\(\int \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [38]

Optimal result

Integrand size = 23, antiderivative size = 269 \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=2 (p+q)^2 r^2 x-\frac {2 (b c-a d) q (p+q) r^2 \log (c+d x)}{b d}-\frac {2 (b c-a d) p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d}-\frac {(b c-a d) q^2 r^2 \log ^2(c+d x)}{b d}-\frac {2 (p+q) r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac {2 (b c-a d) q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}+\frac {(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {2 (b c-a d) p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2579, 2580, 2441, 2440, 2438, 2437, 2338, 31, 8} \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {2 r (p+q) (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac {2 q r (b c-a d) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {2 p q r^2 (b c-a d) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d}-\frac {2 q r^2 (p+q) (b c-a d) \log (c+d x)}{b d}-\frac {2 p q r^2 (b c-a d) \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b d}-\frac {q^2 r^2 (b c-a d) \log ^2(c+d x)}{b d}+2 r^2 x (p+q)^2 \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 2437

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

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 2579

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[(a
+ b*x)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/b), x] + (Dist[q*r*s*((b*c - a*d)/b), Int[Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1
), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] &&
LtQ[s, 4]

Rule 2580

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.45 \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {2 a d p q r^2+2 b d p^2 r^2 x+4 b d p q r^2 x+2 b d q^2 r^2 x-a d p^2 r^2 \log ^2(a+b x)-2 b c p q r^2 \log (c+d x)+2 a d p q r^2 \log (c+d x)-2 b c q^2 r^2 \log (c+d x)-b c q^2 r^2 \log ^2(c+d x)-2 p r \log (a+b x) \left (b c q r \log (c+d x)+(-b c+a d) q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (q r-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+b d x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 (b c-a d) p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{b d} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.11 \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=x \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} - \frac {2 \, {\left (f {\left (p + q\right )} x - \frac {a f p \log \left (b x + a\right )}{b} - \frac {c f q \log \left (d x + c\right )}{d}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{f} - \frac {{\left (\frac {2 \, {\left (p q + q^{2}\right )} c f^{2} \log \left (d x + c\right )}{d} - \frac {2 \, {\left (b c f^{2} p q - a d f^{2} p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b d} + \frac {a d f^{2} p^{2} \log \left (b x + a\right )^{2} + 2 \, b c f^{2} p q \log \left (b x + a\right ) \log \left (d x + c\right ) + b c f^{2} q^{2} \log \left (d x + c\right )^{2} - 2 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} b d f^{2} x + 2 \, {\left (p^{2} + p q\right )} a d f^{2} \log \left (b x + a\right )}{b d}\right )} r^{2}}{f^{2}} \]

[In]

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

[Out]

x*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 - 2*(f*(p + q)*x - a*f*p*log(b*x + a)/b - c*f*q*log(d*x + c)/d)*r*log
(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f - (2*(p*q + q^2)*c*f^2*log(d*x + c)/d - 2*(b*c*f^2*p*q - a*d*f^2*p*q)*(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*d) + (a*d*f^2*p^2*log(b*x
+ a)^2 + 2*b*c*f^2*p*q*log(b*x + a)*log(d*x + c) + b*c*f^2*q^2*log(d*x + c)^2 - 2*(p^2 + 2*p*q + q^2)*b*d*f^2
*x + 2*(p^2 + p*q)*a*d*f^2*log(b*x + a))/(b*d))*r^2/f^2

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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