Integrand size = 14, antiderivative size = 111 \[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {3 n^{3/2} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 (1+m)^{5/2}}-\frac {3 n x^{1+m} \sqrt {\log \left (a x^n\right )}}{2 (1+m)^2}+\frac {x^{1+m} \log ^{\frac {3}{2}}\left (a x^n\right )}{1+m} \]
x^(1+m)*ln(a*x^n)^(3/2)/(1+m)+3/4*n^(3/2)*x^(1+m)*erfi((1+m)^(1/2)*ln(a*x^ n)^(1/2)/n^(1/2))*Pi^(1/2)/(1+m)^(5/2)/((a*x^n)^((1+m)/n))-3/2*n*x^(1+m)*l n(a*x^n)^(1/2)/(1+m)^2
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91 \[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {x^{1+m} \left (3 n^{3/2} \sqrt {\pi } \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+2 \sqrt {1+m} \sqrt {\log \left (a x^n\right )} \left (-3 n+2 (1+m) \log \left (a x^n\right )\right )\right )}{4 (1+m)^{5/2}} \]
(x^(1 + m)*((3*n^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n ]])/(a*x^n)^((1 + m)/n) + 2*Sqrt[1 + m]*Sqrt[Log[a*x^n]]*(-3*n + 2*(1 + m) *Log[a*x^n])))/(4*(1 + m)^(5/2))
Time = 0.39 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2742, 2742, 2747, 2611, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle \frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n \int x^m \sqrt {\log \left (a x^n\right )}dx}{2 (m+1)}\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle \frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n \left (\frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {n \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}}dx}{2 (m+1)}\right )}{2 (m+1)}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n \left (\frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \int \frac {\left (a x^n\right )^{\frac {m+1}{n}}}{\sqrt {\log \left (a x^n\right )}}d\log \left (a x^n\right )}{2 (m+1)}\right )}{2 (m+1)}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n \left (\frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \int \left (a x^n\right )^{\frac {m+1}{n}}d\sqrt {\log \left (a x^n\right )}}{m+1}\right )}{2 (m+1)}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n \left (\frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {\sqrt {\pi } \sqrt {n} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (m+1)^{3/2}}\right )}{2 (m+1)}\) |
(-3*n*(-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)))/(2*(1 + m)) + (x^(1 + m)*Log[a*x^n]^(3/2))/(1 + m)
3.2.62.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(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]
\[\int x^{m} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}d x\]
\[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
\[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int x^{m} \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]
\[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
\[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int x^m\,{\ln \left (a\,x^n\right )}^{3/2} \,d x \]