\(\int \frac {(a+b \log (c x^n))^2}{(d x)^{5/2}} \, dx\) [100]

Optimal result

Integrand size = 20, antiderivative size = 73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {16 b^2 n^2}{27 d (d x)^{3/2}}-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}} \] Output:

-16/27*b^2*n^2/d/(d*x)^(3/2)-8/9*b*n*(a+b*ln(c*x^n))/d/(d*x)^(3/2)-2/3*(a+ 
b*ln(c*x^n))^2/d/(d*x)^(3/2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {2 x \left (9 a^2+12 a b n+8 b^2 n^2+6 b (3 a+2 b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )\right )}{27 (d x)^{5/2}} \] Input:

Integrate[(a + b*Log[c*x^n])^2/(d*x)^(5/2),x]
 

Output:

(-2*x*(9*a^2 + 12*a*b*n + 8*b^2*n^2 + 6*b*(3*a + 2*b*n)*Log[c*x^n] + 9*b^2 
*Log[c*x^n]^2))/(27*(d*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2742, 2741}

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.

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

\(\Big \downarrow \) 2742

\(\displaystyle \frac {4}{3} b n \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}}dx-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {4}{3} b n \left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac {4 b n}{9 d (d x)^{3/2}}\right )-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}\)

Input:

Int[(a + b*Log[c*x^n])^2/(d*x)^(5/2),x]
 

Output:

(-2*(a + b*Log[c*x^n])^2)/(3*d*(d*x)^(3/2)) + (4*b*n*((-4*b*n)/(9*d*(d*x)^ 
(3/2)) - (2*(a + b*Log[c*x^n]))/(3*d*(d*x)^(3/2))))/3
 

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2742

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.29 (sec) , antiderivative size = 716, normalized size of antiderivative = 9.81

method	result	size
risch	\(-\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{3 d^{2} x \sqrt {d x}}-\frac {2 b \left (3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+6 b \ln \left (c \right )+4 n b +6 a \right ) \ln \left (x^{n}\right )}{9 d^{2} x \sqrt {d x}}-\frac {-36 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-36 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+36 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+48 b^{2} \ln \left (c \right ) n +72 a b \ln \left (c \right )+36 a^{2}+36 b^{2} \ln \left (c \right )^{2}+32 b^{2} n^{2}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}+24 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+24 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+48 a n b -36 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4} \operatorname {csgn}\left (i c \right )-36 i \pi \ln \left (c \right ) b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{5} \operatorname {csgn}\left (i c \right )-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4} \operatorname {csgn}\left (i c \right )^{2}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3} \operatorname {csgn}\left (i c \right )^{2}+18 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} \operatorname {csgn}\left (i c \right )-24 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-9 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )^{2}+36 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{54 d^{2} x \sqrt {d x}}\)	\(716\)

Input:

int((a+b*ln(c*x^n))^2/(d*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3/d^2*b^2/x/(d*x)^(1/2)*ln(x^n)^2-2/9/d^2*b*(3*I*b*Pi*csgn(I*x^n)*csgn( 
I*c*x^n)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-3*I*b*Pi*csgn(I*c* 
x^n)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+6*b*ln(c)+4*n*b+6*a)/x/(d*x)^(1/ 
2)*ln(x^n)-1/54/d^2*(-36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*I 
*Pi*ln(c)*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+18*Pi^2*b^2*csgn(I*x^n)^ 
2*csgn(I*c*x^n)^3*csgn(I*c)-9*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn( 
I*c)^2-36*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+24*I*Pi*b^2*n*csg 
n(I*c*x^n)^2*csgn(I*c)-24*I*Pi*b^2*n*csgn(I*c*x^n)^3+48*b^2*ln(c)*n+72*a*b 
*ln(c)+36*a^2+36*b^2*ln(c)^2+24*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+36* 
I*Pi*ln(c)*b^2*csgn(I*c*x^n)^2*csgn(I*c)+36*I*Pi*a*b*csgn(I*x^n)*csgn(I*c* 
x^n)^2+32*b^2*n^2+18*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)-9*Pi^2*b^2*csgn(I* 
c*x^n)^4*csgn(I*c)^2-9*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+18*Pi^2*b^2* 
csgn(I*x^n)*csgn(I*c*x^n)^5-36*I*Pi*ln(c)*b^2*csgn(I*c*x^n)^3-24*I*Pi*b^2* 
n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*I*Pi*a*b*csgn(I*c*x^n)^3+18*Pi^2* 
b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+36*I*Pi*a*b*csgn(I*c*x^n)^2*cs 
gn(I*c)+36*I*Pi*ln(c)*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+48*a*n*b-9*Pi^2*b^2* 
csgn(I*c*x^n)^6)/x/(d*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (9 \, b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + 9 \, b^{2} \log \left (c\right )^{2} + 12 \, a b n + 9 \, a^{2} + 6 \, {\left (2 \, b^{2} n + 3 \, a b\right )} \log \left (c\right ) + 6 \, {\left (2 \, b^{2} n^{2} + 3 \, b^{2} n \log \left (c\right ) + 3 \, a b n\right )} \log \left (x\right )\right )} \sqrt {d x}}{27 \, d^{3} x^{2}} \] Input:

integrate((a+b*log(c*x^n))^2/(d*x)^(5/2),x, algorithm="fricas")
 

