\(\int \frac {(f x)^{-1+m} (a+b \log (c x^n))^2}{(d+e x^m)^4} \, dx\) [366]

Optimal result

Integrand size = 29, antiderivative size = 346 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log (x)}{3 d^3 e m^2}+\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{3 d^3 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^3 e m^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2377, 2376, 2391, 2379, 2438, 2373, 266, 272, 46} \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=-\frac {2 b n x^{1-m} (f x)^{m-1} \log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e m^2}-\frac {2 b n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{m-1} \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3}+\frac {b^2 n^2 x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{d^3 e m^3}-\frac {b^2 n^2 x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m^2}-\frac {b^2 n^2 x^{1-m} (f x)^{m-1}}{3 d^2 e m^3 \left (d+e x^m\right )} \]

[In]

Int[((f*x)^(-1 + m)*(a + b*Log[c*x^n])^2)/(d + e*x^m)^4,x]

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 266

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2373

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] - Dist[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] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 2376

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*Log[c*x^n])^p/(e*r*(q + 1))), x] - Dist[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] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]

Rule 2377

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :>
Dist[(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 2379

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] + Dist[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 2391

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_))/(x_), x_Symbol] :> Dist[1/d,
Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[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 2438

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]

Rubi steps \begin{align*} \text {integral}& = \left (x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx \\ & = -\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}+\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )^3} \, dx}{3 e m} \\ & = -\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx}{3 d m}+\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )^2} \, dx}{3 d e m} \\ & = \frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^2} \, dx}{3 d^2 m}+\frac {\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )} \, dx}{3 d^2 e m}-\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {1}{x \left (d+e x^m\right )^2} \, dx}{3 d e m^2} \\ & = \frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 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}{3 e m \left (d+e x^m\right )^3}-\frac {2 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 )}{3 d^3 e m^2}-\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \text {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^m\right )}{3 d e m^3}+\frac {\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m}}{d+e x^m} \, dx}{3 d^3 m^2}+\frac {\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {\log \left (1+\frac {d x^{-m}}{e}\right )}{x} \, dx}{3 d^3 e m^2} \\ & = \frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 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}{3 e m \left (d+e x^m\right )^3}-\frac {2 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 )}{3 d^3 e m^2}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{3 d^3 e m^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3}-\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \text {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^m\right )}{3 d e m^3} \\ & = -\frac {b^2 n^2 x^{1-m} (f x)^{-1+m}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log (x)}{3 d^3 e m^2}+\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac {2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 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}{3 e m \left (d+e x^m\right )^3}-\frac {2 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 )}{3 d^3 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^3 e m^3}+\frac {2 b^2 n^2 x^{1-m} (f x)^{-1+m} \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{3 d^3 e m^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.60 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.69 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\frac {x^{-m} (f x)^m \left (\frac {b m n \left (a+b \log \left (c x^n\right )\right )}{d \left (d+e x^m\right )^2}-\frac {m^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3}+\frac {b n \left (2 a m-b n+2 b m \log \left (c x^n\right )\right )}{d^2 \left (d+e x^m\right )}-\frac {2 a b m n \log \left (d-d x^m\right )}{d^3}+\frac {3 b^2 n^2 \log \left (d-d x^m\right )}{d^3}+\frac {2 b^2 m n \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^m\right )}{d^3}+\frac {2 b^2 n^2 \left (\frac {1}{2} m^2 \log ^2(x)+\left (-m \log (x)+\log \left (-\frac {e x^m}{d}\right )\right ) \log \left (d+e x^m\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )\right )}{d^3}\right )}{3 e f m^3} \]

[In]

Integrate[((f*x)^(-1 + m)*(a + b*Log[c*x^n])^2)/(d + e*x^m)^4,x]

[Out]

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

Maple [F]

\[\int \frac {\left (f x \right )^{m -1} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{\left (d +e \,x^{m}\right )^{4}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (331) = 662\).

