3.148 \(\int \frac {1}{x^3 \log ^{\frac {5}{2}}(a x^n)} \, dx\)

Optimal. Leaf size=93 \[ \frac {8 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x^2}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]

[Out]

-2/3/n/x^2/ln(a*x^n)^(3/2)+8/3*(a*x^n)^(2/n)*erf(2^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*2^(1/2)*Pi^(1/2)/n^(5/2)/x^2
+8/3/n^2/x^2/ln(a*x^n)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2306, 2310, 2180, 2205} \[ \frac {8 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {Erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x^2}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Log[a*x^n]^(5/2)),x]

[Out]

(8*Sqrt[2*Pi]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(3*n^(5/2)*x^2) - 2/(3*n*x^2*Log[a*x^n]^(
3/2)) + 8/(3*n^2*x^2*Sqrt[Log[a*x^n]])

Rule 2180

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] &&  !$UseGamma === True

Rule 2205

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

Rule 2306

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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx &=-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n}\\ &=-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}+\frac {16 \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2}\\ &=-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (16 \left (a x^n\right )^{2/n}\right ) \operatorname {Subst}\left (\int \frac {e^{-\frac {2 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{3 n^3 x^2}\\ &=-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (32 \left (a x^n\right )^{2/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {2 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{3 n^3 x^2}\\ &=\frac {8 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x^2}-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 78, normalized size = 0.84 \[ -\frac {2 \left (-4 \log \left (a x^n\right )+4 \sqrt {2} n \left (a x^n\right )^{2/n} \left (\frac {\log \left (a x^n\right )}{n}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right )+n\right )}{3 n^2 x^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Log[a*x^n]^(5/2)),x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/log(a*x^n)^(5/2),x, algorithm="giac")

[Out]

integrate(1/(x^3*log(a*x^n)^(5/2)), x)

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/ln(a*x^n)^(5/2),x)

[Out]

int(1/x^3/ln(a*x^n)^(5/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/log(a*x^n)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x^3*log(a*x^n)^(5/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*log(a*x^n)^(5/2)),x)

[Out]

int(1/(x^3*log(a*x^n)^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/ln(a*x**n)**(5/2),x)

[Out]

Integral(1/(x**3*log(a*x**n)**(5/2)), x)

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