3.80 \(\int (a+b \log (c x^n))^2 \log (d (e+f x)^m) \, dx\)

Optimal. Leaf size=288 \[ \frac{2 b e m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{2 b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{f}-\frac{2 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{f}+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{e m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}-m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d (e+f x)^m\right )-\frac{2 b e m n (a-b n) \log (e+f x)}{f}+2 a b m n x+2 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac{2 b^2 e m n \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )}{f}+4 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-4 b^2 m n^2 x \]

[Out]

2*a*b*m*n*x - 4*b^2*m*n^2*x + 2*b*m*n*(a - b*n)*x + 4*b^2*m*n*x*Log[c*x^n] - m*x*(a + b*Log[c*x^n])^2 - (2*b*e
*m*n*(a - b*n)*Log[e + f*x])/f - 2*a*b*n*x*Log[d*(e + f*x)^m] + 2*b^2*n^2*x*Log[d*(e + f*x)^m] - 2*b^2*n*x*Log
[c*x^n]*Log[d*(e + f*x)^m] + x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] - (2*b^2*e*m*n*Log[c*x^n]*Log[1 + (f*x)
/e])/f + (e*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/f - (2*b^2*e*m*n^2*PolyLog[2, -((f*x)/e)])/f + (2*b*e*m*n
*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f - (2*b^2*e*m*n^2*PolyLog[3, -((f*x)/e)])/f

________________________________________________________________________________________

Rubi [A]  time = 0.35215, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {2296, 2295, 2371, 6, 43, 2351, 2317, 2391, 2353, 2374, 6589} \[ \frac{2 b e m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{2 b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{f}-\frac{2 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{f}+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{e m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f}-m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d (e+f x)^m\right )-\frac{2 b e m n (a-b n) \log (e+f x)}{f}+2 a b m n x+2 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac{2 b^2 e m n \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )}{f}+4 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-4 b^2 m n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*b*m*n*x - 4*b^2*m*n^2*x + 2*b*m*n*(a - b*n)*x + 4*b^2*m*n*x*Log[c*x^n] - m*x*(a + b*Log[c*x^n])^2 - (2*b*e
*m*n*(a - b*n)*Log[e + f*x])/f - 2*a*b*n*x*Log[d*(e + f*x)^m] + 2*b^2*n^2*x*Log[d*(e + f*x)^m] - 2*b^2*n*x*Log
[c*x^n]*Log[d*(e + f*x)^m] + x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] - (2*b^2*e*m*n*Log[c*x^n]*Log[1 + (f*x)
/e])/f + (e*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/f - (2*b^2*e*m*n^2*PolyLog[2, -((f*x)/e)])/f + (2*b*e*m*n
*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f - (2*b^2*e*m*n^2*PolyLog[3, -((f*x)/e)])/f

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2371

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m*r, Int[Dist[x^(m - 1)/(e +
f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && IntegerQ[m]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 43

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2317

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

Rule 2391

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]

