\(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^3} \, dx\) [126]

Optimal result

Integrand size = 28, antiderivative size = 346 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {7 b f k n}{36 e x^{3/2}}+\frac {3 b f^2 k n}{8 e^2 x}-\frac {5 b f^3 k n}{4 e^3 \sqrt {x}}+\frac {b f^4 k n \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}-\frac {b f^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b f^4 k n \log (x)}{8 e^4}+\frac {b f^4 k n \log ^2(x)}{8 e^4}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {b f^4 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^4} \] Output:

-7/36*b*f*k*n/e/x^(3/2)+3/8*b*f^2*k*n/e^2/x-5/4*b*f^3*k*n/e^3/x^(1/2)+1/4* 
b*f^4*k*n*ln(e+f*x^(1/2))/e^4-1/4*b*n*ln(d*(e+f*x^(1/2))^k)/x^2-b*f^4*k*n* 
ln(e+f*x^(1/2))*ln(-f*x^(1/2)/e)/e^4-1/8*b*f^4*k*n*ln(x)/e^4+1/8*b*f^4*k*n 
*ln(x)^2/e^4-1/6*f*k*(a+b*ln(c*x^n))/e/x^(3/2)+1/4*f^2*k*(a+b*ln(c*x^n))/e 
^2/x-1/2*f^3*k*(a+b*ln(c*x^n))/e^3/x^(1/2)+1/2*f^4*k*ln(e+f*x^(1/2))*(a+b* 
ln(c*x^n))/e^4-1/2*ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n))/x^2-1/4*f^4*k*ln( 
x)*(a+b*ln(c*x^n))/e^4-b*f^4*k*n*polylog(2,1+f*x^(1/2)/e)/e^4
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {12 a e^3 f k \sqrt {x}+14 b e^3 f k n \sqrt {x}-18 a e^2 f^2 k x-27 b e^2 f^2 k n x+36 a e f^3 k x^{3/2}+90 b e f^3 k n x^{3/2}+36 a e^4 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b e^4 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 a f^4 k x^2 \log (x)+9 b f^4 k n x^2 \log (x)-36 b f^4 k n x^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-9 b f^4 k n x^2 \log ^2(x)+12 b e^3 f k \sqrt {x} \log \left (c x^n\right )-18 b e^2 f^2 k x \log \left (c x^n\right )+36 b e f^3 k x^{3/2} \log \left (c x^n\right )+36 b e^4 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+18 b f^4 k x^2 \log (x) \log \left (c x^n\right )-18 f^4 k x^2 \log \left (e+f \sqrt {x}\right ) \left (2 a+b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right )-72 b f^4 k n x^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{72 e^4 x^2} \] Input:

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^3,x]
 

Output:

-1/72*(12*a*e^3*f*k*Sqrt[x] + 14*b*e^3*f*k*n*Sqrt[x] - 18*a*e^2*f^2*k*x - 
27*b*e^2*f^2*k*n*x + 36*a*e*f^3*k*x^(3/2) + 90*b*e*f^3*k*n*x^(3/2) + 36*a* 
e^4*Log[d*(e + f*Sqrt[x])^k] + 18*b*e^4*n*Log[d*(e + f*Sqrt[x])^k] + 18*a* 
f^4*k*x^2*Log[x] + 9*b*f^4*k*n*x^2*Log[x] - 36*b*f^4*k*n*x^2*Log[1 + (f*Sq 
rt[x])/e]*Log[x] - 9*b*f^4*k*n*x^2*Log[x]^2 + 12*b*e^3*f*k*Sqrt[x]*Log[c*x 
^n] - 18*b*e^2*f^2*k*x*Log[c*x^n] + 36*b*e*f^3*k*x^(3/2)*Log[c*x^n] + 36*b 
*e^4*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 18*b*f^4*k*x^2*Log[x]*Log[c*x^n 
] - 18*f^4*k*x^2*Log[e + f*Sqrt[x]]*(2*a + b*n - 2*b*n*Log[x] + 2*b*Log[c* 
x^n]) - 72*b*f^4*k*n*x^2*PolyLog[2, -((f*Sqrt[x])/e)])/(e^4*x^2)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 331, 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.

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

\(\Big \downarrow \) 2823

\(\displaystyle -b n \int \left (\frac {k \log \left (e+f \sqrt {x}\right ) f^4}{2 e^4 x}-\frac {k \log (x) f^4}{4 e^4 x}-\frac {k f^3}{2 e^3 x^{3/2}}+\frac {k f^2}{4 e^2 x^2}-\frac {k f}{6 e x^{5/2}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 x^3}\right )dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 x^2}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 x^2}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}-b n \left (\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}+\frac {f^4 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{e^4}-\frac {f^4 k \log ^2(x)}{8 e^4}-\frac {f^4 k \log \left (e+f \sqrt {x}\right )}{4 e^4}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {f^4 k \log (x)}{8 e^4}+\frac {5 f^3 k}{4 e^3 \sqrt {x}}-\frac {3 f^2 k}{8 e^2 x}+\frac {7 f k}{36 e x^{3/2}}\right )\)

Input:

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^3,x]
 

Output:

-1/6*(f*k*(a + b*Log[c*x^n]))/(e*x^(3/2)) + (f^2*k*(a + b*Log[c*x^n]))/(4* 
e^2*x) - (f^3*k*(a + b*Log[c*x^n]))/(2*e^3*Sqrt[x]) + (f^4*k*Log[e + f*Sqr 
t[x]]*(a + b*Log[c*x^n]))/(2*e^4) - (Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c 
*x^n]))/(2*x^2) - (f^4*k*Log[x]*(a + b*Log[c*x^n]))/(4*e^4) - b*n*((7*f*k) 
/(36*e*x^(3/2)) - (3*f^2*k)/(8*e^2*x) + (5*f^3*k)/(4*e^3*Sqrt[x]) - (f^4*k 
*Log[e + f*Sqrt[x]])/(4*e^4) + Log[d*(e + f*Sqrt[x])^k]/(4*x^2) + (f^4*k*L 
og[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^4 + (f^4*k*Log[x])/(8*e^4) - (f 
^4*k*Log[x]^2)/(8*e^4) + (f^4*k*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^4)
 

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]

\[\int \frac {\ln \left (d \left (e +f \sqrt {x}\right )^{k}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^3,x, algorithm="fricas 
")
 

Output:

integral((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Timed out} \] Input:

integrate(ln(d*(e+f*x**(1/2))**k)*(a+b*ln(c*x**n))/x**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^3,x, algorithm="maxima 
")
 

Output:

-1/36*(18*b*e*log(d)*log(x^n) + 18*a*e*log(d) + 9*(e*n*log(d) + 2*e*log(c) 
*log(d))*b + 9*(2*b*e*log(x^n) + (e*n + 2*e*log(c))*b + 2*a*e)*log((f*sqrt 
(x) + e)^k) + (6*b*f*k*x*log(x^n) + (6*a*f*k + (7*f*k*n + 6*f*k*log(c))*b) 
*x)/sqrt(x))/(e*x^2) - integrate(1/8*(2*b*f^2*k*log(x^n) + 2*a*f^2*k + (f^ 
2*k*n + 2*f^2*k*log(c))*b)/(e*f*x^(5/2) + e^2*x^2), x)
 

Giac [F]

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

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^3,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {-12 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,e^{3} f \,k^{2} n -36 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b e \,f^{3} k^{2} n x -12 \sqrt {x}\, a \,e^{3} f \,k^{2} n -36 \sqrt {x}\, a e \,f^{3} k^{2} n x -14 \sqrt {x}\, b \,e^{3} f \,k^{2} n^{2}-90 \sqrt {x}\, b e \,f^{3} k^{2} n^{2} x -36 \left (\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b \,e^{2} f^{4} k \,n^{2} x^{2}+36 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b e \,f^{5} k \,n^{2} x^{2}+36 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) a \,f^{4} k^{2} n \,x^{2}+168 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) b \,f^{4} k^{2} n^{2} x^{2}+150 \,\mathrm {log}\left (\sqrt {x}\right ) b \,f^{4} k^{2} n^{2} x^{2}-36 \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )^{2} b \,f^{4} n^{2} x^{2}-36 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,e^{4} k n +36 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,f^{4} k n \,x^{2}-36 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) a \,e^{4} k n -18 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,e^{4} k \,n^{2}-150 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,f^{4} k \,n^{2} x^{2}-9 \mathrm {log}\left (x^{n} c \right )^{2} b \,f^{4} k^{2} x^{2}-18 \,\mathrm {log}\left (x^{n} c \right ) a \,f^{4} k^{2} x^{2}+18 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{2} f^{2} k^{2} n x -84 \,\mathrm {log}\left (x^{n} c \right ) b \,f^{4} k^{2} n \,x^{2}+18 a \,e^{2} f^{2} k^{2} n x +27 b \,e^{2} f^{2} k^{2} n^{2} x}{72 e^{4} k n \,x^{2}} \] Input:

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

Output:

( - 12*sqrt(x)*log(x**n*c)*b*e**3*f*k**2*n - 36*sqrt(x)*log(x**n*c)*b*e*f* 
*3*k**2*n*x - 12*sqrt(x)*a*e**3*f*k**2*n - 36*sqrt(x)*a*e*f**3*k**2*n*x - 
14*sqrt(x)*b*e**3*f*k**2*n**2 - 90*sqrt(x)*b*e*f**3*k**2*n**2*x - 36*int(l 
og((sqrt(x)*f + e)**k*d)/(e**2*x - f**2*x**2),x)*b*e**2*f**4*k*n**2*x**2 + 
 36*int((sqrt(x)*log((sqrt(x)*f + e)**k*d))/(e**2*x - f**2*x**2),x)*b*e*f* 
*5*k*n**2*x**2 + 36*log(sqrt(x)*f + e)*a*f**4*k**2*n*x**2 + 168*log(sqrt(x 
)*f + e)*b*f**4*k**2*n**2*x**2 + 150*log(sqrt(x))*b*f**4*k**2*n**2*x**2 - 
36*log((sqrt(x)*f + e)**k*d)**2*b*f**4*n**2*x**2 - 36*log((sqrt(x)*f + e)* 
*k*d)*log(x**n*c)*b*e**4*k*n + 36*log((sqrt(x)*f + e)**k*d)*log(x**n*c)*b* 
f**4*k*n*x**2 - 36*log((sqrt(x)*f + e)**k*d)*a*e**4*k*n - 18*log((sqrt(x)* 
f + e)**k*d)*b*e**4*k*n**2 - 150*log((sqrt(x)*f + e)**k*d)*b*f**4*k*n**2*x 
**2 - 9*log(x**n*c)**2*b*f**4*k**2*x**2 - 18*log(x**n*c)*a*f**4*k**2*x**2 
+ 18*log(x**n*c)*b*e**2*f**2*k**2*n*x - 84*log(x**n*c)*b*f**4*k**2*n*x**2 
+ 18*a*e**2*f**2*k**2*n*x + 27*b*e**2*f**2*k**2*n**2*x)/(72*e**4*k*n*x**2)