])/(1 + m)

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {\log \left (a x^n\right )} \, dx=-\frac {\sqrt {n} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (1+m)^{3/2}}+\frac {x^{1+m} \sqrt {\log \left (a x^n\right )}}{1+m} \]

[In]

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

[Out]

-1/2*(Sqrt[n]*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((1 + m)^(3/2)*(a*x^n)^((1 + m)
/n)) + (x^(1 + m)*Sqrt[Log[a*x^n]])/(1 + m)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^m*sqrt(log(a*x^n)), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**m*sqrt(log(a*x**n)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^m*sqrt(log(a*x^n)), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^m*sqrt(log(a*x^n)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^m*log(a*x^n)^(1/2), x)