\(\int \frac {(d x)^{3/2}}{(a+b \log (c x^n))^2} \, dx\) [108]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int \frac {(d x)^{3/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {5 e^{-\frac {5 a}{2 b n}} (d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \] Output:

5/2*(d*x)^(5/2)*Ei(5/2*(a+b*ln(c*x^n))/b/n)/b^2/d/exp(5/2*a/b/n)/n^2/((c*x 
^n)^(5/2/n))-(d*x)^(5/2)/b/d/n/(a+b*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {(d x)^{3/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x (d x)^{3/2} \left (5 e^{-\frac {5 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2} \] Input:

Integrate[(d*x)^(3/2)/(a + b*Log[c*x^n])^2,x]
 

Output:

(x*(d*x)^(3/2)*((5*ExpIntegralEi[(5*(a + b*Log[c*x^n]))/(2*b*n)])/(E^((5*a 
)/(2*b*n))*(c*x^n)^(5/(2*n))) - (2*b*n)/(a + b*Log[c*x^n])))/(2*b^2*n^2)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2743, 2747, 2609}

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.

\(\displaystyle \int \frac {(d x)^{3/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx\)

\(\Big \downarrow \) 2743

\(\displaystyle \frac {5 \int \frac {(d x)^{3/2}}{a+b \log \left (c x^n\right )}dx}{2 b n}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {5 (d x)^{5/2} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \int \frac {\left (c x^n\right )^{\left .\frac {5}{2}\right /n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 b d n^2}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {5 (d x)^{5/2} e^{-\frac {5 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {5}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {5 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{5/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\)

Input:

Int[(d*x)^(3/2)/(a + b*Log[c*x^n])^2,x]
 

Output:

(5*(d*x)^(5/2)*ExpIntegralEi[(5*(a + b*Log[c*x^n]))/(2*b*n)])/(2*b^2*d*E^( 
(5*a)/(2*b*n))*n^2*(c*x^n)^(5/(2*n))) - (d*x)^(5/2)/(b*d*n*(a + b*Log[c*x^ 
n]))
 

Defintions of rubi rules used

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.54 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.41

method	result	size
risch	\(-\frac {2 x^{3} d^{2}}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )+i \pi b \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i \pi b \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \right )-i \pi b \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+i \pi b \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {5 i \left (b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i \left (\ln \left (x \right )-\ln \left (d x \right )\right ) b n +2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 n b}} \operatorname {expIntegral}_{1}\left (-\frac {5 \ln \left (d x \right )}{2}+\frac {5 i \left (b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i \left (\ln \left (x \right )-\ln \left (d x \right )\right ) b n +2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 n b}\right )}{2 d \,b^{2} n^{2}}\)	\(432\)

Input:

int((d*x)^(3/2)/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
 

Output:

-2/b/n*x^3/(d*x)^(1/2)/(2*a+2*b*ln(c)+2*b*ln(exp(n*ln(x)))+I*Pi*b*csgn(I*e 
xp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2-I*Pi*b*csgn(I*exp(n*ln(x)))*csgn(I*c 
*exp(n*ln(x)))*csgn(I*c)-I*Pi*b*csgn(I*c*exp(n*ln(x)))^3+I*Pi*b*csgn(I*c*e 
xp(n*ln(x)))^2*csgn(I*c))*d^2-5/2/d/b^2/n^2*exp(5/4*I*(b*Pi*csgn(I*exp(n*l 
n(x)))*csgn(I*c*exp(n*ln(x)))*csgn(I*c)-b*Pi*csgn(I*c*exp(n*ln(x)))^2*csgn 
(I*c)-b*Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp 
(n*ln(x)))^3+2*I*(ln(x)-ln(d*x))*b*n+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n 
*ln(x))+2*I*a)/n/b)*Ei(1,-5/2*ln(d*x)+5/4*I*(b*Pi*csgn(I*exp(n*ln(x)))*csg 
n(I*c*exp(n*ln(x)))*csgn(I*c)-b*Pi*csgn(I*c*exp(n*ln(x)))^2*csgn(I*c)-b*Pi 
*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n*ln(x))) 
^3+2*I*(ln(x)-ln(d*x))*b*n+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x))+2* 
I*a)/n/b)
 

Fricas [F]

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

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

Output:

integral(sqrt(d*x)*d*x/(b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2), x)
 

Sympy [F]

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

integrate((d*x)**(3/2)/(a+b*ln(c*x**n))**2,x)
 

Output:

Integral((d*x)**(3/2)/(a + b*log(c*x**n))**2, x)
 

Maxima [F]

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

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

Output:

4*b*d^(3/2)*n*integrate(1/5*x^(3/2)/(b^3*log(c)^3 + b^3*log(x^n)^3 + 3*a*b 
^2*log(c)^2 + 3*a^2*b*log(c) + a^3 + 3*(b^3*log(c) + a*b^2)*log(x^n)^2 + 3 
*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x^n)), x) + 2/5*d^(3/2)*x^(5/ 
2)/(b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*log(c) + a 
*b)*log(x^n))
 

Giac [F]

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

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

Output:

integrate((d*x)^(3/2)/(b*log(c*x^n) + a)^2, x)
 

Mupad [F(-1)]

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

int((d*x)^(3/2)/(a + b*log(c*x^n))^2,x)
 

Output:

int((d*x)^(3/2)/(a + b*log(c*x^n))^2, x)
 

Reduce [F]

\[ \int \frac {(d x)^{3/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\sqrt {d}\, \left (\int \frac {\sqrt {x}\, x}{\mathrm {log}\left (x^{n} c \right )^{2} b^{2}+2 \,\mathrm {log}\left (x^{n} c \right ) a b +a^{2}}d x \right ) d \] Input:

int((d*x)^(3/2)/(a+b*log(c*x^n))^2,x)
 

Output:

sqrt(d)*int((sqrt(x)*x)/(log(x**n*c)**2*b**2 + 2*log(x**n*c)*a*b + a**2),x 
)*d