3.41 \(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^3} \, dx\)

Optimal. Leaf size=1304 \[ \frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) b^2}{h (b g-a h)^2}-\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^2}{h (b g-a h)^2}+\frac {p^2 r^2 \log (g+h x) b^2}{h (b g-a h)^2}+\frac {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 ) \log (g+h x) b^2}{h (b g-a h)^2}-\frac {p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) b^2}{h (b g-a h)^2}-\frac {p^2 r^2 \log (a+b x) \log \left (\frac {b g-a h}{h (a+b x)}+1\right ) b^2}{h (b g-a h)^2}+\frac {p^2 r^2 \text {Li}_2\left (-\frac {b g-a h}{h (a+b x)}\right ) b^2}{h (b g-a h)^2}+\frac {p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right ) b^2}{h (b g-a h)^2}-\frac {p q r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right ) b^2}{h (b g-a h)^2}-\frac {d p q r^2 \log (a+b x) b}{h (b g-a h) (d g-c h)}-\frac {p^2 r^2 (a+b x) \log (a+b x) b}{(b g-a h)^2 (g+h x)}-\frac {d p q r^2 \log (c+d x) b}{h (b g-a h) (d g-c h)}+\frac {p q r^2 \log (c+d x) b}{h (b g-a h) (g+h x)}-\frac {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 ) b}{h (b g-a h) (g+h x)}+\frac {2 d p q r^2 \log (g+h x) b}{h (b g-a h) (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {d^2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {d^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 )}{h (d g-c h)^2}-\frac {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 )}{h (d g-c h) (g+h x)}+\frac {d^2 q^2 r^2 \log (g+h x)}{h (d g-c h)^2}+\frac {d^2 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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {d^2 p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (d g-c h)^2}-\frac {d^2 q^2 r^2 \log (c+d x) \log \left (\frac {d g-c h}{h (c+d x)}+1\right )}{h (d g-c h)^2}+\frac {d^2 p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {d^2 p q r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (d g-c h)^2}+\frac {d^2 q^2 r^2 \text {Li}_2\left (-\frac {d g-c h}{h (c+d x)}\right )}{h (d g-c h)^2} \]

[Out]

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

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Rubi [A]  time = 1.41, antiderivative size = 1362, normalized size of antiderivative = 1.04, number of steps used = 47, number of rules used = 16, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {2498, 2513, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2418, 2394, 2393, 2395, 36, 44} \[ \frac {p^2 r^2 \log ^2(a+b x) b^2}{2 h (b g-a h)^2}+\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) b^2}{h (b g-a h)^2}-\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^2}{h (b g-a h)^2}+\frac {p^2 r^2 \log (g+h x) b^2}{h (b g-a h)^2}+\frac {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 ) \log (g+h x) b^2}{h (b g-a h)^2}-\frac {p^2 r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right ) b^2}{h (b g-a h)^2}-\frac {p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) b^2}{h (b g-a h)^2}-\frac {p^2 r^2 \text {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right ) b^2}{h (b g-a h)^2}+\frac {p q r^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) b^2}{h (b g-a h)^2}-\frac {p q r^2 \text {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) b^2}{h (b g-a h)^2}-\frac {d p q r^2 \log (a+b x) b}{h (b g-a h) (d g-c h)}-\frac {p^2 r^2 (a+b x) \log (a+b x) b}{(b g-a h)^2 (g+h x)}-\frac {d p q r^2 \log (c+d x) b}{h (b g-a h) (d g-c h)}+\frac {p q r^2 \log (c+d x) b}{h (b g-a h) (g+h x)}-\frac {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 ) b}{h (b g-a h) (g+h x)}+\frac {2 d p q r^2 \log (g+h x) b}{h (b g-a h) (d g-c h)}+\frac {d^2 q^2 r^2 \log ^2(c+d x)}{2 h (d g-c h)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {d^2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {d^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 )}{h (d g-c h)^2}-\frac {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 )}{h (d g-c h) (g+h x)}+\frac {d^2 q^2 r^2 \log (g+h x)}{h (d g-c h)^2}+\frac {d^2 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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {d^2 p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (d g-c h)^2}-\frac {d^2 q^2 r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (d g-c h)^2}+\frac {d^2 p q r^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {d^2 p q r^2 \text {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{h (d g-c h)^2}-\frac {d^2 q^2 r^2 \text {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{h (d g-c h)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 2314

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

Rule 2317

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

Rule 2344

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

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 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 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 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]

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 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 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]]

Rubi steps

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

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Mathematica [B]  time = 6.31, size = 15960, normalized size = 12.24 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 1.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^3,x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h^3*x^3 + 3*g*h^2*x^2 + 3*g^2*h*x + g^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^3,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h*x + g)^3, x)

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maple [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}}{\left (h x +g \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.86, size = 1857, normalized size = 1.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^3,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (g+h\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(g + h*x)^3,x)

[Out]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(g + h*x)^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(h*x+g)**3,x)

[Out]

Timed out

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