3.1.80 \(\int (a+b \log (c x^n))^2 \log (d (e+f x)^m) \, dx\) [80]

Optimal. Leaf size=288 \[ 2 a b m n x-4 b^2 m n^2 x+2 b m n (a-b n) x+4 b^2 m n x \log \left (c x^n\right )-m x \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b e m n (a-b n) \log (e+f x)}{f}-2 a b n x \log \left (d (e+f x)^m\right )+2 b^2 n^2 x \log \left (d (e+f x)^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b^2 e m n \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )}{f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{f}-\frac {2 b^2 e m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{f}+\frac {2 b e m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{f}-\frac {2 b^2 e m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{f} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 288, normalized size of antiderivative = 1.00, 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 = {2333, 2332, 2418, 6, 45, 2393, 2354, 2438, 2395, 2421, 6724} \begin {gather*} \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 \end {gather*}

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 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 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 2332

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

Rule 2333

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 2354

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 2393

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 2395

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 2418

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 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]

Rule 6724

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*}

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Mathematica [A]
time = 0.12, size = 507, normalized size = 1.76 \begin {gather*} \frac {-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 \left (c x^n\right )+4 b^2 f m n x \log \left (c x^n\right )-b^2 f m x \log ^2\left (c x^n\right )+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 \log (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 ^2(x) \log (e+f x)+2 a b e m \log \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)+b^2 e m \log ^2\left (c x^n\right ) \log (e+f x)+a^2 f x \log \left (d (e+f x)^m\right )-2 a b f n x \log \left (d (e+f x)^m\right )+2 b^2 f n^2 x \log \left (d (e+f x)^m\right )+2 a b f x \log \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 f x \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+2 a b e m n \log (x) \log \left (1+\frac {f x}{e}\right )-2 b^2 e m n^2 \log (x) \log \left (1+\frac {f x}{e}\right )-b^2 e m n^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )+2 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+2 b e m n \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )-2 b^2 e m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{f} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.49, size = 10356, normalized size = 35.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(10356\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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