\(\int \sqrt {d x} (a+b \log (c x^n))^2 \, dx\) [97]

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

(2*x*Sqrt[d*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
 

Rubi [A] (verified)

Time = 0.24 (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.

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

\(\Big \downarrow \) 2742

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

\(\Big \downarrow \) 2741

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

Input:

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

Output:

(2*(d*x)^(3/2)*(a + b*Log[c*x^n])^2)/(3*d) - (4*b*n*((-4*b*n*(d*x)^(3/2))/ 
(9*d) + (2*(d*x)^(3/2)*(a + b*Log[c*x^n]))/(3*d)))/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 = 0.27 (sec) , antiderivative size = 710, normalized size of antiderivative = 9.73

method	result	size
risch	\(\frac {2 d \,b^{2} x^{2} \ln \left (x^{n}\right )^{2}}{3 \sqrt {d x}}+\frac {2 d b \,x^{2} \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 \sqrt {d x}}+\frac {d \left (-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}\right ) x^{2}}{54 \sqrt {d x}}\)	\(710\)

Input:

int((d*x)^(1/2)*(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{27} \, {\left (9 \, b^{2} n^{2} x \log \left (x\right )^{2} + 9 \, b^{2} x \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{2} n - 3 \, a b\right )} x \log \left (c\right ) + {\left (8 \, b^{2} n^{2} - 12 \, a b n + 9 \, a^{2}\right )} x + 6 \, {\left (3 \, b^{2} n x \log \left (c\right ) - {\left (2 \, b^{2} n^{2} - 3 \, a b n\right )} x\right )} \log \left (x\right )\right )} \sqrt {d x} \] Input:

integrate((d*x)^(1/2)*(a+b*log(c*x^n))^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 a^{2} x \sqrt {d x}}{3} - \frac {8 a b n x \sqrt {d x}}{9} + \frac {4 a b x \sqrt {d x} \log {\left (c x^{n} \right )}}{3} + \frac {16 b^{2} n^{2} x \sqrt {d x}}{27} - \frac {8 b^{2} n x \sqrt {d x} \log {\left (c x^{n} \right )}}{9} + \frac {2 b^{2} x \sqrt {d x} \log {\left (c x^{n} \right )}^{2}}{3} \] Input:

integrate((d*x)**(1/2)*(a+b*ln(c*x**n))**2,x)
                                                                                    
                                                                                    
 

Output:

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

Maxima [A] (verification not implemented)

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

integrate((d*x)^(1/2)*(a+b*log(c*x^n))^2,x, algorithm="maxima")
 

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 383, normalized size of antiderivative = 5.25 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((d*x)^(1/2)*(a+b*log(c*x^n))^2,x, algorithm="giac")
 

Output:

(1/3*I + 1/3)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log( 
x)^2 - (1/3*I - 1/3)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*log(x)^2*sin(1/4 
*pi*sgn(d)) - (4/9*I + 4/9)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(abs(d))*cos(1/4*p 
i*sgn(d))*log(x) + (2/3*I + 2/3)*sqrt(2)*b^2*n*x^(3/2)*sqrt(abs(d))*cos(1/ 
4*pi*sgn(d))*log(c)*log(x) + (4/9*I - 4/9)*sqrt(2)*b^2*n^2*x^(3/2)*sqrt(ab 
s(d))*log(x)*sin(1/4*pi*sgn(d)) - (2/3*I - 2/3)*sqrt(2)*b^2*n*x^(3/2)*sqrt 
(abs(d))*log(c)*log(x)*sin(1/4*pi*sgn(d)) + (8/27*I + 8/27)*sqrt(2)*b^2*n^ 
2*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) - (4/9*I + 4/9)*sqrt(2)*b^2*n*x^ 
(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(c) + (2/3*I + 2/3)*sqrt(2)*a*b*n 
*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d))*log(x) - (8/27*I - 8/27)*sqrt(2)* 
b^2*n^2*x^(3/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + (4/9*I - 4/9)*sqrt(2)*b^ 
2*n*x^(3/2)*sqrt(abs(d))*log(c)*sin(1/4*pi*sgn(d)) - (2/3*I - 2/3)*sqrt(2) 
*a*b*n*x^(3/2)*sqrt(abs(d))*log(x)*sin(1/4*pi*sgn(d)) - (4/9*I + 4/9)*sqrt 
(2)*a*b*n*x^(3/2)*sqrt(abs(d))*cos(1/4*pi*sgn(d)) + (4/9*I - 4/9)*sqrt(2)* 
a*b*n*x^(3/2)*sqrt(abs(d))*sin(1/4*pi*sgn(d)) + 2/3*b^2*sqrt(d)*x^(3/2)*lo 
g(c)^2 + 4/3*a*b*sqrt(d)*x^(3/2)*log(c) + 2/3*a^2*sqrt(d)*x^(3/2)
 

Mupad [F(-1)]

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

int((d*x)^(1/2)*(a + b*log(c*x^n))^2,x)
 

Output:

int((d*x)^(1/2)*(a + b*log(c*x^n))^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 \sqrt {x}\, \sqrt {d}\, x \left (9 \mathrm {log}\left (x^{n} c \right )^{2} b^{2}+18 \,\mathrm {log}\left (x^{n} c \right ) a b -12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} n +9 a^{2}-12 a b n +8 b^{2} n^{2}\right )}{27} \] Input:

int((d*x)^(1/2)*(a+b*log(c*x^n))^2,x)
 

Output:

(2*sqrt(x)*sqrt(d)*x*(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