\(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^2} \, dx\) [125]

Optimal result

Integrand size = 28, antiderivative size = 248 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {3 b f k n}{e \sqrt {x}}+\frac {b f^2 k n \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x}-\frac {2 b f^2 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b f^2 k n \log (x)}{2 e^2}+\frac {b f^2 k n \log ^2(x)}{4 e^2}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {f^2 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b f^2 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {4 a e f k \sqrt {x}+12 b e f k n \sqrt {x}+4 a e^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+4 b e^2 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+2 a f^2 k x \log (x)+2 b f^2 k n x \log (x)-4 b f^2 k n x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-b f^2 k n x \log ^2(x)+4 b e f k \sqrt {x} \log \left (c x^n\right )+4 b e^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+2 b f^2 k x \log (x) \log \left (c x^n\right )-4 f^2 k x \log \left (e+f \sqrt {x}\right ) \left (a+b n-b n \log (x)+b \log \left (c x^n\right )\right )-8 b f^2 k n x \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{4 e^2 x} \] Input:

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

Output:

-1/4*(4*a*e*f*k*Sqrt[x] + 12*b*e*f*k*n*Sqrt[x] + 4*a*e^2*Log[d*(e + f*Sqrt 
[x])^k] + 4*b*e^2*n*Log[d*(e + f*Sqrt[x])^k] + 2*a*f^2*k*x*Log[x] + 2*b*f^ 
2*k*n*x*Log[x] - 4*b*f^2*k*n*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - b*f^2*k*n*x 
*Log[x]^2 + 4*b*e*f*k*Sqrt[x]*Log[c*x^n] + 4*b*e^2*Log[d*(e + f*Sqrt[x])^k 
]*Log[c*x^n] + 2*b*f^2*k*x*Log[x]*Log[c*x^n] - 4*f^2*k*x*Log[e + f*Sqrt[x] 
]*(a + b*n - b*n*Log[x] + b*Log[c*x^n]) - 8*b*f^2*k*n*x*PolyLog[2, -((f*Sq 
rt[x])/e)])/(e^2*x)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2823, 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.

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

\(\Big \downarrow \) 2823

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2823

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-(b*e*log(d)*log(x^n) + a*e*log(d) + (e*n*log(d) + e*log(c)*log(d))*b + (b 
*e*log(x^n) + (e*n + e*log(c))*b + a*e)*log((f*sqrt(x) + e)^k) + (b*f*k*x* 
log(x^n) + (a*f*k + (3*f*k*n + f*k*log(c))*b)*x)/sqrt(x))/(e*x) - integrat 
e(1/2*(b*f^2*k*log(x^n) + a*f^2*k + (f^2*k*n + f^2*k*log(c))*b)/(e*f*x^(3/ 
2) + e^2*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {-4 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b e f \,k^{2} n -4 \sqrt {x}\, a e f \,k^{2} n -12 \sqrt {x}\, b e f \,k^{2} n^{2}-4 \left (\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b \,e^{2} f^{2} k \,n^{2} x +4 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b e \,f^{3} k \,n^{2} x +4 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) a \,f^{2} k^{2} n x +16 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) b \,f^{2} k^{2} n^{2} x +12 \,\mathrm {log}\left (\sqrt {x}\right ) b \,f^{2} k^{2} n^{2} x -4 \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )^{2} b \,f^{2} n^{2} x -4 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,e^{2} k n +4 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,f^{2} k n x -4 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) a \,e^{2} k n -4 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,e^{2} k \,n^{2}-12 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,f^{2} k \,n^{2} x -\mathrm {log}\left (x^{n} c \right )^{2} b \,f^{2} k^{2} x -2 \,\mathrm {log}\left (x^{n} c \right ) a \,f^{2} k^{2} x -8 \,\mathrm {log}\left (x^{n} c \right ) b \,f^{2} k^{2} n x}{4 e^{2} k n x} \] Input:

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

Output:

( - 4*sqrt(x)*log(x**n*c)*b*e*f*k**2*n - 4*sqrt(x)*a*e*f*k**2*n - 12*sqrt( 
x)*b*e*f*k**2*n**2 - 4*int(log((sqrt(x)*f + e)**k*d)/(e**2*x - f**2*x**2), 
x)*b*e**2*f**2*k*n**2*x + 4*int((sqrt(x)*log((sqrt(x)*f + e)**k*d))/(e**2* 
x - f**2*x**2),x)*b*e*f**3*k*n**2*x + 4*log(sqrt(x)*f + e)*a*f**2*k**2*n*x 
 + 16*log(sqrt(x)*f + e)*b*f**2*k**2*n**2*x + 12*log(sqrt(x))*b*f**2*k**2* 
n**2*x - 4*log((sqrt(x)*f + e)**k*d)**2*b*f**2*n**2*x - 4*log((sqrt(x)*f + 
 e)**k*d)*log(x**n*c)*b*e**2*k*n + 4*log((sqrt(x)*f + e)**k*d)*log(x**n*c) 
*b*f**2*k*n*x - 4*log((sqrt(x)*f + e)**k*d)*a*e**2*k*n - 4*log((sqrt(x)*f 
+ e)**k*d)*b*e**2*k*n**2 - 12*log((sqrt(x)*f + e)**k*d)*b*f**2*k*n**2*x - 
log(x**n*c)**2*b*f**2*k**2*x - 2*log(x**n*c)*a*f**2*k**2*x - 8*log(x**n*c) 
*b*f**2*k**2*n*x)/(4*e**2*k*n*x)