\(\int \frac {(f x)^{-1+m} (a+b \log (c x^n))^2}{(d+e x^m)^3} \, dx\) [363]

Optimal result

Integrand size = 29, antiderivative size = 214 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=-\frac {b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{d^2 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}-\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{d^2 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^2 e m^3}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{d^2 e m^3} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.97 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\frac {x^{-m} (f x)^m \left (\frac {2 b m n \left (a+b \log \left (c x^n\right )\right )}{d \left (d+e x^m\right )}-\frac {m^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^2}-\frac {2 a b m n \log \left (d-d x^m\right )}{d^2}+\frac {2 b^2 n^2 \log \left (d-d x^m\right )}{d^2}+\frac {2 b^2 m n \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^m\right )}{d^2}+\frac {2 b^2 n^2 \left (\frac {1}{2} m^2 \log ^2(x)+\left (-m \log (x)+\log \left (-\frac {e x^m}{d}\right )\right ) \log \left (d+e x^m\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )\right )}{d^2}\right )}{2 e f m^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2777, 2776, 2791, 2773, 792, 2779, 2838}

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.

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

\(\Big \downarrow \) 2777

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

\(\Big \downarrow \) 2776

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

\(\Big \downarrow \) 2791

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

\(\Big \downarrow \) 2773

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

\(\Big \downarrow \) 792

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

\(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{d m^2}-\frac {\log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d m}}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}\right )}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\)

Input:

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

Output:

x^(1 - m)*(f*x)^(-1 + m)*(-1/2*(a + b*Log[c*x^n])^2/(e*m*(d + e*x^m)^2) + 
(b*n*(-((e*((x^m*(a + b*Log[c*x^n]))/(d*m*(d + e*x^m)) - (b*n*Log[d + e*x^ 
m])/(d*e*m^2)))/d) + (-(((a + b*Log[c*x^n])*Log[1 + d/(e*x^m)])/(d*m)) + ( 
b*n*PolyLog[2, -(d/(e*x^m))])/(d*m^2))/d))/(e*m))
 

Defintions of rubi rules used

rule 792

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten 
t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
 

rule 2773

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + 
 b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1)))   Int[(f*x)^m*(d 
 + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq 
Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
 

rule 2776

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*L 
og[c*x^n])^p/(e*r*(q + 1))), x] - Simp[b*f^m*n*(p/(e*r*(q + 1)))   Int[(d + 
 e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d 
, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || G 
tQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
 

rule 2777

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_)*(x_))^(m_.)*((d_) + ( 
e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[(f*x)^m/x^m   Int[x^m*(d + e*x^r)^ 
q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] 
&& EqQ[m, r - 1] && IGtQ[p, 0] &&  !(IntegerQ[m] || GtQ[f, 0])
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2791

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^ 
(q_))/(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x 
^n])^p/x), x], x] - Simp[e/d   Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n 
])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1 
]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (211) = 422\).

Time = 0.08 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.50 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\frac {{\left (b^{2} e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 2 \, {\left (b^{2} e^{2} m^{2} n \log \left (c\right ) + a b e^{2} m^{2} n - b^{2} e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 2 \, {\left (b^{2} d e m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d e m n \log \left (c\right ) + a b d e m n + {\left (2 \, b^{2} d e m^{2} n \log \left (c\right ) + 2 \, a b d e m^{2} n - b^{2} d e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m} - {\left (b^{2} d^{2} m^{2} \log \left (c\right )^{2} + a^{2} d^{2} m^{2} - 2 \, a b d^{2} m n + 2 \, {\left (a b d^{2} m^{2} - b^{2} d^{2} m n\right )} \log \left (c\right )\right )} f^{m - 1} - 2 \, {\left (b^{2} e^{2} f^{m - 1} n^{2} x^{2 \, m} + 2 \, b^{2} d e f^{m - 1} n^{2} x^{m} + b^{2} d^{2} f^{m - 1} n^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - 2 \, {\left ({\left (b^{2} e^{2} m n \log \left (c\right ) + a b e^{2} m n - b^{2} e^{2} n^{2}\right )} f^{m - 1} x^{2 \, m} + 2 \, {\left (b^{2} d e m n \log \left (c\right ) + a b d e m n - b^{2} d e n^{2}\right )} f^{m - 1} x^{m} + {\left (b^{2} d^{2} m n \log \left (c\right ) + a b d^{2} m n - b^{2} d^{2} n^{2}\right )} f^{m - 1}\right )} \log \left (e x^{m} + d\right ) - 2 \, {\left (b^{2} e^{2} f^{m - 1} m n^{2} x^{2 \, m} \log \left (x\right ) + 2 \, b^{2} d e f^{m - 1} m n^{2} x^{m} \log \left (x\right ) + b^{2} d^{2} f^{m - 1} m n^{2} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{2 \, {\left (d^{2} e^{3} m^{3} x^{2 \, m} + 2 \, d^{3} e^{2} m^{3} x^{m} + d^{4} e m^{3}\right )}} \] Input:

