\(\int (f x)^{-1+m} (d+e x^m)^3 (a+b \log (c x^n))^2 \, dx\) [359]

Optimal result

Integrand size = 29, antiderivative size = 372 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 b^2 d^3 n^2 x (f x)^{-1+m}}{m^3}+\frac {3 b^2 d^2 e n^2 x^{1+m} (f x)^{-1+m}}{4 m^3}+\frac {2 b^2 d e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{9 m^3}+\frac {b^2 e^3 n^2 x^{1+3 m} (f x)^{-1+m}}{32 m^3}+\frac {b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m} \]

[Out]

2*b^2*d^3*n^2*x*(f*x)^(-1+m)/m^3+3/4*b^2*d^2*e*n^2*x^(1+m)*(f*x)^(-1+m)/m^3+2/9*b^2*d*e^2*n^2*x^(1+2*m)*(f*x)^
(-1+m)/m^3+1/32*b^2*e^3*n^2*x^(1+3*m)*(f*x)^(-1+m)/m^3+1/4*b^2*d^4*n^2*x^(1-m)*(f*x)^(-1+m)*ln(x)^2/e/m-2*b*d^
3*n*x*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-3/2*b*d^2*e*n*x^(1+m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-2/3*b*d*e^2*n*x^
(1+2*m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-1/8*b*e^3*n*x^(1+3*m)*(f*x)^(-1+m)*(a+b*ln(c*x^n))/m^2-1/2*b*d^4*n*x^
(1-m)*(f*x)^(-1+m)*ln(x)*(a+b*ln(c*x^n))/e/m+1/4*x^(1-m)*(f*x)^(-1+m)*(d+e*x^m)^4*(a+b*ln(c*x^n))^2/e/m

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2377, 2376, 272, 45, 2372, 14, 2338} \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {b d^4 n x^{1-m} \log (x) (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{2 e m}-\frac {2 b d^3 n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{2 m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}+\frac {x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac {b e^3 n x^{3 m+1} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}+\frac {b^2 d^4 n^2 x^{1-m} \log ^2(x) (f x)^{m-1}}{4 e m}+\frac {2 b^2 d^3 n^2 x (f x)^{m-1}}{m^3}+\frac {3 b^2 d^2 e n^2 x^{m+1} (f x)^{m-1}}{4 m^3}+\frac {2 b^2 d e^2 n^2 x^{2 m+1} (f x)^{m-1}}{9 m^3}+\frac {b^2 e^3 n^2 x^{3 m+1} (f x)^{m-1}}{32 m^3} \]

[In]

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

[Out]

(2*b^2*d^3*n^2*x*(f*x)^(-1 + m))/m^3 + (3*b^2*d^2*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(4*m^3) + (2*b^2*d*e^2*n^2*x
^(1 + 2*m)*(f*x)^(-1 + m))/(9*m^3) + (b^2*e^3*n^2*x^(1 + 3*m)*(f*x)^(-1 + m))/(32*m^3) + (b^2*d^4*n^2*x^(1 - m
)*(f*x)^(-1 + m)*Log[x]^2)/(4*e*m) - (2*b*d^3*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (3*b*d^2*e*n*x^(1 +
m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(2*m^2) - (2*b*d*e^2*n*x^(1 + 2*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(
3*m^2) - (b*e^3*n*x^(1 + 3*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(8*m^2) - (b*d^4*n*x^(1 - m)*(f*x)^(-1 + m)*L
og[x]*(a + b*Log[c*x^n]))/(2*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^4*(a + b*Log[c*x^n])^2)/(4*e*m)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[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])

