\(\int x^3 \log ^{\frac {3}{2}}(a x^n) \, dx\) [121]

Optimal result

Integrand size = 14, antiderivative size = 82 \[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {3}{128} n^{3/2} \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {3}{32} n x^4 \sqrt {\log \left (a x^n\right )}+\frac {1}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right ) \]

[Out]

1/4*x^4*ln(a*x^n)^(3/2)+3/128*n^(3/2)*x^4*erfi(2*ln(a*x^n)^(1/2)/n^(1/2))*Pi^(1/2)/((a*x^n)^(4/n))-3/32*n*x^4*
ln(a*x^n)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2342, 2347, 2211, 2235} \[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {3}{128} \sqrt {\pi } n^{3/2} x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {3}{32} n x^4 \sqrt {\log \left (a x^n\right )} \]

[In]

Int[x^3*Log[a*x^n]^(3/2),x]

[Out]

(3*n^(3/2)*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(128*(a*x^n)^(4/n)) - (3*n*x^4*Sqrt[Log[a*x^n]])/3
2 + (x^4*Log[a*x^n]^(3/2))/4

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}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {1}{8} (3 n) \int x^3 \sqrt {\log \left (a x^n\right )} \, dx \\ & = -\frac {3}{32} n x^4 \sqrt {\log \left (a x^n\right )}+\frac {1}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{64} \left (3 n^2\right ) \int \frac {x^3}{\sqrt {\log \left (a x^n\right )}} \, dx \\ & = -\frac {3}{32} n x^4 \sqrt {\log \left (a x^n\right )}+\frac {1}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{64} \left (3 n x^4 \left (a x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right ) \\ & = -\frac {3}{32} n x^4 \sqrt {\log \left (a x^n\right )}+\frac {1}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{32} \left (3 n x^4 \left (a x^n\right )^{-4/n}\right ) \text {Subst}\left (\int e^{\frac {4 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right ) \\ & = \frac {3}{128} n^{3/2} \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {3}{32} n x^4 \sqrt {\log \left (a x^n\right )}+\frac {1}{4} x^4 \log ^{\frac {3}{2}}\left (a x^n\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {1}{128} x^4 \left (3 n^{3/2} \sqrt {\pi } \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+4 \sqrt {\log \left (a x^n\right )} \left (-3 n+8 \log \left (a x^n\right )\right )\right ) \]

[In]

Integrate[x^3*Log[a*x^n]^(3/2),x]

[Out]

(x^4*((3*n^(3/2)*Sqrt[Pi]*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(4/n) + 4*Sqrt[Log[a*x^n]]*(-3*n + 8*Log
[a*x^n])))/128

Maple [F]

\[\int x^{3} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}d x\]

[In]

int(x^3*ln(a*x^n)^(3/2),x)

[Out]

int(x^3*ln(a*x^n)^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*log(a*x^n)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int x^{3} \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]

[In]

integrate(x**3*ln(a*x**n)**(3/2),x)

[Out]

Integral(x**3*log(a*x**n)**(3/2), x)

Maxima [F]

\[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{3} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate(x^3*log(a*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3*log(a*x^n)^(3/2), x)

Giac [F]

\[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{3} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate(x^3*log(a*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(x^3*log(a*x^n)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int x^3\,{\ln \left (a\,x^n\right )}^{3/2} \,d x \]

[In]

int(x^3*log(a*x^n)^(3/2),x)

[Out]

int(x^3*log(a*x^n)^(3/2), x)