\(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))}{x^4} \, dx\) [58]

Optimal result

Integrand size = 28, antiderivative size = 372 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {11 b d f n}{225 x^{5/2}}+\frac {5 b d^2 f^2 n}{72 x^2}-\frac {b d^3 f^3 n}{9 x^{3/2}}+\frac {2 b d^4 f^4 n}{9 x}-\frac {7 b d^5 f^5 n}{9 \sqrt {x}}+\frac {1}{9} b d^6 f^6 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 x^3}-\frac {1}{18} b d^6 f^6 n \log (x)+\frac {1}{12} b d^6 f^6 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} b d^6 f^6 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \] Output:

-11/225*b*d*f*n/x^(5/2)+5/72*b*d^2*f^2*n/x^2-1/9*b*d^3*f^3*n/x^(3/2)+2/9*b 
*d^4*f^4*n/x-7/9*b*d^5*f^5*n/x^(1/2)+1/9*b*d^6*f^6*n*ln(1+d*f*x^(1/2))-1/9 
*b*n*ln(1+d*f*x^(1/2))/x^3-1/18*b*d^6*f^6*n*ln(x)+1/12*b*d^6*f^6*n*ln(x)^2 
-1/15*d*f*(a+b*ln(c*x^n))/x^(5/2)+1/12*d^2*f^2*(a+b*ln(c*x^n))/x^2-1/9*d^3 
*f^3*(a+b*ln(c*x^n))/x^(3/2)+1/6*d^4*f^4*(a+b*ln(c*x^n))/x-1/3*d^5*f^5*(a+ 
b*ln(c*x^n))/x^(1/2)+1/3*d^6*f^6*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))-1/3*ln( 
1+d*f*x^(1/2))*(a+b*ln(c*x^n))/x^3-1/6*d^6*f^6*ln(x)*(a+b*ln(c*x^n))+2/3*b 
*d^6*f^6*n*polylog(2,-d*f*x^(1/2))
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.77 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {\left (-1+d^6 f^6 x^3\right ) \log \left (1+d f \sqrt {x}\right ) \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{9 x^3}-\frac {d f \left (120 a+88 b n-150 a d f \sqrt {x}-125 b d f n \sqrt {x}+200 a d^2 f^2 x+200 b d^2 f^2 n x-300 a d^3 f^3 x^{3/2}-400 b d^3 f^3 n x^{3/2}+600 a d^4 f^4 x^2+1400 b d^4 f^4 n x^2-150 b d^5 f^5 n x^{5/2} \log ^2(x)+10 b \left (12-15 d f \sqrt {x}+20 d^2 f^2 x-30 d^3 f^3 x^{3/2}+60 d^4 f^4 x^2\right ) \log \left (c x^n\right )+100 d^5 f^5 x^{5/2} \log (x) \left (3 a+b n+3 b \log \left (c x^n\right )\right )\right )}{1800 x^{5/2}}+\frac {2}{3} b d^6 f^6 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \] Input:

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

Output:

((-1 + d^6*f^6*x^3)*Log[1 + d*f*Sqrt[x]]*(3*a + b*n + 3*b*Log[c*x^n]))/(9* 
x^3) - (d*f*(120*a + 88*b*n - 150*a*d*f*Sqrt[x] - 125*b*d*f*n*Sqrt[x] + 20 
0*a*d^2*f^2*x + 200*b*d^2*f^2*n*x - 300*a*d^3*f^3*x^(3/2) - 400*b*d^3*f^3* 
n*x^(3/2) + 600*a*d^4*f^4*x^2 + 1400*b*d^4*f^4*n*x^2 - 150*b*d^5*f^5*n*x^( 
5/2)*Log[x]^2 + 10*b*(12 - 15*d*f*Sqrt[x] + 20*d^2*f^2*x - 30*d^3*f^3*x^(3 
/2) + 60*d^4*f^4*x^2)*Log[c*x^n] + 100*d^5*f^5*x^(5/2)*Log[x]*(3*a + b*n + 
 3*b*Log[c*x^n])))/(1800*x^(5/2)) + (2*b*d^6*f^6*n*PolyLog[2, -(d*f*Sqrt[x 
])])/3
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 357, 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.