Rubi steps \begin{align*} \text {integral}& = \left (x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx \\ & = \frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {\left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e m} \\ & = -\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \left (\frac {e x^{-1+m} \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )}{12 m}+\frac {d^4 \log (x)}{x}\right ) \, dx}{2 e m} \\ & = -\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right ) \, dx}{24 m^2}+\frac {\left (b^2 d^4 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac {\log (x)}{x} \, dx}{2 e m} \\ & = \frac {b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m}+\frac {\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \left (48 d^3 x^{-1+m}+36 d^2 e x^{-1+2 m}+16 d e^2 x^{-1+3 m}+3 e^3 x^{-1+4 m}\right ) \, dx}{24 m^2} \\ & = \frac {2 b^2 d^3 n^2 x (f x)^{-1+m}}{m^3}+\frac {3 b^2 d^2 e n^2 x^{1+m} (f x)^{-1+m}}{4 m^3}+\frac {2 b^2 d e^2 n^2 x^{1+2 m} (f x)^{-1+m}}{9 m^3}+\frac {b^2 e^3 n^2 x^{1+3 m} (f x)^{-1+m}}{32 m^3}+\frac {b^2 d^4 n^2 x^{1-m} (f x)^{-1+m} \log ^2(x)}{4 e m}-\frac {2 b d^3 n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 m^2}-\frac {2 b d e^2 n x^{1+2 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{8 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )^2}{4 e m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.77 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {(f x)^m \left (72 a^2 m^2 \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-12 a b m n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )+b^2 n^2 \left (576 d^3+216 d^2 e x^m+64 d e^2 x^{2 m}+9 e^3 x^{3 m}\right )+12 b m \left (12 a m \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )\right ) \log \left (c x^n\right )+72 b^2 m^2 \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right ) \log ^2\left (c x^n\right )\right )}{288 f m^3} \]

[In]

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

[Out]

((f*x)^m*(72*a^2*m^2*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m)) - 12*a*b*m*n*(48*d^3 + 36*d^2*e*x^m
+ 16*d*e^2*x^(2*m) + 3*e^3*x^(3*m)) + b^2*n^2*(576*d^3 + 216*d^2*e*x^m + 64*d*e^2*x^(2*m) + 9*e^3*x^(3*m)) +
12*b*m*(12*a*m*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m)) - b*n*(48*d^3 + 36*d^2*e*x^m + 16*d*e^2*x
^(2*m) + 3*e^3*x^(3*m)))*Log[c*x^n] + 72*b^2*m^2*(4*d^3 + 6*d^2*e*x^m + 4*d*e^2*x^(2*m) + e^3*x^(3*m))*Log[c*x
^n]^2))/(288*f*m^3)

Maple [A] (verified)

Time = 205.35 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.66

method	result	size
parallelrisch	\(-\frac {-288 b^{2} d^{3} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x \,m^{2}-72 x \,x^{3 m} \left (f x \right )^{m -1} a^{2} e^{3} m^{2}-9 x \,x^{3 m} \left (f x \right )^{m -1} b^{2} e^{3} n^{2}-432 b^{2} d^{2} e \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x^{m} x \,m^{2}-864 x \,x^{m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,d^{2} e \,m^{2}+432 x \,x^{m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d^{2} e m n -576 x \,x^{2 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b d \,e^{2} m^{2}+192 x \,x^{2 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d \,e^{2} m n +432 x \,x^{m} \left (f x \right )^{m -1} a b \,d^{2} e m n +192 x \,x^{2 m} \left (f x \right )^{m -1} a b d \,e^{2} m n -288 x \left (f x \right )^{m -1} a^{2} d^{3} m^{2}-576 x \left (f x \right )^{m -1} b^{2} d^{3} n^{2}-216 x \,x^{m} \left (f x \right )^{m -1} b^{2} d^{2} e \,n^{2}-288 x \,x^{2 m} \left (f x \right )^{m -1} a^{2} d \,e^{2} m^{2}-64 x \,x^{2 m} \left (f x \right )^{m -1} b^{2} d \,e^{2} n^{2}-72 b^{2} e^{3} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x \,x^{3 m} m^{2}-576 x \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,d^{3} m^{2}+576 x \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} d^{3} m n +576 x \left (f x \right )^{m -1} a b \,d^{3} m n -432 x \,x^{m} \left (f x \right )^{m -1} a^{2} d^{2} e \,m^{2}+36 x \,x^{3 m} \left (f x \right )^{m -1} a b \,e^{3} m n -144 x \,x^{3 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} a b \,e^{3} m^{2}+36 x \,x^{3 m} \ln \left (c \,x^{n}\right ) \left (f x \right )^{m -1} b^{2} e^{3} m n -288 b^{2} d \,e^{2} \left (f x \right )^{m -1} \ln \left (c \,x^{n}\right )^{2} x^{2 m} x \,m^{2}}{288 m^{3}}\)	\(617\)
risch	\(\text {Expression too large to display}\)	\(4156\)

