3.4.69 \(\int \log (f x^m) (a+b \log (c (d+e x)^n))^2 \, dx\) [369]

Optimal. Leaf size=309 \[ 2 a b m n x-4 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac {4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {2 b^2 d m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {2 b^2 d m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {2 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {2 b^2 d m n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2436, 2333, 2332, 2470, 2458, 45, 2393, 2354, 2438, 2395, 2421, 6724} \begin {gather*} -\frac {2 b d m n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {2 b^2 d m n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e}+\frac {2 b^2 d m n^2 \text {PolyLog}\left (3,\frac {e x}{d}+1\right )}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x \log \left (f x^m\right )+2 a b m n x+2 b m n x (a-b n)-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {2 b^2 d m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \log \left (f x^m\right )-4 b^2 m n^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

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 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 2436

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

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 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2470

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_), x_Symbol] :> With[{u = In
tHide[(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - Dist[m, Int[Dist[1/x, u, x], x], x]] /; Free
Q[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 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 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-m \int \left (-2 a b n+2 b^2 n^2-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e x}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e x}\right ) \, dx\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m \int \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{e}+\frac {\left (2 b^2 m n\right ) \int \frac {(d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{e}\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m \text {Subst}\left (\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{e^2}+\frac {\left (2 b^2 m n\right ) \text {Subst}\left (\int \frac {x \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{e^2}\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m \text {Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^2-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}+\frac {\left (2 b^2 m n\right ) \text {Subst}\left (\int \left (e \log \left (c x^n\right )-\frac {d e \log \left (c x^n\right )}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {m \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}+\frac {(d m) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{e}+\frac {\left (2 b^2 m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (2 b^2 d m n\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{e}\\ &=-2 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac {2 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {2 b^2 d m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(2 b m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac {(2 b d m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac {\left (2 b^2 d m n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=2 a b m n x-2 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac {2 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {2 b^2 d m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {2 b^2 d m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {2 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {\left (2 b^2 m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac {\left (2 b^2 d m n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=2 a b m n x-4 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac {4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {2 b^2 d m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac {m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {d m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {2 b^2 d m n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}-\frac {2 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {2 b^2 d m n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 549, normalized size = 1.78 \begin {gather*} b^2 n^2 \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (x \log ^2(d+e x)-2 e \left (-\frac {x}{e}+\frac {d \log (d+e x)}{e^2}+\frac {x \log (d+e x)}{e}-\frac {d \log ^2(d+e x)}{2 e^2}\right )\right )-x \left (m-\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 b n x \left (-m-m \log (x)+\log \left (f x^m\right )\right ) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )+2 b n x \left (-m+\log \left (f x^m\right )\right ) \log (d+e x) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )+\frac {2 b d n \left (-m-m \log (x)+\log \left (f x^m\right )\right ) \log (d+e x) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )}{e}-2 b e m n \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right ) \left (\frac {x (-1+\log (x))}{e}-\frac {d \left (\frac {\log (x) \log \left (\frac {d+e x}{d}\right )}{e}+\frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{e}\right )}{e}\right )+b^2 m n^2 \left (-x \log ^2(d+e x)+x \log (x) \log ^2(d+e x)+2 e \left (-\frac {x}{e}+\frac {d \log (d+e x)}{e^2}+\frac {x \log (d+e x)}{e}-\frac {d \log ^2(d+e x)}{2 e^2}\right )-2 e \left (\frac {2 e x-d \log (d+e x)-e x \log (d+e x)+\log (x) \left (-e x+e x \log (d+e x)+d \log \left (1+\frac {e x}{d}\right )\right )+d \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}-\frac {d \left (\frac {1}{2} \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log ^2(d+e x)-\log (d+e x) \text {Li}_2\left (\frac {d+e x}{d}\right )+\text {Li}_3\left (\frac {d+e x}{d}\right )\right )}{e^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \ln \left (f \,x^{m}\right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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