\(\int \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^2 \, dx\) [61]

Optimal result

Integrand size = 27, antiderivative size = 374 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {14 b^2 n^2 \sqrt {x}}{d f}+a b n x-3 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2} \] Output:

14*b^2*n^2*x^(1/2)/d/f+a*b*n*x-3*b^2*n^2*x+2*b^2*n^2*x*ln(d*(1/d+f*x^(1/2) 
))-2*b^2*n^2*ln(1+d*f*x^(1/2))/d^2/f^2+b^2*n*x*ln(c*x^n)-6*b*n*x^(1/2)*(a+ 
b*ln(c*x^n))/d/f+b*n*x*(a+b*ln(c*x^n))-2*b*n*x*ln(d*(1/d+f*x^(1/2)))*(a+b* 
ln(c*x^n))+2*b*n*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/d^2/f^2+x^(1/2)*(a+b*ln 
(c*x^n))^2/d/f-1/2*x*(a+b*ln(c*x^n))^2+x*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x 
^n))^2-ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^2/d^2/f^2+4*b^2*n^2*polylog(2,-d* 
f*x^(1/2))/d^2/f^2-4*b*n*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))/d^2/f^2+8 
*b^2*n^2*polylog(3,-d*f*x^(1/2))/d^2/f^2
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.41 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {-2 a^2 d f \sqrt {x}+12 a b d f n \sqrt {x}-28 b^2 d f n^2 \sqrt {x}+a^2 d^2 f^2 x-4 a b d^2 f^2 n x+6 b^2 d^2 f^2 n^2 x+2 a^2 \log \left (1+d f \sqrt {x}\right )-4 a b n \log \left (1+d f \sqrt {x}\right )+4 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-2 a^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right )+4 a b d^2 f^2 n x \log \left (1+d f \sqrt {x}\right )-4 b^2 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right )-4 a b d f \sqrt {x} \log \left (c x^n\right )+12 b^2 d f n \sqrt {x} \log \left (c x^n\right )+2 a b d^2 f^2 x \log \left (c x^n\right )-4 b^2 d^2 f^2 n x \log \left (c x^n\right )+4 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-4 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-4 a b d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+4 b^2 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-2 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+b^2 d^2 f^2 x \log ^2\left (c x^n\right )+2 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-2 b^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+8 b n \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-16 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{2 d^2 f^2} \] Input:

Integrate[Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2,x]
 

Output:

-1/2*(-2*a^2*d*f*Sqrt[x] + 12*a*b*d*f*n*Sqrt[x] - 28*b^2*d*f*n^2*Sqrt[x] + 
 a^2*d^2*f^2*x - 4*a*b*d^2*f^2*n*x + 6*b^2*d^2*f^2*n^2*x + 2*a^2*Log[1 + d 
*f*Sqrt[x]] - 4*a*b*n*Log[1 + d*f*Sqrt[x]] + 4*b^2*n^2*Log[1 + d*f*Sqrt[x] 
] - 2*a^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] + 4*a*b*d^2*f^2*n*x*Log[1 + d*f*S 
qrt[x]] - 4*b^2*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt[x]] - 4*a*b*d*f*Sqrt[x]*Log 
[c*x^n] + 12*b^2*d*f*n*Sqrt[x]*Log[c*x^n] + 2*a*b*d^2*f^2*x*Log[c*x^n] - 4 
*b^2*d^2*f^2*n*x*Log[c*x^n] + 4*a*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 4*b^ 
2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 4*a*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] 
*Log[c*x^n] + 4*b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 2*b^2*d* 
f*Sqrt[x]*Log[c*x^n]^2 + b^2*d^2*f^2*x*Log[c*x^n]^2 + 2*b^2*Log[1 + d*f*Sq 
rt[x]]*Log[c*x^n]^2 - 2*b^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 
8*b*n*(a - b*n + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 16*b^2*n^2*Pol 
yLog[3, -(d*f*Sqrt[x])])/(d^2*f^2)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2817, 2009}

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.

\(\displaystyle \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\)

\(\Big \downarrow \) 2817

\(\displaystyle -2 b n \int \left (\frac {1}{2} \left (-a-b \log \left (c x^n\right )\right )+\log \left (d \left (\sqrt {x} f+\frac {1}{d}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2 x}+\frac {a+b \log \left (c x^n\right )}{d f \sqrt {x}}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 b n \left (\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )-\frac {a x}{2}-\frac {1}{2} b x \log \left (c x^n\right )-\frac {2 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {7 b n \sqrt {x}}{d f}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {3 b n x}{2}\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\)

Input:

Int[Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2,x]
 

Output:

(Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - (x*(a + b*Log[c*x^n])^2)/2 + x*Log[ 
d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + 
b*Log[c*x^n])^2)/(d^2*f^2) - 2*b*n*((-7*b*n*Sqrt[x])/(d*f) - (a*x)/2 + (3* 
b*n*x)/2 - b*n*x*Log[d*(d^(-1) + f*Sqrt[x])] + (b*n*Log[1 + d*f*Sqrt[x]])/ 
(d^2*f^2) - (b*x*Log[c*x^n])/2 + (3*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) - (x 
*(a + b*Log[c*x^n]))/2 + x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]) 
- (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(d^2*f^2) - (2*b*n*PolyLog[2, 
-(d*f*Sqrt[x])])/(d^2*f^2) + (2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x 
])])/(d^2*f^2) - (4*b*n*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2817

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, 
Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p 
- 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] 
&& RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 
 1] && EqQ[m, 1] && EqQ[d*e, 1]))
 

Maple [F]

\[\int \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*log(d*f*sqrt(x) + 1), 
 x)
 

Sympy [F(-1)]

Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

(b^2*x*log(x^n)^2 - 2*(b^2*(n - log(c)) - a*b)*x*log(x^n) + ((2*n^2 - 2*n* 
log(c) + log(c)^2)*b^2 - 2*a*b*(n - log(c)) + a^2)*x)*log(d*f*sqrt(x) + 1) 
 - 1/27*(9*b^2*d*f*x^2*log(x^n)^2 + 6*(3*a*b*d*f - (5*d*f*n - 3*d*f*log(c) 
)*b^2)*x^2*log(x^n) + (9*a^2*d*f - 6*(5*d*f*n - 3*d*f*log(c))*a*b + (38*d* 
f*n^2 - 30*d*f*n*log(c) + 9*d*f*log(c)^2)*b^2)*x^2)/sqrt(x) + integrate(1/ 
2*(b^2*d^2*f^2*x*log(x^n)^2 + 2*(a*b*d^2*f^2 - (d^2*f^2*n - d^2*f^2*log(c) 
)*b^2)*x*log(x^n) + (a^2*d^2*f^2 - 2*(d^2*f^2*n - d^2*f^2*log(c))*a*b + (2 
*d^2*f^2*n^2 - 2*d^2*f^2*n*log(c) + d^2*f^2*log(c)^2)*b^2)*x)/(d*f*sqrt(x) 
 + 1), x)
 

Giac [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(x)*log(x**n*c)**2*b**2*d*f*n + 12*sqrt(x)*log(x**n*c)*a*b*d*f*n - 
36*sqrt(x)*log(x**n*c)*b**2*d*f*n**2 + 6*sqrt(x)*a**2*d*f*n - 36*sqrt(x)*a 
*b*d*f*n**2 + 84*sqrt(x)*b**2*d*f*n**3 - 3*int(log(x**n*c)**2/(d**2*f**2*x 
**2 - x),x)*b**2*n - 6*int(log(x**n*c)/(d**2*f**2*x**2 - x),x)*a*b*n + 6*i 
nt(log(x**n*c)/(d**2*f**2*x**2 - x),x)*b**2*n**2 + 3*int((sqrt(x)*log(x**n 
*c)**2)/(d**2*f**2*x**2 - x),x)*b**2*d*f*n + 6*int((sqrt(x)*log(x**n*c))/( 
d**2*f**2*x**2 - x),x)*a*b*d*f*n - 6*int((sqrt(x)*log(x**n*c))/(d**2*f**2* 
x**2 - x),x)*b**2*d*f*n**2 + 6*log(sqrt(x)*d*f + 1)*log(x**n*c)**2*b**2*d* 
*2*f**2*n*x + 12*log(sqrt(x)*d*f + 1)*log(x**n*c)*a*b*d**2*f**2*n*x - 12*l 
og(sqrt(x)*d*f + 1)*log(x**n*c)*b**2*d**2*f**2*n**2*x + 6*log(sqrt(x)*d*f 
+ 1)*a**2*d**2*f**2*n*x - 6*log(sqrt(x)*d*f + 1)*a**2*n - 12*log(sqrt(x)*d 
*f + 1)*a*b*d**2*f**2*n**2*x + 12*log(sqrt(x)*d*f + 1)*a*b*n**2 + 12*log(s 
qrt(x)*d*f + 1)*b**2*d**2*f**2*n**3*x - 12*log(sqrt(x)*d*f + 1)*b**2*n**3 
- log(x**n*c)**3*b**2 - 3*log(x**n*c)**2*a*b - 3*log(x**n*c)**2*b**2*d**2* 
f**2*n*x + 3*log(x**n*c)**2*b**2*n - 6*log(x**n*c)*a*b*d**2*f**2*n*x + 12* 
log(x**n*c)*b**2*d**2*f**2*n**2*x - 3*a**2*d**2*f**2*n*x + 12*a*b*d**2*f** 
2*n**2*x - 18*b**2*d**2*f**2*n**3*x)/(6*d**2*f**2*n)