[In]

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

[Out]

-1/288*(-288*b^2*d^3*(f*x)^(m-1)*ln(c*x^n)^2*x*m^2-72*x*(x^m)^3*(f*x)^(m-1)*a^2*e^3*m^2-9*x*(x^m)^3*(f*x)^(m-1
)*b^2*e^3*n^2-432*b^2*d^2*e*(f*x)^(m-1)*ln(c*x^n)^2*x^m*x*m^2-864*x*x^m*ln(c*x^n)*(f*x)^(m-1)*a*b*d^2*e*m^2+43
2*x*x^m*ln(c*x^n)*(f*x)^(m-1)*b^2*d^2*e*m*n-576*x*(x^m)^2*ln(c*x^n)*(f*x)^(m-1)*a*b*d*e^2*m^2+192*x*(x^m)^2*ln
(c*x^n)*(f*x)^(m-1)*b^2*d*e^2*m*n+432*x*x^m*(f*x)^(m-1)*a*b*d^2*e*m*n+192*x*(x^m)^2*(f*x)^(m-1)*a*b*d*e^2*m*n-
288*x*(f*x)^(m-1)*a^2*d^3*m^2-576*x*(f*x)^(m-1)*b^2*d^3*n^2-216*x*x^m*(f*x)^(m-1)*b^2*d^2*e*n^2-288*x*(x^m)^2*
(f*x)^(m-1)*a^2*d*e^2*m^2-64*x*(x^m)^2*(f*x)^(m-1)*b^2*d*e^2*n^2-72*b^2*e^3*(f*x)^(m-1)*ln(c*x^n)^2*x*(x^m)^3*
m^2-576*x*ln(c*x^n)*(f*x)^(m-1)*a*b*d^3*m^2+576*x*ln(c*x^n)*(f*x)^(m-1)*b^2*d^3*m*n+576*x*(f*x)^(m-1)*a*b*d^3*
m*n-432*x*x^m*(f*x)^(m-1)*a^2*d^2*e*m^2+36*x*(x^m)^3*(f*x)^(m-1)*a*b*e^3*m*n-144*x*(x^m)^3*ln(c*x^n)*(f*x)^(m-
1)*a*b*e^3*m^2+36*x*(x^m)^3*ln(c*x^n)*(f*x)^(m-1)*b^2*e^3*m*n-288*b^2*d*e^2*(f*x)^(m-1)*ln(c*x^n)^2*(x^m)^2*x*
m^2)/m^3

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.59 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {9 \, {\left (8 \, b^{2} e^{3} m^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} e^{3} m^{2} \log \left (c\right )^{2} + 8 \, a^{2} e^{3} m^{2} - 4 \, a b e^{3} m n + b^{2} e^{3} n^{2} + 4 \, {\left (4 \, a b e^{3} m^{2} - b^{2} e^{3} m n\right )} \log \left (c\right ) + 4 \, {\left (4 \, b^{2} e^{3} m^{2} n \log \left (c\right ) + 4 \, a b e^{3} m^{2} n - b^{2} e^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{4 \, m} + 32 \, {\left (9 \, b^{2} d e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 9 \, b^{2} d e^{2} m^{2} \log \left (c\right )^{2} + 9 \, a^{2} d e^{2} m^{2} - 6 \, a b d e^{2} m n + 2 \, b^{2} d e^{2} n^{2} + 6 \, {\left (3 \, a b d e^{2} m^{2} - b^{2} d e^{2} m n\right )} \log \left (c\right ) + 6 \, {\left (3 \, b^{2} d e^{2} m^{2} n \log \left (c\right ) + 3 \, a b d e^{2} m^{2} n - b^{2} d e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{3 \, m} + 216 \, {\left (2 \, b^{2} d^{2} e m^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} d^{2} e m^{2} \log \left (c\right )^{2} + 2 \, a^{2} d^{2} e m^{2} - 2 \, a b d^{2} e m n + b^{2} d^{2} e n^{2} + 2 \, {\left (2 \, a b d^{2} e m^{2} - b^{2} d^{2} e m n\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} d^{2} e m^{2} n \log \left (c\right ) + 2 \, a b d^{2} e m^{2} n - b^{2} d^{2} e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 288 \, {\left (b^{2} d^{3} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{3} m^{2} \log \left (c\right )^{2} + a^{2} d^{3} m^{2} - 2 \, a b d^{3} m n + 2 \, b^{2} d^{3} n^{2} + 2 \, {\left (a b d^{3} m^{2} - b^{2} d^{3} m n\right )} \log \left (c\right ) + 2 \, {\left (b^{2} d^{3} m^{2} n \log \left (c\right ) + a b d^{3} m^{2} n - b^{2} d^{3} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m}}{288 \, m^{3}} \]

