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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 x^4 \left (a x^n\right )^{-4/n} \left (16 n \Gamma \left (\frac {1}{2},-\frac {4 \log \left (a x^n\right )}{n}\right ) \left (-\frac {\log \left (a x^n\right )}{n}\right )^{3/2}+\left (a x^n\right )^{4/n} \left (n+8 \log \left (a x^n\right )\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \] Input:

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

Output:

(-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))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2743, 2743, 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.

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

\(\Big \downarrow \) 2743

\(\displaystyle \frac {8 \int \frac {x^3}{\log ^{\frac {3}{2}}\left (a x^n\right )}dx}{3 n}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}\)

\(\Big \downarrow \) 2743

\(\displaystyle \frac {8 \left (\frac {8 \int \frac {x^3}{\sqrt {\log \left (a x^n\right )}}dx}{n}-\frac {2 x^4}{n \sqrt {\log \left (a x^n\right )}}\right )}{3 n}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {8 \left (\frac {8 x^4 \left (a x^n\right )^{-4/n} \int \frac {\left (a x^n\right )^{4/n}}{\sqrt {\log \left (a x^n\right )}}d\log \left (a x^n\right )}{n^2}-\frac {2 x^4}{n \sqrt {\log \left (a x^n\right )}}\right )}{3 n}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {8 \left (\frac {16 x^4 \left (a x^n\right )^{-4/n} \int \left (a x^n\right )^{4/n}d\sqrt {\log \left (a x^n\right )}}{n^2}-\frac {2 x^4}{n \sqrt {\log \left (a x^n\right )}}\right )}{3 n}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {8 \left (\frac {4 \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^4}{n \sqrt {\log \left (a x^n\right )}}\right )}{3 n}-\frac {2 x^4}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2611

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]
 

rule 2633

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 2743

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] - 
Simp[(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 2747

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {-\frac {16 \sqrt {\mathrm {log}\left (x^{n} a \right )}\, \mathrm {log}\left (x^{n} a \right ) x^{4}}{3}-\frac {2 \sqrt {\mathrm {log}\left (x^{n} a \right )}\, n \,x^{4}}{3}+\frac {64 \left (\int \frac {\sqrt {\mathrm {log}\left (x^{n} a \right )}\, x^{3}}{\mathrm {log}\left (x^{n} a \right )}d x \right ) \mathrm {log}\left (x^{n} a \right )^{2}}{3}}{\mathrm {log}\left (x^{n} a \right )^{2} n^{2}} \] Input:

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

Output:

(2*( - 8*sqrt(log(x**n*a))*log(x**n*a)*x**4 - sqrt(log(x**n*a))*n*x**4 + 3 
2*int((sqrt(log(x**n*a))*x**3)/log(x**n*a),x)*log(x**n*a)**2))/(3*log(x**n 
*a)**2*n**2)