Time = 0.31 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.34 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\frac {{\left (b^{2} e^{3} m^{2} n^{2} \log \left (x\right )^{2} + {\left (2 \, b^{2} e^{3} m^{2} n \log \left (c\right ) + 2 \, a b e^{3} m^{2} n - 3 \, b^{2} e^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + {\left (3 \, b^{2} d e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d e^{2} m n \log \left (c\right ) + 2 \, a b d e^{2} m n - b^{2} d e^{2} n^{2} + {\left (6 \, b^{2} d e^{2} m^{2} n \log \left (c\right ) + 6 \, a b d e^{2} m^{2} n - 7 \, b^{2} d e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + {\left (3 \, b^{2} d^{2} e m^{2} n^{2} \log \left (x\right )^{2} + 5 \, b^{2} d^{2} e m n \log \left (c\right ) + 5 \, a b d^{2} e m n - 2 \, b^{2} d^{2} e n^{2} + 2 \, {\left (3 \, b^{2} d^{2} e m^{2} n \log \left (c\right ) + 3 \, a b d^{2} e m^{2} n - 2 \, b^{2} d^{2} e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m} - {\left (b^{2} d^{3} m^{2} \log \left (c\right )^{2} + a^{2} d^{3} m^{2} - 3 \, a b d^{3} m n + b^{2} d^{3} n^{2} + {\left (2 \, a b d^{3} m^{2} - 3 \, b^{2} d^{3} m n\right )} \log \left (c\right )\right )} f^{m - 1} - 2 \, {\left (b^{2} e^{3} f^{m - 1} n^{2} x^{3 \, m} + 3 \, b^{2} d e^{2} f^{m - 1} n^{2} x^{2 \, m} + 3 \, b^{2} d^{2} e f^{m - 1} n^{2} x^{m} + b^{2} d^{3} f^{m - 1} n^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - {\left ({\left (2 \, b^{2} e^{3} m n \log \left (c\right ) + 2 \, a b e^{3} m n - 3 \, b^{2} e^{3} n^{2}\right )} f^{m - 1} x^{3 \, m} + 3 \, {\left (2 \, b^{2} d e^{2} m n \log \left (c\right ) + 2 \, a b d e^{2} m n - 3 \, b^{2} d e^{2} n^{2}\right )} f^{m - 1} x^{2 \, m} + 3 \, {\left (2 \, b^{2} d^{2} e m n \log \left (c\right ) + 2 \, a b d^{2} e m n - 3 \, b^{2} d^{2} e n^{2}\right )} f^{m - 1} x^{m} + {\left (2 \, b^{2} d^{3} m n \log \left (c\right ) + 2 \, a b d^{3} m n - 3 \, b^{2} d^{3} n^{2}\right )} f^{m - 1}\right )} \log \left (e x^{m} + d\right ) - 2 \, {\left (b^{2} e^{3} f^{m - 1} m n^{2} x^{3 \, m} \log \left (x\right ) + 3 \, b^{2} d e^{2} f^{m - 1} m n^{2} x^{2 \, m} \log \left (x\right ) + 3 \, b^{2} d^{2} e f^{m - 1} m n^{2} x^{m} \log \left (x\right ) + b^{2} d^{3} f^{m - 1} m n^{2} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{3 \, {\left (d^{3} e^{4} m^{3} x^{3 \, m} + 3 \, d^{4} e^{3} m^{3} x^{2 \, m} + 3 \, d^{5} e^{2} m^{3} x^{m} + d^{6} e m^{3}\right )}} \]

[In]

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

[Out]