[In]

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

[Out]

1/288*(9*(8*b^2*e^3*m^2*n^2*log(x)^2 + 8*b^2*e^3*m^2*log(c)^2 + 8*a^2*e^3*m^2 - 4*a*b*e^3*m*n + b^2*e^3*n^2 +
4*(4*a*b*e^3*m^2 - b^2*e^3*m*n)*log(c) + 4*(4*b^2*e^3*m^2*n*log(c) + 4*a*b*e^3*m^2*n - b^2*e^3*m*n^2)*log(x))*
f^(m - 1)*x^(4*m) + 32*(9*b^2*d*e^2*m^2*n^2*log(x)^2 + 9*b^2*d*e^2*m^2*log(c)^2 + 9*a^2*d*e^2*m^2 - 6*a*b*d*e^
2*m*n + 2*b^2*d*e^2*n^2 + 6*(3*a*b*d*e^2*m^2 - b^2*d*e^2*m*n)*log(c) + 6*(3*b^2*d*e^2*m^2*n*log(c) + 3*a*b*d*e
^2*m^2*n - b^2*d*e^2*m*n^2)*log(x))*f^(m - 1)*x^(3*m) + 216*(2*b^2*d^2*e*m^2*n^2*log(x)^2 + 2*b^2*d^2*e*m^2*lo
g(c)^2 + 2*a^2*d^2*e*m^2 - 2*a*b*d^2*e*m*n + b^2*d^2*e*n^2 + 2*(2*a*b*d^2*e*m^2 - b^2*d^2*e*m*n)*log(c) + 2*(2
*b^2*d^2*e*m^2*n*log(c) + 2*a*b*d^2*e*m^2*n - b^2*d^2*e*m*n^2)*log(x))*f^(m - 1)*x^(2*m) + 288*(b^2*d^3*m^2*n^
2*log(x)^2 + b^2*d^3*m^2*log(c)^2 + a^2*d^3*m^2 - 2*a*b*d^3*m*n + 2*b^2*d^3*n^2 + 2*(a*b*d^3*m^2 - b^2*d^3*m*n
)*log(c) + 2*(b^2*d^3*m^2*n*log(c) + a*b*d^3*m^2*n - b^2*d^3*m*n^2)*log(x))*f^(m - 1)*x^m)/m^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (364) = 728\).

