\(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^3}{x^2} \, dx\) [68]

Optimal result

Integrand size = 30, antiderivative size = 610 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {90 b^3 d f n^3}{\sqrt {x}}+6 b^3 d^2 f^2 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}-3 b^3 d^2 f^2 n^3 \log (x)+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right ) \] Output:

-90*b^3*d*f*n^3/x^(1/2)+6*b^3*d^2*f^2*n^3*ln(1+d*f*x^(1/2))-6*b^3*n^3*ln(1 
+d*f*x^(1/2))/x-3*b^3*d^2*f^2*n^3*ln(x)+3/2*b^3*d^2*f^2*n^3*ln(x)^2-42*b^2 
*d*f*n^2*(a+b*ln(c*x^n))/x^(1/2)+6*b^2*d^2*f^2*n^2*ln(1+d*f*x^(1/2))*(a+b* 
ln(c*x^n))-6*b^2*n^2*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/x-3*b^2*d^2*f^2*n^2 
*ln(x)*(a+b*ln(c*x^n))-9*b*d*f*n*(a+b*ln(c*x^n))^2/x^(1/2)+3*b*d^2*f^2*n*l 
n(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^2-3*b*n*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n)) 
^2/x-1/2*d^2*f^2*(a+b*ln(c*x^n))^3-d*f*(a+b*ln(c*x^n))^3/x^(1/2)+d^2*f^2*l 
n(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^3-ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^3/x-1 
/8*d^2*f^2*(a+b*ln(c*x^n))^4/b/n+12*b^3*d^2*f^2*n^3*polylog(2,-d*f*x^(1/2) 
)+12*b^2*d^2*f^2*n^2*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))+6*b*d^2*f^2*n 
*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^(1/2))-24*b^3*d^2*f^2*n^3*polylog(3,-d 
*f*x^(1/2))-24*b^2*d^2*f^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))+48* 
b^3*d^2*f^2*n^3*polylog(4,-d*f*x^(1/2))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1455\) vs. \(2(610)=1220\).

Time = 1.07 (sec) , antiderivative size = 1455, normalized size of antiderivative = 2.39 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

d^2*f^2*Log[1 + d*f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 
3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) 
+ 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n] 
)^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n] 
)^3) - d^2*f^2*Log[Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3 
*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 
 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n]) 
^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n]) 
^3) - (Log[1 + d*f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3 
*a^2*b*n*Log[x] + 6*a*b^2*n^2*Log[x] + 6*b^3*n^3*Log[x] + 3*a*b^2*n^2*Log[ 
x]^2 + 3*b^3*n^3*Log[x]^2 + b^3*n^3*Log[x]^3 + 3*a^2*b*(-(n*Log[x]) + Log[ 
c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + 
Log[c*x^n]) + 6*a*b^2*n*Log[x]*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*Log[ 
x]*(-(n*Log[x]) + Log[c*x^n]) + 3*b^3*n^2*Log[x]^2*(-(n*Log[x]) + Log[c*x^ 
n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c* 
x^n])^2 + 3*b^3*n*Log[x]*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + 
 Log[c*x^n])^3))/x + (-(a^3*d*f) - 3*a^2*b*d*f*n - 6*a*b^2*d*f*n^2 - 6*b^3 
*d*f*n^3 - 3*a^2*b*d*f*(-(n*Log[x]) + Log[c*x^n]) - 6*a*b^2*d*f*n*(-(n*Log 
[x]) + Log[c*x^n]) - 6*b^3*d*f*n^2*(-(n*Log[x]) + Log[c*x^n]) - 3*a*b^2*d* 
f*(-(n*Log[x]) + Log[c*x^n])^2 - 3*b^3*d*f*n*(-(n*Log[x]) + Log[c*x^n])...
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2824, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

\(\displaystyle -3 b n \left (\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{24 b^2 n^2}-2 d^2 f^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-4 b d^2 f^2 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+8 b d^2 f^2 n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}-d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b d^2 f^2 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {3 d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {14 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-4 b^2 d^2 f^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+8 b^2 d^2 f^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )-16 b^2 d^2 f^2 n^2 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right )-\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-2 b^2 d^2 f^2 n^2 \log \left (d f \sqrt {x}+1\right )+b^2 d^2 f^2 n^2 \log (x)+\frac {30 b^2 d f n^2}{\sqrt {x}}+\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{x}\right )+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}\)

Input:

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

Output:

-((d*f*(a + b*Log[c*x^n])^3)/Sqrt[x]) + d^2*f^2*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 - (d^2*f^2 
*Log[x]*(a + b*Log[c*x^n])^3)/2 - 3*b*n*((30*b^2*d*f*n^2)/Sqrt[x] - 2*b^2* 
d^2*f^2*n^2*Log[1 + d*f*Sqrt[x]] + (2*b^2*n^2*Log[1 + d*f*Sqrt[x]])/x + b^ 
2*d^2*f^2*n^2*Log[x] - (b^2*d^2*f^2*n^2*Log[x]^2)/2 + (14*b*d*f*n*(a + b*L 
og[c*x^n]))/Sqrt[x] - 2*b*d^2*f^2*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n] 
) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/x + b*d^2*f^2*n*Log[x] 
*(a + b*Log[c*x^n]) + (3*d*f*(a + b*Log[c*x^n])^2)/Sqrt[x] - d^2*f^2*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 + (d^2*f^2*(a + b*Log[c*x^n])^3)/(6*b*n) - (d^2*f^2*Log[x]*(a + 
 b*Log[c*x^n])^3)/(6*b*n) + (d^2*f^2*(a + b*Log[c*x^n])^4)/(24*b^2*n^2) - 
4*b^2*d^2*f^2*n^2*PolyLog[2, -(d*f*Sqrt[x])] - 4*b*d^2*f^2*n*(a + b*Log[c* 
x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 2*d^2*f^2*(a + b*Log[c*x^n])^2*PolyLog[ 
2, -(d*f*Sqrt[x])] + 8*b^2*d^2*f^2*n^2*PolyLog[3, -(d*f*Sqrt[x])] + 8*b*d^ 
2*f^2*n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])] - 16*b^2*d^2*f^2*n^2 
*PolyLog[4, -(d*f*Sqrt[x])])
 

Defintions of rubi rules used

rule 2009

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

rule 2824

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ 
.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)], 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, g, m, n, 
 q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ 
[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ 
(q + 1)/m] && EqQ[d*e, 1]))
 

Maple [F]

\[\int \frac {\ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(x)*a**3*d*f - 6*sqrt(x)*a**2*b*d*f*n - 12*sqrt(x)*a*b**2*d*f*n* 
*2 - 12*sqrt(x)*b**3*d*f*n**3 + int(log(x**n*c)**3/(d**2*f**2*x**3 - x**2) 
,x)*b**3*x + 3*int(log(x**n*c)**2/(d**2*f**2*x**3 - x**2),x)*a*b**2*x + 3* 
int(log(x**n*c)**2/(d**2*f**2*x**3 - x**2),x)*b**3*n*x + 3*int(log(x**n*c) 
/(d**2*f**2*x**3 - x**2),x)*a**2*b*x + 6*int(log(x**n*c)/(d**2*f**2*x**3 - 
 x**2),x)*a*b**2*n*x + 6*int(log(x**n*c)/(d**2*f**2*x**3 - x**2),x)*b**3*n 
**2*x - int((sqrt(x)*log(x**n*c)**3)/(d**2*f**2*x**3 - x**2),x)*b**3*d*f*x 
 - 3*int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**3 - x**2),x)*a*b**2*d*f*x 
- 3*int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**3 - x**2),x)*b**3*d*f*n*x - 
 3*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**3 - x**2),x)*a**2*b*d*f*x - 6*i 
nt((sqrt(x)*log(x**n*c))/(d**2*f**2*x**3 - x**2),x)*a*b**2*d*f*n*x - 6*int 
((sqrt(x)*log(x**n*c))/(d**2*f**2*x**3 - x**2),x)*b**3*d*f*n**2*x - 2*log( 
sqrt(x)*d*f + 1)*log(x**n*c)**3*b**3 - 6*log(sqrt(x)*d*f + 1)*log(x**n*c)* 
*2*a*b**2 - 6*log(sqrt(x)*d*f + 1)*log(x**n*c)**2*b**3*n - 6*log(sqrt(x)*d 
*f + 1)*log(x**n*c)*a**2*b - 12*log(sqrt(x)*d*f + 1)*log(x**n*c)*a*b**2*n 
- 12*log(sqrt(x)*d*f + 1)*log(x**n*c)*b**3*n**2 + 2*log(sqrt(x)*d*f + 1)*a 
**3*d**2*f**2*x - 2*log(sqrt(x)*d*f + 1)*a**3 + 6*log(sqrt(x)*d*f + 1)*a** 
2*b*d**2*f**2*n*x - 6*log(sqrt(x)*d*f + 1)*a**2*b*n + 12*log(sqrt(x)*d*f + 
 1)*a*b**2*d**2*f**2*n**2*x - 12*log(sqrt(x)*d*f + 1)*a*b**2*n**2 + 12*log 
(sqrt(x)*d*f + 1)*b**3*d**2*f**2*n**3*x - 12*log(sqrt(x)*d*f + 1)*b**3*...