1/3*((b^2*e^3*m^2*n^2*log(x)^2 + (2*b^2*e^3*m^2*n*log(c) + 2*a*b*e^3*m^2*n - 3*b^2*e^3*m*n^2)*log(x))*f^(m - 1
)*x^(3*m) + (3*b^2*d*e^2*m^2*n^2*log(x)^2 + 2*b^2*d*e^2*m*n*log(c) + 2*a*b*d*e^2*m*n - b^2*d*e^2*n^2 + (6*b^2*
d*e^2*m^2*n*log(c) + 6*a*b*d*e^2*m^2*n - 7*b^2*d*e^2*m*n^2)*log(x))*f^(m - 1)*x^(2*m) + (3*b^2*d^2*e*m^2*n^2*l
og(x)^2 + 5*b^2*d^2*e*m*n*log(c) + 5*a*b*d^2*e*m*n - 2*b^2*d^2*e*n^2 + 2*(3*b^2*d^2*e*m^2*n*log(c) + 3*a*b*d^2
*e*m^2*n - 2*b^2*d^2*e*m*n^2)*log(x))*f^(m - 1)*x^m - (b^2*d^3*m^2*log(c)^2 + a^2*d^3*m^2 - 3*a*b*d^3*m*n + b^
2*d^3*n^2 + (2*a*b*d^3*m^2 - 3*b^2*d^3*m*n)*log(c))*f^(m - 1) - 2*(b^2*e^3*f^(m - 1)*n^2*x^(3*m) + 3*b^2*d*e^2
*f^(m - 1)*n^2*x^(2*m) + 3*b^2*d^2*e*f^(m - 1)*n^2*x^m + b^2*d^3*f^(m - 1)*n^2)*dilog(-(e*x^m + d)/d + 1) - ((
2*b^2*e^3*m*n*log(c) + 2*a*b*e^3*m*n - 3*b^2*e^3*n^2)*f^(m - 1)*x^(3*m) + 3*(2*b^2*d*e^2*m*n*log(c) + 2*a*b*d*
e^2*m*n - 3*b^2*d*e^2*n^2)*f^(m - 1)*x^(2*m) + 3*(2*b^2*d^2*e*m*n*log(c) + 2*a*b*d^2*e*m*n - 3*b^2*d^2*e*n^2)*
f^(m - 1)*x^m + (2*b^2*d^3*m*n*log(c) + 2*a*b*d^3*m*n - 3*b^2*d^3*n^2)*f^(m - 1))*log(e*x^m + d) - 2*(b^2*e^3*
f^(m - 1)*m*n^2*x^(3*m)*log(x) + 3*b^2*d*e^2*f^(m - 1)*m*n^2*x^(2*m)*log(x) + 3*b^2*d^2*e*f^(m - 1)*m*n^2*x^m*
log(x) + b^2*d^3*f^(m - 1)*m*n^2*log(x))*log((e*x^m + d)/d))/(d^3*e^4*m^3*x^(3*m) + 3*d^4*e^3*m^3*x^(2*m) + 3*
d^5*e^2*m^3*x^m + d^6*e*m^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/3*a*b*f^m*n*((2*e*x^m + 3*d)/((d^2*e^3*f*m*x^(2*m) + 2*d^3*e^2*f*m*x^m + d^4*e*f*m)*m) + 2*log(x)/(d^3*e*f*m
) - 2*log(e*x^m + d)/(d^3*e*f*m^2)) - 1/3*(f^m*log(x^n)^2/(e^4*f*m*x^(3*m) + 3*d*e^3*f*m*x^(2*m) + 3*d^2*e^2*f
*m*x^m + d^3*e*f*m) - 3*integrate(1/3*(3*e*f^m*m*x^m*log(c)^2 + 2*(d*f^m*n + (3*e*f^m*m*log(c) + e*f^m*n)*x^m)
*log(x^n))/(e^5*f*m*x*x^(4*m) + 4*d*e^4*f*m*x*x^(3*m) + 6*d^2*e^3*f*m*x*x^(2*m) + 4*d^3*e^2*f*m*x*x^m + d^4*e*
f*m*x), x))*b^2 - 2/3*a*b*f^m*log(c*x^n)/(e^4*f*m*x^(3*m) + 3*d*e^3*f*m*x^(2*m) + 3*d^2*e^2*f*m*x^m + d^3*e*f*
m) - 1/3*a^2*f^m/(e^4*f*m*x^(3*m) + 3*d*e^3*f*m*x^(2*m) + 3*d^2*e^2*f*m*x^m + d^3*e*f*m)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx=\int \frac {{\left (f\,x\right )}^{m-1}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x^m\right )}^4} \,d x \]

[In]

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

[Out]

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