Rule 2353

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rubi steps

\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx &=-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (-\frac{2 a b n x}{e+f x}+\frac{2 b^2 n^2 x}{e+f x}-\frac{2 b^2 n x \log \left (c x^n\right )}{e+f x}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x}\right ) \, dx\\ &=-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{\left (-2 a b n+2 b^2 n^2\right ) x}{e+f x}-\frac{2 b^2 n x \log \left (c x^n\right )}{e+f x}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x}\right ) \, dx\\ &=-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\left (2 b^2 f m n\right ) \int \frac{x \log \left (c x^n\right )}{e+f x} \, dx+(2 b f m n (a-b n)) \int \frac{x}{e+f x} \, dx\\ &=-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f (e+f x)}\right ) \, dx+\left (2 b^2 f m n\right ) \int \left (\frac{\log \left (c x^n\right )}{f}-\frac{e \log \left (c x^n\right )}{f (e+f x)}\right ) \, dx+(2 b f m n (a-b n)) \int \left (\frac{1}{f}-\frac{e}{f (e+f x)}\right ) \, dx\\ &=2 b m n (a-b n) x-\frac{2 b e m n (a-b n) \log (e+f x)}{f}-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-m \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+(e m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\left (2 b^2 m n\right ) \int \log \left (c x^n\right ) \, dx-\left (2 b^2 e m n\right ) \int \frac{\log \left (c x^n\right )}{e+f x} \, dx\\ &=-2 b^2 m n^2 x+2 b m n (a-b n) x+2 b^2 m n x \log \left (c x^n\right )-m x \left (a+b \log \left (c x^n\right )\right )^2-\frac{2 b e m n (a-b n) \log (e+f x)}{f}-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2 b^2 e m n \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}+(2 b m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{(2 b e m n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{f}+\frac{\left (2 b^2 e m n^2\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{f}\\ &=2 a b m n x-2 b^2 m n^2 x+2 b m n (a-b n) x+2 b^2 m n x \log \left (c x^n\right )-m x \left (a+b \log \left (c x^n\right )\right )^2-\frac{2 b e m n (a-b n) \log (e+f x)}{f}-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2 b^2 e m n \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}-\frac{2 b^2 e m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{f}+\frac{2 b e m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{f}+\left (2 b^2 m n\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (2 b^2 e m n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{f}\\ &=2 a b m n x-4 b^2 m n^2 x+2 b m n (a-b n) x+4 b^2 m n x \log \left (c x^n\right )-m x \left (a+b \log \left (c x^n\right )\right )^2-\frac{2 b e m n (a-b n) \log (e+f x)}{f}-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{2 b^2 e m n \log \left (c x^n\right ) \log \left (1+\frac{f x}{e}\right )}{f}+\frac{e m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{f}-\frac{2 b^2 e m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{f}+\frac{2 b e m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{f}-\frac{2 b^2 e m n^2 \text{Li}_3\left (-\frac{f x}{e}\right )}{f}\\ \end{align*}

Mathematica [A]  time = 0.195642, size = 507, normalized size = 1.76 \[ \frac{2 b e m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )-b n\right )-2 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )+a^2 f x \log \left (d (e+f x)^m\right )+a^2 e m \log (e+f x)+a^2 (-f) m x+2 a b f x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+2 a b e m \log \left (c x^n\right ) \log (e+f x)-2 a b f m x \log \left (c x^n\right )-2 a b f n x \log \left (d (e+f x)^m\right )-2 a b e m n \log (e+f x)-2 a b e m n \log (x) \log (e+f x)+2 a b e m n \log (x) \log \left (\frac{f x}{e}+1\right )+4 a b f m n x+b^2 f x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-2 b^2 f n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^2 e m \log ^2\left (c x^n\right ) \log (e+f x)-2 b^2 e m n \log \left (c x^n\right ) \log (e+f x)-2 b^2 e m n \log (x) \log \left (c x^n\right ) \log (e+f x)+2 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )-b^2 f m x \log ^2\left (c x^n\right )+4 b^2 f m n x \log \left (c x^n\right )+2 b^2 f n^2 x \log \left (d (e+f x)^m\right )+b^2 e m n^2 \log ^2(x) \log (e+f x)-b^2 e m n^2 \log ^2(x) \log \left (\frac{f x}{e}+1\right )+2 b^2 e m n^2 \log (e+f x)+2 b^2 e m n^2 \log (x) \log (e+f x)-2 b^2 e m n^2 \log (x) \log \left (\frac{f x}{e}+1\right )-6 b^2 f m n^2 x}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2*f*m*x) + 4*a*b*f*m*n*x - 6*b^2*f*m*n^2*x - 2*a*b*f*m*x*Log[c*x^n] + 4*b^2*f*m*n*x*Log[c*x^n] - b^2*f*m*
x*Log[c*x^n]^2 + a^2*e*m*Log[e + f*x] - 2*a*b*e*m*n*Log[e + f*x] + 2*b^2*e*m*n^2*Log[e + f*x] - 2*a*b*e*m*n*Lo
g[x]*Log[e + f*x] + 2*b^2*e*m*n^2*Log[x]*Log[e + f*x] + b^2*e*m*n^2*Log[x]^2*Log[e + f*x] + 2*a*b*e*m*Log[c*x^
n]*Log[e + f*x] - 2*b^2*e*m*n*Log[c*x^n]*Log[e + f*x] - 2*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[e + f*x] + b^2*e*m*L
og[c*x^n]^2*Log[e + f*x] + a^2*f*x*Log[d*(e + f*x)^m] - 2*a*b*f*n*x*Log[d*(e + f*x)^m] + 2*b^2*f*n^2*x*Log[d*(
e + f*x)^m] + 2*a*b*f*x*Log[c*x^n]*Log[d*(e + f*x)^m] - 2*b^2*f*n*x*Log[c*x^n]*Log[d*(e + f*x)^m] + b^2*f*x*Lo
g[c*x^n]^2*Log[d*(e + f*x)^m] + 2*a*b*e*m*n*Log[x]*Log[1 + (f*x)/e] - 2*b^2*e*m*n^2*Log[x]*Log[1 + (f*x)/e] -
b^2*e*m*n^2*Log[x]^2*Log[1 + (f*x)/e] + 2*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 2*b*e*m*n*(a - b*n +
b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] - 2*b^2*e*m*n^2*PolyLog[3, -((f*x)/e)])/f