Time = 38.32 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.04 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\begin {cases} \frac {a^{2} d^{3} x \left (f x\right )^{m - 1}}{m} + \frac {3 a^{2} d^{2} e x x^{m} \left (f x\right )^{m - 1}}{2 m} + \frac {a^{2} d e^{2} x x^{2 m} \left (f x\right )^{m - 1}}{m} + \frac {a^{2} e^{3} x x^{3 m} \left (f x\right )^{m - 1}}{4 m} + \frac {2 a b d^{3} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {2 a b d^{3} n x \left (f x\right )^{m - 1}}{m^{2}} + \frac {3 a b d^{2} e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {3 a b d^{2} e n x x^{m} \left (f x\right )^{m - 1}}{2 m^{2}} + \frac {2 a b d e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m} - \frac {2 a b d e^{2} n x x^{2 m} \left (f x\right )^{m - 1}}{3 m^{2}} + \frac {a b e^{3} x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{2 m} - \frac {a b e^{3} n x x^{3 m} \left (f x\right )^{m - 1}}{8 m^{2}} + \frac {b^{2} d^{3} x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{m} - \frac {2 b^{2} d^{3} n x \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{m^{2}} + \frac {2 b^{2} d^{3} n^{2} x \left (f x\right )^{m - 1}}{m^{3}} + \frac {3 b^{2} d^{2} e x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{2 m} - \frac {3 b^{2} d^{2} e n x x^{m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{2 m^{2}} + \frac {3 b^{2} d^{2} e n^{2} x x^{m} \left (f x\right )^{m - 1}}{4 m^{3}} + \frac {b^{2} d e^{2} x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{m} - \frac {2 b^{2} d e^{2} n x x^{2 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{3 m^{2}} + \frac {2 b^{2} d e^{2} n^{2} x x^{2 m} \left (f x\right )^{m - 1}}{9 m^{3}} + \frac {b^{2} e^{3} x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}^{2}}{4 m} - \frac {b^{2} e^{3} n x x^{3 m} \left (f x\right )^{m - 1} \log {\left (c x^{n} \right )}}{8 m^{2}} + \frac {b^{2} e^{3} n^{2} x x^{3 m} \left (f x\right )^{m - 1}}{32 m^{3}} & \text {for}\: m \neq 0 \\\frac {\left (d + e\right )^{3} \left (\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{f} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*d**3*x*(f*x)**(m - 1)/m + 3*a**2*d**2*e*x*x**m*(f*x)**(m - 1)/(2*m) + a**2*d*e**2*x*x**(2*m)*(
f*x)**(m - 1)/m + a**2*e**3*x*x**(3*m)*(f*x)**(m - 1)/(4*m) + 2*a*b*d**3*x*(f*x)**(m - 1)*log(c*x**n)/m - 2*a*
b*d**3*n*x*(f*x)**(m - 1)/m**2 + 3*a*b*d**2*e*x*x**m*(f*x)**(m - 1)*log(c*x**n)/m - 3*a*b*d**2*e*n*x*x**m*(f*x
)**(m - 1)/(2*m**2) + 2*a*b*d*e**2*x*x**(2*m)*(f*x)**(m - 1)*log(c*x**n)/m - 2*a*b*d*e**2*n*x*x**(2*m)*(f*x)**
(m - 1)/(3*m**2) + a*b*e**3*x*x**(3*m)*(f*x)**(m - 1)*log(c*x**n)/(2*m) - a*b*e**3*n*x*x**(3*m)*(f*x)**(m - 1)
/(8*m**2) + b**2*d**3*x*(f*x)**(m - 1)*log(c*x**n)**2/m - 2*b**2*d**3*n*x*(f*x)**(m - 1)*log(c*x**n)/m**2 + 2*
b**2*d**3*n**2*x*(f*x)**(m - 1)/m**3 + 3*b**2*d**2*e*x*x**m*(f*x)**(m - 1)*log(c*x**n)**2/(2*m) - 3*b**2*d**2*
e*n*x*x**m*(f*x)**(m - 1)*log(c*x**n)/(2*m**2) + 3*b**2*d**2*e*n**2*x*x**m*(f*x)**(m - 1)/(4*m**3) + b**2*d*e*
*2*x*x**(2*m)*(f*x)**(m - 1)*log(c*x**n)**2/m - 2*b**2*d*e**2*n*x*x**(2*m)*(f*x)**(m - 1)*log(c*x**n)/(3*m**2)
+ 2*b**2*d*e**2*n**2*x*x**(2*m)*(f*x)**(m - 1)/(9*m**3) + b**2*e**3*x*x**(3*m)*(f*x)**(m - 1)*log(c*x**n)**2/
(4*m) - b**2*e**3*n*x*x**(3*m)*(f*x)**(m - 1)*log(c*x**n)/(8*m**2) + b**2*e**3*n**2*x*x**(3*m)*(f*x)**(m - 1)/
(32*m**3), Ne(m, 0)), ((d + e)**3*Piecewise(((a**2*log(c*x**n) + a*b*log(c*x**n)**2 + b**2*log(c*x**n)**3/3)/n
, Ne(n, 0)), ((a**2 + 2*a*b*log(c) + b**2*log(c)**2)*log(x), True))/f, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.55 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} e^{3} f^{m - 1} x^{4 \, m} \log \left (c x^{n}\right )^{2}}{4 \, m} + \frac {b^{2} d e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )^{2}}{m} + \frac {3 \, b^{2} d^{2} e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )^{2}}{2 \, m} + \frac {a b e^{3} f^{m - 1} x^{4 \, m} \log \left (c x^{n}\right )}{2 \, m} + \frac {2 \, a b d e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )}{m} + \frac {3 \, a b d^{2} e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )}{m} - 2 \, {\left (\frac {f^{m - 1} n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{m}}{m^{3}}\right )} b^{2} d^{3} - \frac {3}{4} \, {\left (\frac {2 \, f^{m - 1} n x^{2 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{2 \, m}}{m^{3}}\right )} b^{2} d^{2} e - \frac {2}{9} \, {\left (\frac {3 \, f^{m - 1} n x^{3 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{3 \, m}}{m^{3}}\right )} b^{2} d e^{2} - \frac {1}{32} \, {\left (\frac {4 \, f^{m - 1} n x^{4 \, m} \log \left (c x^{n}\right )}{m^{2}} - \frac {f^{m - 1} n^{2} x^{4 \, m}}{m^{3}}\right )} b^{2} e^{3} + \frac {a^{2} e^{3} f^{m - 1} x^{4 \, m}}{4 \, m} - \frac {a b e^{3} f^{m - 1} n x^{4 \, m}}{8 \, m^{2}} + \frac {a^{2} d e^{2} f^{m - 1} x^{3 \, m}}{m} - \frac {2 \, a b d e^{2} f^{m - 1} n x^{3 \, m}}{3 \, m^{2}} + \frac {3 \, a^{2} d^{2} e f^{m - 1} x^{2 \, m}}{2 \, m} - \frac {3 \, a b d^{2} e f^{m - 1} n x^{2 \, m}}{2 \, m^{2}} - \frac {2 \, a b d^{3} f^{m - 1} n x^{m}}{m^{2}} + \frac {\left (f x\right )^{m} b^{2} d^{3} \log \left (c x^{n}\right )^{2}}{f m} + \frac {2 \, \left (f x\right )^{m} a b d^{3} \log \left (c x^{n}\right )}{f m} + \frac {\left (f x\right )^{m} a^{2} d^{3}}{f m} \]

