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

Optimal. Leaf size=269 \[ -\frac{2 p q r^2 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac{2 r (p+q) (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 q r (b c-a d) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{2 q r^2 (p+q) (b c-a d) \log (c+d x)}{b d}-\frac{2 p q r^2 (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d}-\frac{q^2 r^2 (b c-a d) \log ^2(c+d x)}{b d}+2 r^2 x (p+q)^2 \]

[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)

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Rubi [A]  time = 0.153009, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2487, 2494, 2394, 2393, 2391, 2390, 2301, 31, 8} \[ -\frac{2 p q r^2 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d}+\frac{(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac{2 r (p+q) (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{2 q r (b c-a d) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{2 q r^2 (p+q) (b c-a d) \log (c+d x)}{b d}-\frac{2 p q r^2 (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d}-\frac{q^2 r^2 (b c-a d) \log ^2(c+d x)}{b d}+2 r^2 x (p+q)^2 \]

Antiderivative was successfully verified.

[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 2487

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 2494

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]

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 2390

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 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 31

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

Rule 8

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

Rubi steps

\begin{align*} \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 (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 ) \operatorname{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 ) \operatorname{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]  time = 0.215741, size = 437, normalized size = 1.62 \[ \frac{2 p q r^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+b d x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 p r \log (a+b x) \left (d \left (a \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a r (p-q)+b p r x\right )+q r (b c-a d) \log \left (\frac{b (c+d x)}{b c-a d}\right )-b c q r \log (c+d x)\right )-d p^2 r^2 (2 a+b x) \log ^2(a+b x)+2 a d p q r^2 \log (c+d x)+2 a d p q r^2-2 b c p q r^2 \log (c+d x)-b c q^2 r^2 \log ^2(c+d x)-2 b c q^2 r^2 \log (c+d x)+4 b d p q r^2 x+2 b d q^2 r^2 x}{b d}+\frac{p^2 r^2 \left ((a+b x) \log ^2(a+b x)-2 (a+b x) \log (a+b x)+2 b x\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \begin{align*} \int \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(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)

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Maxima [A]  time = 1.36595, size = 402, normalized size = 1.49 \begin{align*} 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}} \end{align*}

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,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

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

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

[Out]

Timed out

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