\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx\)

\(\Big \downarrow \) 2823

\(\displaystyle -b n \int \left (\frac {d^6 \log \left (d \sqrt {x} f+1\right ) f^6}{3 x}-\frac {d^6 \log (x) f^6}{6 x}-\frac {d^5 f^5}{3 x^{3/2}}+\frac {d^4 f^4}{6 x^2}-\frac {d^3 f^3}{9 x^{5/2}}+\frac {d^2 f^2}{12 x^3}-\frac {d f}{15 x^{7/2}}-\frac {\log \left (d \sqrt {x} f+1\right )}{3 x^4}\right )dx+\frac {1}{3} d^6 f^6 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} d^6 f^6 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-b n \left (-\frac {2}{3} d^6 f^6 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-\frac {1}{12} d^6 f^6 \log ^2(x)-\frac {1}{9} d^6 f^6 \log \left (d f \sqrt {x}+1\right )+\frac {1}{18} d^6 f^6 \log (x)+\frac {7 d^5 f^5}{9 \sqrt {x}}-\frac {2 d^4 f^4}{9 x}+\frac {d^3 f^3}{9 x^{3/2}}-\frac {5 d^2 f^2}{72 x^2}+\frac {11 d f}{225 x^{5/2}}+\frac {\log \left (d f \sqrt {x}+1\right )}{9 x^3}\right )\)

Input:

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

Output:

-1/15*(d*f*(a + b*Log[c*x^n]))/x^(5/2) + (d^2*f^2*(a + b*Log[c*x^n]))/(12* 
x^2) - (d^3*f^3*(a + b*Log[c*x^n]))/(9*x^(3/2)) + (d^4*f^4*(a + b*Log[c*x^ 
n]))/(6*x) - (d^5*f^5*(a + b*Log[c*x^n]))/(3*Sqrt[x]) + (d^6*f^6*Log[1 + d 
*f*Sqrt[x]]*(a + b*Log[c*x^n]))/3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n 
]))/(3*x^3) - (d^6*f^6*Log[x]*(a + b*Log[c*x^n]))/6 - b*n*((11*d*f)/(225*x 
^(5/2)) - (5*d^2*f^2)/(72*x^2) + (d^3*f^3)/(9*x^(3/2)) - (2*d^4*f^4)/(9*x) 
 + (7*d^5*f^5)/(9*Sqrt[x]) - (d^6*f^6*Log[1 + d*f*Sqrt[x]])/9 + Log[1 + d* 
f*Sqrt[x]]/(9*x^3) + (d^6*f^6*Log[x])/18 - (d^6*f^6*Log[x]^2)/12 - (2*d^6* 
f^6*PolyLog[2, -(d*f*Sqrt[x])])/3)
 

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]

\[\int \frac {\ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{4}}d x\]

Input:

int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))/x^4,x)
 

Output:

int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))/x^4,x)
 

Fricas [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \] Input:

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x^4,x, algorithm="fricas 
")
 

Output:

integral((b*log(c*x^n) + a)*log(d*f*sqrt(x) + 1)/x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\text {Timed out} \] Input:

integrate(ln(d*(1/d+f*x**(1/2)))*(a+b*ln(c*x**n))/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \] Input:

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x^4,x, algorithm="maxima 
")
 

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^4, x)
 

Giac [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \] Input:

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \] Input:

int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n)))/x^4,x)
 

Output:

int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n)))/x^4, x)
 

