\(\int (g+h x) \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [37]

Optimal result

Integrand size = 29, antiderivative size = 1063 \[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {3 (b g-a h) p q r^2 x}{2 b}+\frac {3 (d g-c h) p q r^2 x}{2 d}+\frac {p q r^2 (g+h x)^2}{2 h}+\frac {p^2 r^2 (4 b g-3 a h+b h x)^2}{4 b^2 h}+\frac {q^2 r^2 (4 d g-3 c h+d h x)^2}{4 d^2 h}+\frac {(b g-a h)^2 p q r^2 \log (a+b x)}{2 b^2 h}-\frac {2 (b g-a h) p^2 r^2 (a+b x) \log (a+b x)}{b^2}-\frac {(d g-c h) p q r^2 (a+b x) \log (a+b x)}{b d}-\frac {h p^2 r^2 (a+b x)^2 \log (a+b x)}{2 b^2}-\frac {p q r^2 (g+h x)^2 \log (a+b x)}{2 h}-\frac {(b g-a h)^2 p^2 r^2 \log ^2(a+b x)}{2 b^2 h}+\frac {(d g-c h)^2 p q r^2 \log (c+d x)}{2 d^2 h}-\frac {(b g-a h) p q r^2 (c+d x) \log (c+d x)}{b d}-\frac {2 (d g-c h) q^2 r^2 (c+d x) \log (c+d x)}{d^2}-\frac {h q^2 r^2 (c+d x)^2 \log (c+d x)}{2 d^2}-\frac {p q r^2 (g+h x)^2 \log (c+d x)}{2 h}-\frac {(b g-a h)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 h}-\frac {(d g-c h)^2 q^2 r^2 \log ^2(c+d x)}{2 d^2 h}-\frac {(d g-c h)^2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 h}+\frac {(b g-a h) p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{b}+\frac {(d g-c h) q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{d}+\frac {p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 h}+\frac {q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 h}+\frac {(b g-a h)^2 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{b^2 h}+\frac {(d g-c h)^2 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{d^2 h}+\frac {(g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h}-\frac {(d g-c h)^2 p q r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 h}-\frac {(b g-a h)^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b^2 h} \]

[Out]

-(-c*h+d*g)*p*q*r^2*(b*x+a)*ln(b*x+a)/b/d-(-a*h+b*g)*p*q*r^2*(d*x+c)*ln(d*x+c)/b/d-(-a*h+b*g)^2*p*q*r^2*ln(-d*
(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b^2/h-(-c*h+d*g)^2*p*q*r^2*ln(b*x+a)*ln(b*(d*x+c)/(-a*d+b*c))/d^2/h+1/2*(h*x+g)^
2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/h-(-a*h+b*g)^2*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))/b^2/h+(-a*h+b*g)^2*
p*r*ln(b*x+a)*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/b^2/h+(-c*h+d*g)^2*q*r*ln(d*x+c)*(
p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/d^2/h-(-c*h+d*g)^2*p*q*r^2*polylog(2,-d*(b*x+a)/(
-a*d+b*c))/d^2/h+(-a*h+b*g)*p*r*x*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/b+(-c*h+d*g)*q
*r*x*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/d+1/2*(-a*h+b*g)^2*p*q*r^2*ln(b*x+a)/b^2/h+
1/2*(-c*h+d*g)^2*p*q*r^2*ln(d*x+c)/d^2/h+3/2*(-a*h+b*g)*p*q*r^2*x/b+3/2*(-c*h+d*g)*p*q*r^2*x/d+1/2*p*r*(h*x+g)
^2*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/h+1/2*q*r*(h*x+g)^2*(p*r*ln(b*x+a)+q*r*ln(d*x
+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/h+1/2*p*q*r^2*(h*x+g)^2/h+1/4*p^2*r^2*(b*h*x-3*a*h+4*b*g)^2/b^2/h+1/4*q^2
*r^2*(d*h*x-3*c*h+4*d*g)^2/d^2/h-2*(-a*h+b*g)*p^2*r^2*(b*x+a)*ln(b*x+a)/b^2-1/2*h*p^2*r^2*(b*x+a)^2*ln(b*x+a)/
b^2-1/2*p*q*r^2*(h*x+g)^2*ln(b*x+a)/h-1/2*(-a*h+b*g)^2*p^2*r^2*ln(b*x+a)^2/b^2/h-2*(-c*h+d*g)*q^2*r^2*(d*x+c)*
ln(d*x+c)/d^2-1/2*h*q^2*r^2*(d*x+c)^2*ln(d*x+c)/d^2-1/2*p*q*r^2*(h*x+g)^2*ln(d*x+c)/h-1/2*(-c*h+d*g)^2*q^2*r^2
*ln(d*x+c)^2/d^2/h