[In]

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

[Out]

1/4*b^2*e^3*f^(m - 1)*x^(4*m)*log(c*x^n)^2/m + b^2*d*e^2*f^(m - 1)*x^(3*m)*log(c*x^n)^2/m + 3/2*b^2*d^2*e*f^(m
- 1)*x^(2*m)*log(c*x^n)^2/m + 1/2*a*b*e^3*f^(m - 1)*x^(4*m)*log(c*x^n)/m + 2*a*b*d*e^2*f^(m - 1)*x^(3*m)*log(
c*x^n)/m + 3*a*b*d^2*e*f^(m - 1)*x^(2*m)*log(c*x^n)/m - 2*(f^(m - 1)*n*x^m*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^m/
m^3)*b^2*d^3 - 3/4*(2*f^(m - 1)*n*x^(2*m)*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^(2*m)/m^3)*b^2*d^2*e - 2/9*(3*f^(m
- 1)*n*x^(3*m)*log(c*x^n)/m^2 - f^(m - 1)*n^2*x^(3*m)/m^3)*b^2*d*e^2 - 1/32*(4*f^(m - 1)*n*x^(4*m)*log(c*x^n)/
m^2 - f^(m - 1)*n^2*x^(4*m)/m^3)*b^2*e^3 + 1/4*a^2*e^3*f^(m - 1)*x^(4*m)/m - 1/8*a*b*e^3*f^(m - 1)*n*x^(4*m)/m
^2 + a^2*d*e^2*f^(m - 1)*x^(3*m)/m - 2/3*a*b*d*e^2*f^(m - 1)*n*x^(3*m)/m^2 + 3/2*a^2*d^2*e*f^(m - 1)*x^(2*m)/m
- 3/2*a*b*d^2*e*f^(m - 1)*n*x^(2*m)/m^2 - 2*a*b*d^3*f^(m - 1)*n*x^m/m^2 + (f*x)^m*b^2*d^3*log(c*x^n)^2/(f*m)
+ 2*(f*x)^m*a*b*d^3*log(c*x^n)/(f*m) + (f*x)^m*a^2*d^3/(f*m)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 995 vs. \(2 (354) = 708\).