Reduce [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {-600 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a n -200 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,n^{2}+150 a \,d^{2} f^{2} n x +125 b \,d^{2} f^{2} n^{2} x -600 \sqrt {x}\, a \,d^{5} f^{5} n \,x^{2}-120 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b d f n +150 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} n x -600 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,d^{5} f^{5} n \,x^{2}+600 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b \,d^{6} f^{6} n \,x^{3}-120 \sqrt {x}\, a d f n -88 \sqrt {x}\, b d f \,n^{2}+600 \left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b \,d^{6} f^{6} n^{2} x^{3}-600 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b \,d^{7} f^{7} n^{2} x^{3}-150 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{6} f^{6} x^{3}-300 \,\mathrm {log}\left (x^{n} c \right ) a \,d^{6} f^{6} x^{3}+300 a \,d^{4} f^{4} n \,x^{2}+400 b \,d^{4} f^{4} n^{2} x^{2}-600 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b n -200 \sqrt {x}\, a \,d^{3} f^{3} n x -200 \sqrt {x}\, b \,d^{3} f^{3} n^{2} x +300 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{4} f^{4} n \,x^{2}-200 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,d^{3} f^{3} n x -1400 \sqrt {x}\, b \,d^{5} f^{5} n^{2} x^{2}-600 \mathrm {log}\left (\sqrt {x}\, d f +1\right )^{2} b \,d^{6} f^{6} n^{2} x^{3}+600 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a \,d^{6} f^{6} n \,x^{3}+200 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,d^{6} f^{6} n^{2} x^{3}+2940 \,\mathrm {log}\left (\sqrt {x}\right ) b \,d^{6} f^{6} n^{2} x^{3}-1570 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{6} f^{6} n \,x^{3}}{1800 n \,x^{3}} \] Input:

int(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x^4,x)
 

Output:

( - 600*sqrt(x)*log(x**n*c)*b*d**5*f**5*n*x**2 - 200*sqrt(x)*log(x**n*c)*b 
*d**3*f**3*n*x - 120*sqrt(x)*log(x**n*c)*b*d*f*n - 600*sqrt(x)*a*d**5*f**5 
*n*x**2 - 200*sqrt(x)*a*d**3*f**3*n*x - 120*sqrt(x)*a*d*f*n - 1400*sqrt(x) 
*b*d**5*f**5*n**2*x**2 - 200*sqrt(x)*b*d**3*f**3*n**2*x - 88*sqrt(x)*b*d*f 
*n**2 + 600*int(log(sqrt(x)*d*f + 1)/(d**2*f**2*x**2 - x),x)*b*d**6*f**6*n 
**2*x**3 - 600*int((sqrt(x)*log(sqrt(x)*d*f + 1))/(d**2*f**2*x**2 - x),x)* 
b*d**7*f**7*n**2*x**3 - 600*log(sqrt(x)*d*f + 1)**2*b*d**6*f**6*n**2*x**3 
+ 600*log(sqrt(x)*d*f + 1)*log(x**n*c)*b*d**6*f**6*n*x**3 - 600*log(sqrt(x 
)*d*f + 1)*log(x**n*c)*b*n + 600*log(sqrt(x)*d*f + 1)*a*d**6*f**6*n*x**3 - 
 600*log(sqrt(x)*d*f + 1)*a*n + 200*log(sqrt(x)*d*f + 1)*b*d**6*f**6*n**2* 
x**3 - 200*log(sqrt(x)*d*f + 1)*b*n**2 + 2940*log(sqrt(x))*b*d**6*f**6*n** 
2*x**3 - 150*log(x**n*c)**2*b*d**6*f**6*x**3 - 300*log(x**n*c)*a*d**6*f**6 
*x**3 - 1570*log(x**n*c)*b*d**6*f**6*n*x**3 + 300*log(x**n*c)*b*d**4*f**4* 
n*x**2 + 150*log(x**n*c)*b*d**2*f**2*n*x + 300*a*d**4*f**4*n*x**2 + 150*a* 
d**2*f**2*n*x + 400*b*d**4*f**4*n**2*x**2 + 125*b*d**2*f**2*n**2*x)/(1800* 
n*x**3)