\(\int \frac {\log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3}{x^3} \, dx\) [138]

Optimal result

Integrand size = 28, antiderivative size = 914 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Output:

-3/2*b^3*f^4*n^3*ln(e+f*x^(1/2))*ln(-f*x^(1/2)/e)/e^4-12*b^2*f^4*n^2*(a+b* 
ln(c*x^n))*polylog(3,-f*x^(1/2)/e)/e^4+3*b^2*f^4*n^2*(a+b*ln(c*x^n))*polyl 
og(2,-f*x^(1/2)/e)/e^4+3*b*f^4*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x^(1/2)/e) 
/e^4+3/4*b*f^4*n*ln(1+f*x^(1/2)/e)*(a+b*ln(c*x^n))^2/e^4+3/4*b^2*f^4*n^2*l 
n(e+f*x^(1/2))*(a+b*ln(c*x^n))/e^4-63/4*b^2*f^3*n^2*(a+b*ln(c*x^n))/e^3/x^ 
(1/2)-15/4*b*f^3*n*(a+b*ln(c*x^n))^2/e^3/x^(1/2)-3/2*b^3*f^4*n^3*polylog(2 
,1+f*x^(1/2)/e)/e^4+3/8*b^3*f^4*n^3*ln(e+f*x^(1/2))/e^4-3/4*b*n*ln(d*(e+f* 
x^(1/2)))*(a+b*ln(c*x^n))^2/x^2-3/4*b^2*n^2*ln(d*(e+f*x^(1/2)))*(a+b*ln(c* 
x^n))/x^2-255/8*b^3*f^3*n^3/e^3/x^(1/2)-6*b^3*f^4*n^3*polylog(3,-f*x^(1/2) 
/e)/e^4+24*b^3*f^4*n^3*polylog(4,-f*x^(1/2)/e)/e^4-1/2*ln(d*(e+f*x^(1/2))) 
*(a+b*ln(c*x^n))^3/x^2-3/8*b^3*n^3*ln(d*(e+f*x^(1/2)))/x^2+1/2*f^4*ln(1+f* 
x^(1/2)/e)*(a+b*ln(c*x^n))^3/e^4-1/2*f^3*(a+b*ln(c*x^n))^3/e^3/x^(1/2)-3/1 
6*b^3*f^4*n^3*ln(x)/e^4-1/16*f^4*(a+b*ln(c*x^n))^4/b/e^4/n+3/16*b^3*f^4*n^ 
3*ln(x)^2/e^4-175/216*b^3*f*n^3/e/x^(3/2)+45/16*b^3*f^2*n^3/e^2/x-37/36*b^ 
2*f*n^2*(a+b*ln(c*x^n))/e/x^(3/2)+21/8*b^2*f^2*n^2*(a+b*ln(c*x^n))/e^2/x-7 
/12*b*f*n*(a+b*ln(c*x^n))^2/e/x^(3/2)+9/8*b*f^2*n*(a+b*ln(c*x^n))^2/e^2/x- 
3/8*b^2*f^4*n^2*ln(x)*(a+b*ln(c*x^n))/e^4-1/6*f*(a+b*ln(c*x^n))^3/e/x^(3/2 
)+1/4*f^2*(a+b*ln(c*x^n))^3/e^2/x-1/8*f^4*(a+b*ln(c*x^n))^3/e^4
 

Mathematica [A] (verified)

Time = 2.39 (sec) , antiderivative size = 1549, normalized size of antiderivative = 1.69 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/432*(54*e^4*Log[d*(e + f*Sqrt[x])]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3 
*b^3*n^3 + 6*b*(2*a^2 + 2*a*b*n + b^2*n^2)*Log[c*x^n] + 6*b^2*(2*a + b*n)* 
Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3) + 18*e^3*f*Sqrt[x]*(4*a^3 + 6*a^2*b*n + 
 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2* 
n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a 
*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 
 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 27*e^2*f^2*x*(4*a^3 + 6*a^2*b*n + 6 
*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n* 
(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b 
^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4 
*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 54*e*f^3*x^(3/2)*(4*a^3 + 6*a^2*b*n + 
 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2* 
n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a 
*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 
 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 54*f^4*x^2*Log[e + f*Sqrt[x]]*(4*a^ 
3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^ 
n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log 
[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + 
Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 27*f^4*x^2*Log[x]*(4 
*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Lo...
 