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

Output:

1/2*((b^2*e^2*m^2*n^2*log(x)^2 + 2*(b^2*e^2*m^2*n*log(c) + a*b*e^2*m^2*n - 
 b^2*e^2*m*n^2)*log(x))*f^(m - 1)*x^(2*m) + 2*(b^2*d*e*m^2*n^2*log(x)^2 + 
b^2*d*e*m*n*log(c) + a*b*d*e*m*n + (2*b^2*d*e*m^2*n*log(c) + 2*a*b*d*e*m^2 
*n - b^2*d*e*m*n^2)*log(x))*f^(m - 1)*x^m - (b^2*d^2*m^2*log(c)^2 + a^2*d^ 
2*m^2 - 2*a*b*d^2*m*n + 2*(a*b*d^2*m^2 - b^2*d^2*m*n)*log(c))*f^(m - 1) - 
2*(b^2*e^2*f^(m - 1)*n^2*x^(2*m) + 2*b^2*d*e*f^(m - 1)*n^2*x^m + b^2*d^2*f 
^(m - 1)*n^2)*dilog(-(e*x^m + d)/d + 1) - 2*((b^2*e^2*m*n*log(c) + a*b*e^2 
*m*n - b^2*e^2*n^2)*f^(m - 1)*x^(2*m) + 2*(b^2*d*e*m*n*log(c) + a*b*d*e*m* 
n - b^2*d*e*n^2)*f^(m - 1)*x^m + (b^2*d^2*m*n*log(c) + a*b*d^2*m*n - b^2*d 
^2*n^2)*f^(m - 1))*log(e*x^m + d) - 2*(b^2*e^2*f^(m - 1)*m*n^2*x^(2*m)*log 
(x) + 2*b^2*d*e*f^(m - 1)*m*n^2*x^m*log(x) + b^2*d^2*f^(m - 1)*m*n^2*log(x 
))*log((e*x^m + d)/d))/(d^2*e^3*m^3*x^(2*m) + 2*d^3*e^2*m^3*x^m + d^4*e*m^ 
3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