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Maple [F]  time = 1.948, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{2} e m \log \left (f x + e\right ) -{\left (f m - f \log \left (d\right )\right )} b^{2} x\right )} \log \left (x^{n}\right )^{2} +{\left (b^{2} f x \log \left (x^{n}\right )^{2} - 2 \,{\left ({\left (f n - f \log \left (c\right )\right )} b^{2} - a b f\right )} x \log \left (x^{n}\right ) -{\left (2 \,{\left (f n - f \log \left (c\right )\right )} a b -{\left (2 \, f n^{2} - 2 \, f n \log \left (c\right ) + f \log \left (c\right )^{2}\right )} b^{2} - a^{2} f\right )} x\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{f} - \int \frac{{\left ({\left (f^{2} m - f^{2} \log \left (d\right )\right )} a^{2} - 2 \,{\left (f^{2} m n -{\left (f^{2} m - f^{2} \log \left (d\right )\right )} \log \left (c\right )\right )} a b +{\left (2 \, f^{2} m n^{2} - 2 \, f^{2} m n \log \left (c\right ) +{\left (f^{2} m - f^{2} \log \left (d\right )\right )} \log \left (c\right )^{2}\right )} b^{2}\right )} x^{2} -{\left (b^{2} e f \log \left (c\right )^{2} \log \left (d\right ) + 2 \, a b e f \log \left (c\right ) \log \left (d\right ) + a^{2} e f \log \left (d\right )\right )} x + 2 \,{\left ({\left ({\left (f^{2} m - f^{2} \log \left (d\right )\right )} a b -{\left (2 \, f^{2} m n - f^{2} n \log \left (d\right ) -{\left (f^{2} m - f^{2} \log \left (d\right )\right )} \log \left (c\right )\right )} b^{2}\right )} x^{2} -{\left (a b e f \log \left (d\right ) +{\left (e f m n - e f n \log \left (d\right ) + e f \log \left (c\right ) \log \left (d\right )\right )} b^{2}\right )} x +{\left (b^{2} e f m n x + b^{2} e^{2} m n\right )} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{f^{2} x^{2} + e f x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*e*m*log(f*x + e) - (f*m - f*log(d))*b^2*x)*log(x^n)^2 + (b^2*f*x*log(x^n)^2 - 2*((f*n - f*log(c))*b^2 -
a*b*f)*x*log(x^n) - (2*(f*n - f*log(c))*a*b - (2*f*n^2 - 2*f*n*log(c) + f*log(c)^2)*b^2 - a^2*f)*x)*log((f*x +
e)^m))/f - integrate((((f^2*m - f^2*log(d))*a^2 - 2*(f^2*m*n - (f^2*m - f^2*log(d))*log(c))*a*b + (2*f^2*m*n^
2 - 2*f^2*m*n*log(c) + (f^2*m - f^2*log(d))*log(c)^2)*b^2)*x^2 - (b^2*e*f*log(c)^2*log(d) + 2*a*b*e*f*log(c)*l
og(d) + a^2*e*f*log(d))*x + 2*(((f^2*m - f^2*log(d))*a*b - (2*f^2*m*n - f^2*n*log(d) - (f^2*m - f^2*log(d))*lo
g(c))*b^2)*x^2 - (a*b*e*f*log(d) + (e*f*m*n - e*f*n*log(d) + e*f*log(c)*log(d))*b^2)*x + (b^2*e*f*m*n*x + b^2*
e^2*m*n)*log(f*x + e))*log(x^n))/(f^2*x^2 + e*f*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*log((f*x + e)^m*d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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