Rubi [A] (verified)

Time = 1.90 (sec) , antiderivative size = 1011, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{2 e^4}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{4 e^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 e^3 \sqrt {x}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 e^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f}{6 e x^{3/2}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-3 b n \left (\frac {\left (a+b \log \left (c x^n\right )\right )^4 f^4}{48 b^2 e^4 n^2}+\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{6 b e^4 n}-\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{6 b e^4 n}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{12 b e^4 n}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^4}{24 b e^4 n}-\frac {b^2 n^2 \log ^2(x) f^4}{16 e^4}-\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{4 e^4}-\frac {b^2 n^2 \log \left (e+f \sqrt {x}\right ) f^4}{8 e^4}+\frac {b^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right ) f^4}{2 e^4}+\frac {b^2 n^2 \log (x) f^4}{16 e^4}-\frac {b n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) f^4}{4 e^4}+\frac {b n \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4}{8 e^4}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right ) f^4}{2 e^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {8 b^2 n^2 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {5 \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 e^3 \sqrt {x}}+\frac {21 b n \left (a+b \log \left (c x^n\right )\right ) f^3}{4 e^3 \sqrt {x}}+\frac {85 b^2 n^2 f^3}{8 e^3 \sqrt {x}}-\frac {3 \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 e^2 x}-\frac {7 b n \left (a+b \log \left (c x^n\right )\right ) f^2}{8 e^2 x}-\frac {15 b^2 n^2 f^2}{16 e^2 x}+\frac {7 \left (a+b \log \left (c x^n\right )\right )^2 f}{36 e x^{3/2}}+\frac {37 b n \left (a+b \log \left (c x^n\right )\right ) f}{108 e x^{3/2}}+\frac {175 b^2 n^2 f}{648 e x^{3/2}}+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{8 x^2}+\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}\right )\)

Input:

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

Output:

-1/6*(f*(a + b*Log[c*x^n])^3)/(e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^3)/(4* 
e^2*x) - (f^3*(a + b*Log[c*x^n])^3)/(2*e^3*Sqrt[x]) + (f^4*Log[e + f*Sqrt[ 
x]]*(a + b*Log[c*x^n])^3)/(2*e^4) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x 
^n])^3)/(2*x^2) - (f^4*Log[x]*(a + b*Log[c*x^n])^3)/(4*e^4) - 3*b*n*((175* 
b^2*f*n^2)/(648*e*x^(3/2)) - (15*b^2*f^2*n^2)/(16*e^2*x) + (85*b^2*f^3*n^2 
)/(8*e^3*Sqrt[x]) - (b^2*f^4*n^2*Log[e + f*Sqrt[x]])/(8*e^4) + (b^2*n^2*Lo 
g[d*(e + f*Sqrt[x])])/(8*x^2) + (b^2*f^4*n^2*Log[e + f*Sqrt[x]]*Log[-((f*S 
qrt[x])/e)])/(2*e^4) + (b^2*f^4*n^2*Log[x])/(16*e^4) - (b^2*f^4*n^2*Log[x] 
^2)/(16*e^4) + (37*b*f*n*(a + b*Log[c*x^n]))/(108*e*x^(3/2)) - (7*b*f^2*n* 
(a + b*Log[c*x^n]))/(8*e^2*x) + (21*b*f^3*n*(a + b*Log[c*x^n]))/(4*e^3*Sqr 
t[x]) - (b*f^4*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*e^4) + (b*n*Log 
[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) + (b*f^4*n*Log[x]*(a + b*L 
og[c*x^n]))/(8*e^4) + (7*f*(a + b*Log[c*x^n])^2)/(36*e*x^(3/2)) - (3*f^2*( 
a + b*Log[c*x^n])^2)/(8*e^2*x) + (5*f^3*(a + b*Log[c*x^n])^2)/(4*e^3*Sqrt[ 
x]) + (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(4*x^2) - (f^4*Log[1 + 
 (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*e^4) + (f^4*(a + b*Log[c*x^n])^3) 
/(24*b*e^4*n) + (f^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(6*b*e^4*n) 
- (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(6*b*e^4*n) - (f^4*Log 
[x]*(a + b*Log[c*x^n])^3)/(12*b*e^4*n) + (f^4*(a + b*Log[c*x^n])^4)/(48*b^ 
2*e^4*n^2) + (b^2*f^4*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*e^4) - (b*f...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

