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

Optimal. Leaf size=123 \[ \frac{2 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.265072, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2396, 2433, 2374, 6589, 2445} \[ \frac{2 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{g+h x} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{(2 b f p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{(2 b p q) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac{f \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{\left (2 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\frac{2 b^2 p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}\\ \end{align*}

Mathematica [B]  time = 0.164413, size = 324, normalized size = 2.63 \[ \frac{2 b p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 b^2 p^2 q^2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )+a^2 \log (g+h x)+2 a b \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-2 a b p q \log (e+f x) \log (g+h x)+2 a b p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+b^2 \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-2 b^2 p q \log (e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.697, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}}{hx+g}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \log \left (h x + g\right )}{h} + \int \frac{b^{2} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} +{\left (\log \left (c\right )^{2} + 2 \, \log \left (c\right ) \log \left (d^{q}\right ) + \log \left (d^{q}\right )^{2}\right )} b^{2} + 2 \, a b{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + 2 \,{\left (b^{2}{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + a b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{2}}{h x + g}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{g + h x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{h x + g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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