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 1063, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {2584, 2593, 2458, 45, 2372, 12, 14, 2338, 2465, 2436, 2332, 2441, 2440, 2438, 2442} \[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {p^2 r^2 \log ^2(a+b x) (b g-a h)^2}{2 b^2 h}+\frac {p q r^2 \log (a+b x) (b g-a h)^2}{2 b^2 h}-\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b g-a h)^2}{b^2 h}+\frac {p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) (b g-a h)^2}{b^2 h}-\frac {p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b g-a h)^2}{b^2 h}+\frac {3 p q r^2 x (b g-a h)}{2 b}-\frac {2 p^2 r^2 (a+b x) \log (a+b x) (b g-a h)}{b^2}-\frac {p q r^2 (c+d x) \log (c+d x) (b g-a h)}{b d}+\frac {p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) (b g-a h)}{b}+\frac {p q r^2 (g+h x)^2}{2 h}+\frac {p^2 r^2 (4 b g-3 a h+b h x)^2}{4 b^2 h}+\frac {q^2 r^2 (4 d g-3 c h+d h x)^2}{4 d^2 h}-\frac {(d g-c h)^2 q^2 r^2 \log ^2(c+d x)}{2 d^2 h}+\frac {(g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h}+\frac {3 (d g-c h) p q r^2 x}{2 d}-\frac {h p^2 r^2 (a+b x)^2 \log (a+b x)}{2 b^2}-\frac {p q r^2 (g+h x)^2 \log (a+b x)}{2 h}-\frac {(d g-c h) p q r^2 (a+b x) \log (a+b x)}{b d}+\frac {(d g-c h)^2 p q r^2 \log (c+d x)}{2 d^2 h}-\frac {h q^2 r^2 (c+d x)^2 \log (c+d x)}{2 d^2}-\frac {p q r^2 (g+h x)^2 \log (c+d x)}{2 h}-\frac {2 (d g-c h) q^2 r^2 (c+d x) \log (c+d x)}{d^2}-\frac {(d g-c h)^2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 h}+\frac {p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 h}+\frac {q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{2 h}+\frac {(d g-c h) q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{d}+\frac {(d g-c h)^2 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{d^2 h}-\frac {(d g-c h)^2 p q r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 h} \]

[In]

Int[(g + h*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]

[Out]

(3*(b*g - a*h)*p*q*r^2*x)/(2*b) + (3*(d*g - c*h)*p*q*r^2*x)/(2*d) + (p*q*r^2*(g + h*x)^2)/(2*h) + (p^2*r^2*(4*
b*g - 3*a*h + b*h*x)^2)/(4*b^2*h) + (q^2*r^2*(4*d*g - 3*c*h + d*h*x)^2)/(4*d^2*h) + ((b*g - a*h)^2*p*q*r^2*Log
[a + b*x])/(2*b^2*h) - (2*(b*g - a*h)*p^2*r^2*(a + b*x)*Log[a + b*x])/b^2 - ((d*g - c*h)*p*q*r^2*(a + b*x)*Log
[a + b*x])/(b*d) - (h*p^2*r^2*(a + b*x)^2*Log[a + b*x])/(2*b^2) - (p*q*r^2*(g + h*x)^2*Log[a + b*x])/(2*h) - (
(b*g - a*h)^2*p^2*r^2*Log[a + b*x]^2)/(2*b^2*h) + ((d*g - c*h)^2*p*q*r^2*Log[c + d*x])/(2*d^2*h) - ((b*g - a*h
)*p*q*r^2*(c + d*x)*Log[c + d*x])/(b*d) - (2*(d*g - c*h)*q^2*r^2*(c + d*x)*Log[c + d*x])/d^2 - (h*q^2*r^2*(c +
d*x)^2*Log[c + d*x])/(2*d^2) - (p*q*r^2*(g + h*x)^2*Log[c + d*x])/(2*h) - ((b*g - a*h)^2*p*q*r^2*Log[-((d*(a
+ b*x))/(b*c - a*d))]*Log[c + d*x])/(b^2*h) - ((d*g - c*h)^2*q^2*r^2*Log[c + d*x]^2)/(2*d^2*h) - ((d*g - c*h)^
2*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(d^2*h) + ((b*g - a*h)*p*r*x*(p*r*Log[a + b*x] + q*r*Lo
g[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/b + ((d*g - c*h)*q*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x
] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/d + (p*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(
f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*h) + (q*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r]))/(2*h) + ((b*g - a*h)^2*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*
(f*(a + b*x)^p*(c + d*x)^q)^r]))/(b^2*h) + ((d*g - c*h)^2*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x
] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(d^2*h) + ((g + h*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(2*h
) - ((d*g - c*h)^2*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^2*h) - ((b*g - a*h)^2*p*q*r^2*PolyLog[
2, (b*(c + d*x))/(b*c - a*d)])/(b^2*h)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2332