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) + d*e)/x^3, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 16*sqrt(x)*a**3*e**3*f - 48*sqrt(x)*a**3*e*f**3*x - 24*sqrt(x)*a**2*b* 
e**3*f*n - 72*sqrt(x)*a**2*b*e*f**3*n*x - 24*sqrt(x)*a*b**2*e**3*f*n**2 - 
72*sqrt(x)*a*b**2*e*f**3*n**2*x - 12*sqrt(x)*b**3*e**3*f*n**3 - 36*sqrt(x) 
*b**3*e*f**3*n**3*x - 24*int(log(x**n*c)**3/(e**2*x**3 - f**2*x**4),x)*b** 
3*e**6*x**2 - 72*int(log(x**n*c)**2/(e**2*x**3 - f**2*x**4),x)*a*b**2*e**6 
*x**2 - 36*int(log(x**n*c)**2/(e**2*x**3 - f**2*x**4),x)*b**3*e**6*n*x**2 
- 72*int(log(x**n*c)/(e**2*x**3 - f**2*x**4),x)*a**2*b*e**6*x**2 - 72*int( 
log(x**n*c)/(e**2*x**3 - f**2*x**4),x)*a*b**2*e**6*n*x**2 - 36*int(log(x** 
n*c)/(e**2*x**3 - f**2*x**4),x)*b**3*e**6*n**2*x**2 + 24*int((sqrt(x)*log( 
x**n*c)**3)/(e**2*x**3 - f**2*x**4),x)*b**3*e**5*f*x**2 + 72*int((sqrt(x)* 
log(x**n*c)**2)/(e**2*x**3 - f**2*x**4),x)*a*b**2*e**5*f*x**2 + 36*int((sq 
rt(x)*log(x**n*c)**2)/(e**2*x**3 - f**2*x**4),x)*b**3*e**5*f*n*x**2 + 72*i 
nt((sqrt(x)*log(x**n*c))/(e**2*x**3 - f**2*x**4),x)*a**2*b*e**5*f*x**2 + 7 
2*int((sqrt(x)*log(x**n*c))/(e**2*x**3 - f**2*x**4),x)*a*b**2*e**5*f*n*x** 
2 + 36*int((sqrt(x)*log(x**n*c))/(e**2*x**3 - f**2*x**4),x)*b**3*e**5*f*n* 
*2*x**2 - 48*log(sqrt(x)*d*f + d*e)*log(x**n*c)**3*b**3*e**4 - 144*log(sqr 
t(x)*d*f + d*e)*log(x**n*c)**2*a*b**2*e**4 - 72*log(sqrt(x)*d*f + d*e)*log 
(x**n*c)**2*b**3*e**4*n - 144*log(sqrt(x)*d*f + d*e)*log(x**n*c)*a**2*b*e* 
*4 - 144*log(sqrt(x)*d*f + d*e)*log(x**n*c)*a*b**2*e**4*n - 72*log(sqrt(x) 
*d*f + d*e)*log(x**n*c)*b**3*e**4*n**2 - 48*log(sqrt(x)*d*f + d*e)*a**3...