Time = 0.62 (sec) , antiderivative size = 995, normalized size of antiderivative = 2.67 \[ \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {b^{2} e^{3} f^{m} n^{2} x^{4 \, m} \log \left (x\right )^{2}}{4 \, f m} + \frac {b^{2} d e^{2} f^{m} n^{2} x^{3 \, m} \log \left (x\right )^{2}}{f m} + \frac {3 \, b^{2} d^{2} e f^{m} n^{2} x^{2 \, m} \log \left (x\right )^{2}}{2 \, f m} + \frac {b^{2} d^{3} f^{m} n^{2} x^{m} \log \left (x\right )^{2}}{f m} + \frac {b^{2} e^{3} f^{m} n x^{4 \, m} \log \left (c\right ) \log \left (x\right )}{2 \, f m} + \frac {2 \, b^{2} d e^{2} f^{m} n x^{3 \, m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {3 \, b^{2} d^{2} e f^{m} n x^{2 \, m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {2 \, b^{2} d^{3} f^{m} n x^{m} \log \left (c\right ) \log \left (x\right )}{f m} + \frac {b^{2} e^{3} f^{m} x^{4 \, m} \log \left (c\right )^{2}}{4 \, f m} + \frac {b^{2} d e^{2} f^{m} x^{3 \, m} \log \left (c\right )^{2}}{f m} + \frac {3 \, b^{2} d^{2} e f^{m} x^{2 \, m} \log \left (c\right )^{2}}{2 \, f m} + \frac {b^{2} d^{3} f^{m} x^{m} \log \left (c\right )^{2}}{f m} + \frac {a b e^{3} f^{m} n x^{4 \, m} \log \left (x\right )}{2 \, f m} - \frac {b^{2} e^{3} f^{m} n^{2} x^{4 \, m} \log \left (x\right )}{8 \, f m^{2}} + \frac {2 \, a b d e^{2} f^{m} n x^{3 \, m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d e^{2} f^{m} n^{2} x^{3 \, m} \log \left (x\right )}{3 \, f m^{2}} + \frac {3 \, a b d^{2} e f^{m} n x^{2 \, m} \log \left (x\right )}{f m} - \frac {3 \, b^{2} d^{2} e f^{m} n^{2} x^{2 \, m} \log \left (x\right )}{2 \, f m^{2}} + \frac {2 \, a b d^{3} f^{m} n x^{m} \log \left (x\right )}{f m} - \frac {2 \, b^{2} d^{3} f^{m} n^{2} x^{m} \log \left (x\right )}{f m^{2}} + \frac {a b e^{3} f^{m} x^{4 \, m} \log \left (c\right )}{2 \, f m} - \frac {b^{2} e^{3} f^{m} n x^{4 \, m} \log \left (c\right )}{8 \, f m^{2}} + \frac {2 \, a b d e^{2} f^{m} x^{3 \, m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d e^{2} f^{m} n x^{3 \, m} \log \left (c\right )}{3 \, f m^{2}} + \frac {3 \, a b d^{2} e f^{m} x^{2 \, m} \log \left (c\right )}{f m} - \frac {3 \, b^{2} d^{2} e f^{m} n x^{2 \, m} \log \left (c\right )}{2 \, f m^{2}} + \frac {2 \, a b d^{3} f^{m} x^{m} \log \left (c\right )}{f m} - \frac {2 \, b^{2} d^{3} f^{m} n x^{m} \log \left (c\right )}{f m^{2}} + \frac {a^{2} e^{3} f^{m} x^{4 \, m}}{4 \, f m} - \frac {a b e^{3} f^{m} n x^{4 \, m}}{8 \, f m^{2}} + \frac {b^{2} e^{3} f^{m} n^{2} x^{4 \, m}}{32 \, f m^{3}} + \frac {a^{2} d e^{2} f^{m} x^{3 \, m}}{f m} - \frac {2 \, a b d e^{2} f^{m} n x^{3 \, m}}{3 \, f m^{2}} + \frac {2 \, b^{2} d e^{2} f^{m} n^{2} x^{3 \, m}}{9 \, f m^{3}} + \frac {3 \, a^{2} d^{2} e f^{m} x^{2 \, m}}{2 \, f m} - \frac {3 \, a b d^{2} e f^{m} n x^{2 \, m}}{2 \, f m^{2}} + \frac {3 \, b^{2} d^{2} e f^{m} n^{2} x^{2 \, m}}{4 \, f m^{3}} + \frac {a^{2} d^{3} f^{m} x^{m}}{f m} - \frac {2 \, a b d^{3} f^{m} n x^{m}}{f m^{2}} + \frac {2 \, b^{2} d^{3} f^{m} n^{2} x^{m}}{f m^{3}} \]

