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

Optimal. Leaf size=1063 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.16285, antiderivative size = 1097, normalized size of antiderivative = 1.03, number of steps used = 39, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {2498, 2513, 2411, 43, 2334, 12, 14, 2301, 2418, 2389, 2295, 2394, 2393, 2391, 2395} \[ \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 \text{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{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{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{p^2 r^2 \log (a+b x) \left (\frac{2 \log (a+b x) (b g-a h)^2}{b^2}+\frac{4 h (a+b x) (b g-a h)}{b^2}+\frac{h^2 (a+b x)^2}{b^2}\right )}{2 h}+\frac{(d g-c h)^2 p q r^2 \log (c+d x)}{2 d^2 h}-\frac{p q r^2 (g+h x)^2 \log (c+d x)}{2 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{2 \log (c+d x) (d g-c h)^2}{d^2}+\frac{4 h (c+d x) (d g-c h)}{d^2}+\frac{h^2 (c+d x)^2}{d^2}\right )}{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{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 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 h} \]

Antiderivative was successfully verified.

[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) - ((d*g - c*h)*p*q*r^2*(a + b*x)*Log[a + b*x])/(b*d) - (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) - (p^2*r^2*Log[a + b*x]*((4*h*(b*g - a*h)*(a + b*x))
/b^2 + (h^2*(a + b*x)^2)/b^2 + (2*(b*g - a*h)^2*Log[a + b*x])/b^2))/(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) - (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) - (q^2*r^2*Log[c + d*x]*((4*h*(d*g - c*h)*(c + d*x))/d^2 + (h^2*(c + d*x)^2)/d^2 + (2*(d*
g - c*h)^2*Log[c + d*x])/d^2))/(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*Log[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 2498

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 2513

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]]

Rule 2411

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 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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2301

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 2418

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 2389

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 2295

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

Rule 2394

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 2393

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 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 2395

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]

Rubi steps

\begin{align*} \int (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\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 ) \operatorname{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 ) \operatorname{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{p^2 r^2 \log (a+b x) \left (\frac{4 h (b g-a h) (a+b x)}{b^2}+\frac{h^2 (a+b x)^2}{b^2}+\frac{2 (b g-a h)^2 \log (a+b x)}{b^2}\right )}{2 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{4 h (d g-c h) (c+d x)}{d^2}+\frac{h^2 (c+d x)^2}{d^2}+\frac{2 (d g-c h)^2 \log (c+d x)}{d^2}\right )}{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 ) \operatorname{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 ) \operatorname{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{p q r^2 (g+h x)^2 \log (a+b x)}{2 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{4 h (b g-a h) (a+b x)}{b^2}+\frac{h^2 (a+b x)^2}{b^2}+\frac{2 (b g-a h)^2 \log (a+b x)}{b^2}\right )}{2 h}-\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{q^2 r^2 \log (c+d x) \left (\frac{4 h (d g-c h) (c+d x)}{d^2}+\frac{h^2 (c+d x)^2}{d^2}+\frac{2 (d g-c h)^2 \log (c+d x)}{d^2}\right )}{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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{(d g-c h) p q r^2 (a+b x) \log (a+b x)}{b d}-\frac{p q r^2 (g+h x)^2 \log (a+b x)}{2 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{4 h (b g-a h) (a+b x)}{b^2}+\frac{h^2 (a+b x)^2}{b^2}+\frac{2 (b g-a h)^2 \log (a+b x)}{b^2}\right )}{2 h}-\frac{(b g-a h) p q r^2 (c+d x) \log (c+d x)}{b d}-\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{q^2 r^2 \log (c+d x) \left (\frac{4 h (d g-c h) (c+d x)}{d^2}+\frac{h^2 (c+d x)^2}{d^2}+\frac{2 (d g-c h)^2 \log (c+d x)}{d^2}\right )}{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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{(d g-c h) p q r^2 (a+b x) \log (a+b x)}{b d}-\frac{p q r^2 (g+h x)^2 \log (a+b x)}{2 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{4 h (b g-a h) (a+b x)}{b^2}+\frac{h^2 (a+b x)^2}{b^2}+\frac{2 (b g-a h)^2 \log (a+b x)}{b^2}\right )}{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{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{q^2 r^2 \log (c+d x) \left (\frac{4 h (d g-c h) (c+d x)}{d^2}+\frac{h^2 (c+d x)^2}{d^2}+\frac{2 (d g-c h)^2 \log (c+d x)}{d^2}\right )}{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 ) \operatorname{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 ) \operatorname{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{(d g-c h) p q r^2 (a+b x) \log (a+b x)}{b d}-\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{p^2 r^2 \log (a+b x) \left (\frac{4 h (b g-a h) (a+b x)}{b^2}+\frac{h^2 (a+b x)^2}{b^2}+\frac{2 (b g-a h)^2 \log (a+b x)}{b^2}\right )}{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{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{q^2 r^2 \log (c+d x) \left (\frac{4 h (d g-c h) (c+d x)}{d^2}+\frac{h^2 (c+d x)^2}{d^2}+\frac{2 (d g-c h)^2 \log (c+d x)}{d^2}\right )}{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]  time = 1.12735, size = 480, normalized size = 0.45 \[ \frac{-4 p q r^2 (b c-a d) (a d h+b c h-2 b d g) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+2 p r \log (a+b x) \left (a d \left ((4 b d g-2 a d h) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a d h r (3 p+q)+2 b q r (c h-2 d g)\right )-2 q r (b c-a d) (a d h+b c h-2 b d g) \log \left (\frac{b (c+d x)}{b c-a d}\right )+2 b^2 c q r (c h-2 d g) \log (c+d x)\right )+b \left (d \left (2 b d x (2 g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 r (2 a d p (2 g-h x)+b x (d (p+q) (4 g+h x)-2 c h q)) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+r^2 (b x (p+q) (d (p+q) (8 g+h x)-6 c h q)-2 a p (2 c h q-4 d g q+3 d h x (p+q)))\right )+2 q r \log (c+d x) \left (-2 b c (c h-2 d g) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 a d p r (c h+2 d g)+b c r (c h (p+3 q)-4 d g (p+q))\right )+2 b c q^2 r^2 (c h-2 d g) \log ^2(c+d x)\right )+2 a d^2 p^2 r^2 (a h-2 b g) \log ^2(a+b x)}{4 b^2 d^2} \]

Antiderivative was successfully verified.

[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]  time = 0.176, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 1.37604, size = 841, normalized size = 0.79 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)