a*b*f^m*n*(1/((d*e^2*f*m*x^m + d^2*e*f*m)*m) + log(x)/(d^2*e*f*m) - log(e* 
x^m + d)/(d^2*e*f*m^2)) - 1/2*(f^m*log(x^n)^2/(e^3*f*m*x^(2*m) + 2*d*e^2*f 
*m*x^m + d^2*e*f*m) - 2*integrate((e*f^m*m*x^m*log(c)^2 + (d*f^m*n + (2*e* 
f^m*m*log(c) + e*f^m*n)*x^m)*log(x^n))/(e^4*f*m*x*x^(3*m) + 3*d*e^3*f*m*x* 
x^(2*m) + 3*d^2*e^2*f*m*x*x^m + d^3*e*f*m*x), x))*b^2 - a*b*f^m*log(c*x^n) 
/(e^3*f*m*x^(2*m) + 2*d*e^2*f*m*x^m + d^2*e*f*m) - 1/2*a^2*f^m/(e^3*f*m*x^ 
(2*m) + 2*d*e^2*f*m*x^m + d^2*e*f*m)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\frac {f^{m} \left (4 x^{2 m} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{x^{3 m} e^{3} x +3 x^{2 m} d \,e^{2} x +3 x^{m} d^{2} e x +d^{3} x}d x \right ) b^{2} d^{3} e^{2} m^{2} n -4 x^{2 m} \mathrm {log}\left (x^{m} e +d \right ) a b \,e^{2} m n -2 x^{2 m} \mathrm {log}\left (x^{m} e +d \right ) b^{2} e^{2} n^{2}+4 x^{2 m} \mathrm {log}\left (x \right ) a b \,e^{2} m^{2} n +2 x^{2 m} \mathrm {log}\left (x \right ) b^{2} e^{2} m \,n^{2}-2 x^{2 m} a b \,e^{2} m n -x^{2 m} b^{2} e^{2} n^{2}+8 x^{m} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{x^{3 m} e^{3} x +3 x^{2 m} d \,e^{2} x +3 x^{m} d^{2} e x +d^{3} x}d x \right ) b^{2} d^{4} e \,m^{2} n -8 x^{m} \mathrm {log}\left (x^{m} e +d \right ) a b d e m n -4 x^{m} \mathrm {log}\left (x^{m} e +d \right ) b^{2} d e \,n^{2}+8 x^{m} \mathrm {log}\left (x \right ) a b d e \,m^{2} n +4 x^{m} \mathrm {log}\left (x \right ) b^{2} d e m \,n^{2}+4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{x^{3 m} e^{3} x +3 x^{2 m} d \,e^{2} x +3 x^{m} d^{2} e x +d^{3} x}d x \right ) b^{2} d^{5} m^{2} n -4 \,\mathrm {log}\left (x^{m} e +d \right ) a b \,d^{2} m n -2 \,\mathrm {log}\left (x^{m} e +d \right ) b^{2} d^{2} n^{2}-2 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d^{2} m^{2}-4 \,\mathrm {log}\left (x^{n} c \right ) a b \,d^{2} m^{2}-2 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d^{2} m n +4 \,\mathrm {log}\left (x \right ) a b \,d^{2} m^{2} n +2 \,\mathrm {log}\left (x \right ) b^{2} d^{2} m \,n^{2}-2 a^{2} d^{2} m^{2}+2 a b \,d^{2} m n +b^{2} d^{2} n^{2}\right )}{4 d^{2} e f \,m^{3} \left (x^{2 m} e^{2}+2 x^{m} d e +d^{2}\right )} \] Input:

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

Output:

(f**m*(4*x**(2*m)*int(log(x**n*c)/(x**(3*m)*e**3*x + 3*x**(2*m)*d*e**2*x + 
 3*x**m*d**2*e*x + d**3*x),x)*b**2*d**3*e**2*m**2*n - 4*x**(2*m)*log(x**m* 
e + d)*a*b*e**2*m*n - 2*x**(2*m)*log(x**m*e + d)*b**2*e**2*n**2 + 4*x**(2* 
m)*log(x)*a*b*e**2*m**2*n + 2*x**(2*m)*log(x)*b**2*e**2*m*n**2 - 2*x**(2*m 
)*a*b*e**2*m*n - x**(2*m)*b**2*e**2*n**2 + 8*x**m*int(log(x**n*c)/(x**(3*m 
)*e**3*x + 3*x**(2*m)*d*e**2*x + 3*x**m*d**2*e*x + d**3*x),x)*b**2*d**4*e* 
m**2*n - 8*x**m*log(x**m*e + d)*a*b*d*e*m*n - 4*x**m*log(x**m*e + d)*b**2* 
d*e*n**2 + 8*x**m*log(x)*a*b*d*e*m**2*n + 4*x**m*log(x)*b**2*d*e*m*n**2 + 
4*int(log(x**n*c)/(x**(3*m)*e**3*x + 3*x**(2*m)*d*e**2*x + 3*x**m*d**2*e*x 
 + d**3*x),x)*b**2*d**5*m**2*n - 4*log(x**m*e + d)*a*b*d**2*m*n - 2*log(x* 
*m*e + d)*b**2*d**2*n**2 - 2*log(x**n*c)**2*b**2*d**2*m**2 - 4*log(x**n*c) 
*a*b*d**2*m**2 - 2*log(x**n*c)*b**2*d**2*m*n + 4*log(x)*a*b*d**2*m**2*n + 
2*log(x)*b**2*d**2*m*n**2 - 2*a**2*d**2*m**2 + 2*a*b*d**2*m*n + b**2*d**2* 
n**2))/(4*d**2*e*f*m**3*(x**(2*m)*e**2 + 2*x**m*d*e + d**2))