[In]

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

[Out]

1/4*b^2*e^3*f^m*n^2*x^(4*m)*log(x)^2/(f*m) + b^2*d*e^2*f^m*n^2*x^(3*m)*log(x)^2/(f*m) + 3/2*b^2*d^2*e*f^m*n^2*
x^(2*m)*log(x)^2/(f*m) + b^2*d^3*f^m*n^2*x^m*log(x)^2/(f*m) + 1/2*b^2*e^3*f^m*n*x^(4*m)*log(c)*log(x)/(f*m) +
2*b^2*d*e^2*f^m*n*x^(3*m)*log(c)*log(x)/(f*m) + 3*b^2*d^2*e*f^m*n*x^(2*m)*log(c)*log(x)/(f*m) + 2*b^2*d^3*f^m*
n*x^m*log(c)*log(x)/(f*m) + 1/4*b^2*e^3*f^m*x^(4*m)*log(c)^2/(f*m) + b^2*d*e^2*f^m*x^(3*m)*log(c)^2/(f*m) + 3/
2*b^2*d^2*e*f^m*x^(2*m)*log(c)^2/(f*m) + b^2*d^3*f^m*x^m*log(c)^2/(f*m) + 1/2*a*b*e^3*f^m*n*x^(4*m)*log(x)/(f*
m) - 1/8*b^2*e^3*f^m*n^2*x^(4*m)*log(x)/(f*m^2) + 2*a*b*d*e^2*f^m*n*x^(3*m)*log(x)/(f*m) - 2/3*b^2*d*e^2*f^m*n
^2*x^(3*m)*log(x)/(f*m^2) + 3*a*b*d^2*e*f^m*n*x^(2*m)*log(x)/(f*m) - 3/2*b^2*d^2*e*f^m*n^2*x^(2*m)*log(x)/(f*m
^2) + 2*a*b*d^3*f^m*n*x^m*log(x)/(f*m) - 2*b^2*d^3*f^m*n^2*x^m*log(x)/(f*m^2) + 1/2*a*b*e^3*f^m*x^(4*m)*log(c)
/(f*m) - 1/8*b^2*e^3*f^m*n*x^(4*m)*log(c)/(f*m^2) + 2*a*b*d*e^2*f^m*x^(3*m)*log(c)/(f*m) - 2/3*b^2*d*e^2*f^m*n
*x^(3*m)*log(c)/(f*m^2) + 3*a*b*d^2*e*f^m*x^(2*m)*log(c)/(f*m) - 3/2*b^2*d^2*e*f^m*n*x^(2*m)*log(c)/(f*m^2) +
2*a*b*d^3*f^m*x^m*log(c)/(f*m) - 2*b^2*d^3*f^m*n*x^m*log(c)/(f*m^2) + 1/4*a^2*e^3*f^m*x^(4*m)/(f*m) - 1/8*a*b*
e^3*f^m*n*x^(4*m)/(f*m^2) + 1/32*b^2*e^3*f^m*n^2*x^(4*m)/(f*m^3) + a^2*d*e^2*f^m*x^(3*m)/(f*m) - 2/3*a*b*d*e^2
*f^m*n*x^(3*m)/(f*m^2) + 2/9*b^2*d*e^2*f^m*n^2*x^(3*m)/(f*m^3) + 3/2*a^2*d^2*e*f^m*x^(2*m)/(f*m) - 3/2*a*b*d^2
*e*f^m*n*x^(2*m)/(f*m^2) + 3/4*b^2*d^2*e*f^m*n^2*x^(2*m)/(f*m^3) + a^2*d^3*f^m*x^m/(f*m) - 2*a*b*d^3*f^m*n*x^m
/(f*m^2) + 2*b^2*d^3*f^m*n^2*x^m/(f*m^3)

Mupad [F(-1)]

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

[In]

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

[Out]

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