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

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 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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -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 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2584

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)*((g_.) + (h_.)*(x_))^(
m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1))), x] + (-Dist[b*
p*r*(s/(h*(m + 1))), Int[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(a + b*x)), x], x] -
Dist[d*q*r*(s/(h*(m + 1))), Int[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x)), x]
, x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]

Rule 2593

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.45 \[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {2 a d^2 (-2 b g+a h) p^2 r^2 \log ^2(a+b x)+2 p r \log (a+b x) \left (2 b^2 c (-2 d g+c h) q r \log (c+d x)-2 (b c-a d) (-2 b d g+b c h+a d h) q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (2 b (-2 d g+c h) q r+a d h (3 p+q) r+(4 b d g-2 a d h) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+b \left (2 b c (-2 d g+c h) q^2 r^2 \log ^2(c+d x)+2 q r \log (c+d x) \left (2 a d (2 d g+c h) p r+b c (-4 d g (p+q)+c h (p+3 q)) r-2 b c (-2 d g+c h) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+d \left (r^2 (-2 a p (-4 d g q+2 c h q+3 d h (p+q) x)+b (p+q) x (-6 c h q+d (p+q) (8 g+h x)))-2 r (2 a d p (2 g-h x)+b x (-2 c h q+d (p+q) (4 g+h x))) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b d x (2 g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-4 (b c-a d) (-2 b d g+b c h+a d h) p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{4 b^2 d^2} \]

[In]

Integrate[(g + h*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]

[Out]

(2*a*d^2*(-2*b*g + a*h)*p^2*r^2*Log[a + b*x]^2 + 2*p*r*Log[a + b*x]*(2*b^2*c*(-2*d*g + c*h)*q*r*Log[c + d*x] -
2*(b*c - a*d)*(-2*b*d*g + b*c*h + a*d*h)*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*(2*b*(-2*d*g + c*h)*q*r + a
*d*h*(3*p + q)*r + (4*b*d*g - 2*a*d*h)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])) + b*(2*b*c*(-2*d*g + c*h)*q^2*r^
2*Log[c + d*x]^2 + 2*q*r*Log[c + d*x]*(2*a*d*(2*d*g + c*h)*p*r + b*c*(-4*d*g*(p + q) + c*h*(p + 3*q))*r - 2*b*
c*(-2*d*g + c*h)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) + d*(r^2*(-2*a*p*(-4*d*g*q + 2*c*h*q + 3*d*h*(p + q)*x)
+ b*(p + q)*x*(-6*c*h*q + d*(p + q)*(8*g + h*x))) - 2*r*(2*a*d*p*(2*g - h*x) + b*x*(-2*c*h*q + d*(p + q)*(4*g
+ h*x)))*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 2*b*d*x*(2*g + h*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2))
- 4*(b*c - a*d)*(-2*b*d*g + b*c*h + a*d*h)*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(4*b^2*d^2)

Maple [F]

\[\int \left (h x +g \right ) {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]

[In]

int((h*x+g)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

[Out]

int((h*x+g)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

Fricas [F]

\[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (h x + g\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]

[In]

integrate((h*x+g)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="fricas")

[Out]

integral((h*x + g)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)

Sympy [F]

\[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (g + h x\right ) \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \]

[In]

integrate((h*x+g)*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2,x)

[Out]

