\(\int \frac {x^3}{\log ^{\frac {5}{2}}(a x^n)} \, dx\) [142]

Optimal result

Integrand size = 14, antiderivative size = 87 \[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {32 \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {16 x^4}{3 n^2 \sqrt {\log \left (a x^n\right )}} \]

[Out]

-2/3*x^4/n/ln(a*x^n)^(3/2)+32/3*x^4*erfi(2*ln(a*x^n)^(1/2)/n^(1/2))*Pi^(1/2)/n^(5/2)/((a*x^n)^(4/n))-16/3*x^4/
n^2/ln(a*x^n)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 87, 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 = {2343, 2347, 2211, 2235} \[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {32 \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {16 x^4}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]

[In]

Int[x^3/Log[a*x^n]^(5/2),x]

[Out]

(32*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(3*n^(5/2)*(a*x^n)^(4/n)) - (2*x^4)/(3*n*Log[a*x^n]^(3/2)
) - (16*x^4)/(3*n^2*Sqrt[Log[a*x^n]])

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 2343

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log
[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - Dist[(m + 1)/(b*n*(p + 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] && LtQ[p, -1]

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 {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8 \int \frac {x^3}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n} \\ & = -\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {16 x^4}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {64 \int \frac {x^3}{\sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2} \\ & = -\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {16 x^4}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (64 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 )}{3 n^3} \\ & = -\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {16 x^4}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (128 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 )}{3 n^3} \\ & = \frac {32 \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 )}{3 n^{5/2}}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {16 x^4}{3 n^2 \sqrt {\log \left (a x^n\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 x^4 \left (a x^n\right )^{-4/n} \left (16 n \Gamma \left (\frac {1}{2},-\frac {4 \log \left (a x^n\right )}{n}\right ) \left (-\frac {\log \left (a x^n\right )}{n}\right )^{3/2}+\left (a x^n\right )^{4/n} \left (n+8 \log \left (a x^n\right )\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]

[In]

Integrate[x^3/Log[a*x^n]^(5/2),x]

[Out]

(-2*x^4*(16*n*Gamma[1/2, (-4*Log[a*x^n])/n]*(-(Log[a*x^n]/n))^(3/2) + (a*x^n)^(4/n)*(n + 8*Log[a*x^n])))/(3*n^
2*(a*x^n)^(4/n)*Log[a*x^n]^(3/2))

Maple [F]

\[\int \frac {x^{3}}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}d x\]

[In]

int(x^3/ln(a*x^n)^(5/2),x)

[Out]

int(x^3/ln(a*x^n)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3/log(a*x^n)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {x^{3}}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**3/ln(a*x**n)**(5/2),x)

[Out]

Integral(x**3/log(a*x**n)**(5/2), x)

Maxima [F]

\[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{3}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^3/log(a*x^n)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^3/log(a*x^n)^(5/2), x)

Giac [F]

\[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{3}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^3/log(a*x^n)^(5/2),x, algorithm="giac")

[Out]

integrate(x^3/log(a*x^n)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {x^3}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]

[In]

int(x^3/log(a*x^n)^(5/2),x)

[Out]

int(x^3/log(a*x^n)^(5/2), x)