Output:

-2/27*(9*b^2*n^2*log(x)^2 + 8*b^2*n^2 + 9*b^2*log(c)^2 + 12*a*b*n + 9*a^2 
+ 6*(2*b^2*n + 3*a*b)*log(c) + 6*(2*b^2*n^2 + 3*b^2*n*log(c) + 3*a*b*n)*lo 
g(x))*sqrt(d*x)/(d^3*x^2)
 

Sympy [A] (verification not implemented)

Time = 2.47 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=- \frac {2 a^{2} x}{3 \left (d x\right )^{\frac {5}{2}}} - \frac {8 a b n x}{9 \left (d x\right )^{\frac {5}{2}}} - \frac {4 a b x \log {\left (c x^{n} \right )}}{3 \left (d x\right )^{\frac {5}{2}}} - \frac {16 b^{2} n^{2} x}{27 \left (d x\right )^{\frac {5}{2}}} - \frac {8 b^{2} n x \log {\left (c x^{n} \right )}}{9 \left (d x\right )^{\frac {5}{2}}} - \frac {2 b^{2} x \log {\left (c x^{n} \right )}^{2}}{3 \left (d x\right )^{\frac {5}{2}}} \] Input:

integrate((a+b*ln(c*x**n))**2/(d*x)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

-2*a**2*x/(3*(d*x)**(5/2)) - 8*a*b*n*x/(9*(d*x)**(5/2)) - 4*a*b*x*log(c*x* 
*n)/(3*(d*x)**(5/2)) - 16*b**2*n**2*x/(27*(d*x)**(5/2)) - 8*b**2*n*x*log(c 
*x**n)/(9*(d*x)**(5/2)) - 2*b**2*x*log(c*x**n)**2/(3*(d*x)**(5/2))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {8}{27} \, b^{2} {\left (\frac {2 \, n^{2}}{\left (d x\right )^{\frac {3}{2}} d} + \frac {3 \, n \log \left (c x^{n}\right )}{\left (d x\right )^{\frac {3}{2}} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {8 \, a b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} \] Input:

integrate((a+b*log(c*x^n))^2/(d*x)^(5/2),x, algorithm="maxima")
 

Output:

-8/27*b^2*(2*n^2/((d*x)^(3/2)*d) + 3*n*log(c*x^n)/((d*x)^(3/2)*d)) - 2/3*b 
^2*log(c*x^n)^2/((d*x)^(3/2)*d) - 8/9*a*b*n/((d*x)^(3/2)*d) - 4/3*a*b*log( 
c*x^n)/((d*x)^(3/2)*d) - 2/3*a^2/((d*x)^(3/2)*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (61) = 122\).

Time = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.37 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {9 \, b^{2} n^{2} \log \left (d x\right )^{2}}{\sqrt {d x} d x} - \frac {6 \, {\left (3 \, b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n^{2} - 3 \, b^{2} n \log \left (c\right ) - 3 \, a b n\right )} \log \left (d x\right )}{\sqrt {d x} d x} + \frac {9 \, b^{2} n^{2} \log \left (d\right )^{2} - 12 \, b^{2} n^{2} \log \left (d\right ) - 18 \, b^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} n^{2} + 12 \, b^{2} n \log \left (c\right ) + 9 \, b^{2} \log \left (c\right )^{2} - 18 \, a b n \log \left (d\right ) + 12 \, a b n + 18 \, a b \log \left (c\right ) + 9 \, a^{2}}{\sqrt {d x} d x}\right )}}{27 \, d} \] Input:

integrate((a+b*log(c*x^n))^2/(d*x)^(5/2),x, algorithm="giac")
 

Output:

-2/27*(9*b^2*n^2*log(d*x)^2/(sqrt(d*x)*d*x) - 6*(3*b^2*n^2*log(d) - 2*b^2* 
n^2 - 3*b^2*n*log(c) - 3*a*b*n)*log(d*x)/(sqrt(d*x)*d*x) + (9*b^2*n^2*log( 
d)^2 - 12*b^2*n^2*log(d) - 18*b^2*n*log(c)*log(d) + 8*b^2*n^2 + 12*b^2*n*l 
og(c) + 9*b^2*log(c)^2 - 18*a*b*n*log(d) + 12*a*b*n + 18*a*b*log(c) + 9*a^ 
2)/(sqrt(d*x)*d*x))/d
 

Mupad [F(-1)]

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

int((a + b*log(c*x^n))^2/(d*x)^(5/2),x)
 

Output:

int((a + b*log(c*x^n))^2/(d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx=\frac {2 \sqrt {d}\, \left (-9 \mathrm {log}\left (x^{n} c \right )^{2} b^{2}-18 \,\mathrm {log}\left (x^{n} c \right ) a b -12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} n -9 a^{2}-12 a b n -8 b^{2} n^{2}\right )}{27 \sqrt {x}\, d^{3} x} \] Input:

int((a+b*log(c*x^n))^2/(d*x)^(5/2),x)
 

Output:

(2*sqrt(d)*( - 9*log(x**n*c)**2*b**2 - 18*log(x**n*c)*a*b - 12*log(x**n*c) 
*b**2*n - 9*a**2 - 12*a*b*n - 8*b**2*n**2))/(27*sqrt(x)*d**3*x)