Integral((g + h*x)*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.59 \[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{2} \, {\left (h x^{2} + 2 \, g x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac {r {\left (\frac {2 \, {\left (2 \, a b f g p - a^{2} f h p\right )} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left (2 \, c d f g q - c^{2} f h q\right )} \log \left (d x + c\right )}{d^{2}} - \frac {b d f h {\left (p + q\right )} x^{2} - 2 \, {\left (a d f h p - {\left (2 \, d f g {\left (p + q\right )} - c f h q\right )} b\right )} x}{b d}\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \, f} + \frac {r^{2} {\left (\frac {2 \, {\left (2 \, a c d f^{2} h p q - {\left (4 \, {\left (p q + q^{2}\right )} c d f^{2} g - {\left (p q + 3 \, q^{2}\right )} c^{2} f^{2} h\right )} b\right )} \log \left (d x + c\right )}{b d^{2}} - \frac {4 \, {\left (2 \, a b d^{2} f^{2} g p q - a^{2} d^{2} f^{2} h p q - {\left (2 \, c d f^{2} g p q - c^{2} f^{2} h p q\right )} b^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b^{2} d^{2}} + \frac {{\left (p^{2} + 2 \, p q + q^{2}\right )} b^{2} d^{2} f^{2} h x^{2} - 4 \, {\left (2 \, c d f^{2} g p q - c^{2} f^{2} h p q\right )} b^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - 2 \, {\left (2 \, c d f^{2} g q^{2} - c^{2} f^{2} h q^{2}\right )} b^{2} \log \left (d x + c\right )^{2} - 2 \, {\left (2 \, a b d^{2} f^{2} g p^{2} - a^{2} d^{2} f^{2} h p^{2}\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (p^{2} + p q\right )} a b d^{2} f^{2} h - {\left (4 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} d^{2} f^{2} g - 3 \, {\left (p q + q^{2}\right )} c d f^{2} h\right )} b^{2}\right )} x + 2 \, {\left ({\left (3 \, p^{2} + p q\right )} a^{2} d^{2} f^{2} h + 2 \, {\left (c d f^{2} h p q - 2 \, {\left (p^{2} + p q\right )} d^{2} f^{2} g\right )} a b\right )} \log \left (b x + a\right )}{b^{2} d^{2}}\right )}}{4 \, f^{2}} \]

[In]

integrate((h*x+g)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="maxima")

[Out]

1/2*(h*x^2 + 2*g*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/2*r*(2*(2*a*b*f*g*p - a^2*f*h*p)*log(b*x + a)/b
^2 + 2*(2*c*d*f*g*q - c^2*f*h*q)*log(d*x + c)/d^2 - (b*d*f*h*(p + q)*x^2 - 2*(a*d*f*h*p - (2*d*f*g*(p + q) - c
*f*h*q)*b)*x)/(b*d))*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f + 1/4*r^2*(2*(2*a*c*d*f^2*h*p*q - (4*(p*q + q^2)*c
*d*f^2*g - (p*q + 3*q^2)*c^2*f^2*h)*b)*log(d*x + c)/(b*d^2) - 4*(2*a*b*d^2*f^2*g*p*q - a^2*d^2*f^2*h*p*q - (2*
c*d*f^2*g*p*q - c^2*f^2*h*p*q)*b^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b
*c - a*d)))/(b^2*d^2) + ((p^2 + 2*p*q + q^2)*b^2*d^2*f^2*h*x^2 - 4*(2*c*d*f^2*g*p*q - c^2*f^2*h*p*q)*b^2*log(b
*x + a)*log(d*x + c) - 2*(2*c*d*f^2*g*q^2 - c^2*f^2*h*q^2)*b^2*log(d*x + c)^2 - 2*(2*a*b*d^2*f^2*g*p^2 - a^2*d
^2*f^2*h*p^2)*log(b*x + a)^2 - 2*(3*(p^2 + p*q)*a*b*d^2*f^2*h - (4*(p^2 + 2*p*q + q^2)*d^2*f^2*g - 3*(p*q + q^
2)*c*d*f^2*h)*b^2)*x + 2*((3*p^2 + p*q)*a^2*d^2*f^2*h + 2*(c*d*f^2*h*p*q - 2*(p^2 + p*q)*d^2*f^2*g)*a*b)*log(b
*x + a))/(b^2*d^2))/f^2

Giac [F]

\[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (h x + g\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]

[In]

integrate((h*x+g)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="giac")

[Out]

integrate((h*x + g)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,\left (g+h\,x\right ) \,d x \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(g + h*x),x)

[Out]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(g + h*x), x)