\(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))}{x^3} \, dx\) [57]

Optimal result

Integrand size = 28, antiderivative size = 289 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {7 b d f n}{36 x^{3/2}}+\frac {3 b d^2 f^2 n}{8 x}-\frac {5 b d^3 f^3 n}{4 \sqrt {x}}+\frac {1}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b d^4 f^4 n \log (x)+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.72 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {\left (-1+d^4 f^4 x^2\right ) \log \left (1+d f \sqrt {x}\right ) \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{4 x^2}-\frac {d f \left (12 a+14 b n-18 a d f \sqrt {x}-27 b d f n \sqrt {x}+36 a d^2 f^2 x+90 b d^2 f^2 n x-9 b d^3 f^3 n x^{3/2} \log ^2(x)+6 b \left (2-3 d f \sqrt {x}+6 d^2 f^2 x\right ) \log \left (c x^n\right )+9 d^3 f^3 x^{3/2} \log (x) \left (2 a+b n+2 b \log \left (c x^n\right )\right )\right )}{72 x^{3/2}}+b d^4 f^4 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \] Input:

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

Output:

((-1 + d^4*f^4*x^2)*Log[1 + d*f*Sqrt[x]]*(2*a + b*n + 2*b*Log[c*x^n]))/(4* 
x^2) - (d*f*(12*a + 14*b*n - 18*a*d*f*Sqrt[x] - 27*b*d*f*n*Sqrt[x] + 36*a* 
d^2*f^2*x + 90*b*d^2*f^2*n*x - 9*b*d^3*f^3*n*x^(3/2)*Log[x]^2 + 6*b*(2 - 3 
*d*f*Sqrt[x] + 6*d^2*f^2*x)*Log[c*x^n] + 9*d^3*f^3*x^(3/2)*Log[x]*(2*a + b 
*n + 2*b*Log[c*x^n])))/(72*x^(3/2)) + b*d^4*f^4*n*PolyLog[2, -(d*f*Sqrt[x] 
)]
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.97, 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.

\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 2823

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

\[\int \frac {\ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:

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

Output:

integral((b*log(c*x^n) + a)*log(d*f*sqrt(x) + 1)/x^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^3, x)
 

Giac [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {-36 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,d^{3} f^{3} n x -12 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b d f n -36 \sqrt {x}\, a \,d^{3} f^{3} n x -12 \sqrt {x}\, a d f n -90 \sqrt {x}\, b \,d^{3} f^{3} n^{2} x -14 \sqrt {x}\, b d f \,n^{2}+36 \left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b \,d^{4} f^{4} n^{2} x^{2}-36 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b \,d^{5} f^{5} n^{2} x^{2}-36 \mathrm {log}\left (\sqrt {x}\, d f +1\right )^{2} b \,d^{4} f^{4} n^{2} x^{2}+36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b \,d^{4} f^{4} n \,x^{2}-36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b n +36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a \,d^{4} f^{4} n \,x^{2}-36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a n +18 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,d^{4} f^{4} n^{2} x^{2}-18 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,n^{2}+150 \,\mathrm {log}\left (\sqrt {x}\right ) b \,d^{4} f^{4} n^{2} x^{2}-9 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{4} f^{4} x^{2}-18 \,\mathrm {log}\left (x^{n} c \right ) a \,d^{4} f^{4} x^{2}-84 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{4} f^{4} n \,x^{2}+18 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} n x +18 a \,d^{2} f^{2} n x +27 b \,d^{2} f^{2} n^{2} x}{72 n \,x^{2}} \] Input:

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

Output:

( - 36*sqrt(x)*log(x**n*c)*b*d**3*f**3*n*x - 12*sqrt(x)*log(x**n*c)*b*d*f* 
n - 36*sqrt(x)*a*d**3*f**3*n*x - 12*sqrt(x)*a*d*f*n - 90*sqrt(x)*b*d**3*f* 
*3*n**2*x - 14*sqrt(x)*b*d*f*n**2 + 36*int(log(sqrt(x)*d*f + 1)/(d**2*f**2 
*x**2 - x),x)*b*d**4*f**4*n**2*x**2 - 36*int((sqrt(x)*log(sqrt(x)*d*f + 1) 
)/(d**2*f**2*x**2 - x),x)*b*d**5*f**5*n**2*x**2 - 36*log(sqrt(x)*d*f + 1)* 
*2*b*d**4*f**4*n**2*x**2 + 36*log(sqrt(x)*d*f + 1)*log(x**n*c)*b*d**4*f**4 
*n*x**2 - 36*log(sqrt(x)*d*f + 1)*log(x**n*c)*b*n + 36*log(sqrt(x)*d*f + 1 
)*a*d**4*f**4*n*x**2 - 36*log(sqrt(x)*d*f + 1)*a*n + 18*log(sqrt(x)*d*f + 
1)*b*d**4*f**4*n**2*x**2 - 18*log(sqrt(x)*d*f + 1)*b*n**2 + 150*log(sqrt(x 
))*b*d**4*f**4*n**2*x**2 - 9*log(x**n*c)**2*b*d**4*f**4*x**2 - 18*log(x**n 
*c)*a*d**4*f**4*x**2 - 84*log(x**n*c)*b*d**4*f**4*n*x**2 + 18*log(x**n*c)* 
b*d**2*f**2*n*x + 18*a*d**2*f**2*n*x + 27*b*d**2*f**2*n**2*x)/(72*n*x**2)