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

Optimal. Leaf size=519 \[ \frac{2 b p q (h i-g j)^2 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^3}-\frac{2 b^2 p^2 q^2 (h i-g j)^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h^3}+\frac{j (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac{b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2 h}+\frac{j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac{j (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{(h i-g j)^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^3}-\frac{2 a b j p q x (f i-e j)}{f h}-\frac{2 a b j p q x (h i-g j)}{h^2}-\frac{2 b^2 j p q (e+f x) (f i-e j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}-\frac{2 b^2 j p q (e+f x) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac{b^2 j^2 p^2 q^2 (e+f x)^2}{4 f^2 h}+\frac{2 b^2 j p^2 q^2 x (f i-e j)}{f h}+\frac{2 b^2 j p^2 q^2 x (h i-g j)}{h^2} \]

[Out]

(-2*a*b*j*(f*i - e*j)*p*q*x)/(f*h) - (2*a*b*j*(h*i - g*j)*p*q*x)/h^2 + (2*b^2*j*(f*i - e*j)*p^2*q^2*x)/(f*h) +
(2*b^2*j*(h*i - g*j)*p^2*q^2*x)/h^2 + (b^2*j^2*p^2*q^2*(e + f*x)^2)/(4*f^2*h) - (2*b^2*j*(f*i - e*j)*p*q*(e +
f*x)*Log[c*(d*(e + f*x)^p)^q])/(f^2*h) - (2*b^2*j*(h*i - g*j)*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^2)
- (b*j^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^2*h) + (j*(f*i - e*j)*(e + f*x)*(a + b*Log[c*
(d*(e + f*x)^p)^q])^2)/(f^2*h) + (j*(h*i - g*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h^2) + (j^2*(
e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*f^2*h) + ((h*i - g*j)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*L
og[(f*(g + h*x))/(f*g - e*h)])/h^3 + (2*b*(h*i - g*j)^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(
e + f*x))/(f*g - e*h))])/h^3 - (2*b^2*(h*i - g*j)^2*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^3

________________________________________________________________________________________

Rubi [A]  time = 1.34371, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.371, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 6589, 2401, 2390, 2305, 2304, 2445} \[ \frac{2 b p q (h i-g j)^2 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^3}-\frac{2 b^2 p^2 q^2 (h i-g j)^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h^3}+\frac{j (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac{b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2 h}+\frac{j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac{j (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{(h i-g j)^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^3}-\frac{2 a b j p q x (f i-e j)}{f h}-\frac{2 a b j p q x (h i-g j)}{h^2}-\frac{2 b^2 j p q (e+f x) (f i-e j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}-\frac{2 b^2 j p q (e+f x) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac{b^2 j^2 p^2 q^2 (e+f x)^2}{4 f^2 h}+\frac{2 b^2 j p^2 q^2 x (f i-e j)}{f h}+\frac{2 b^2 j p^2 q^2 x (h i-g j)}{h^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*j*(f*i - e*j)*p*q*x)/(f*h) - (2*a*b*j*(h*i - g*j)*p*q*x)/h^2 + (2*b^2*j*(f*i - e*j)*p^2*q^2*x)/(f*h) +
(2*b^2*j*(h*i - g*j)*p^2*q^2*x)/h^2 + (b^2*j^2*p^2*q^2*(e + f*x)^2)/(4*f^2*h) - (2*b^2*j*(f*i - e*j)*p*q*(e +
f*x)*Log[c*(d*(e + f*x)^p)^q])/(f^2*h) - (2*b^2*j*(h*i - g*j)*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^2)
- (b*j^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^2*h) + (j*(f*i - e*j)*(e + f*x)*(a + b*Log[c*
(d*(e + f*x)^p)^q])^2)/(f^2*h) + (j*(h*i - g*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h^2) + (j^2*(
e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*f^2*h) + ((h*i - g*j)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*L
og[(f*(g + h*x))/(f*g - e*h)])/h^3 + (2*b*(h*i - g*j)^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(
e + f*x))/(f*g - e*h))])/h^3 - (2*b^2*(h*i - g*j)^2*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^3

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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 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 2401

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

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 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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{(530+j x)^2 \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{(530+j x)^2 \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 )\\ &=\operatorname{Subst}\left (\int \left (\frac{j (530 h-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{h^2}+\frac{(530 h-g j)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{h^2 (g+h x)}+\frac{j (530+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{h}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{j \int (530+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(j (530 h-g j)) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(530 h-g j)^2 \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(530 h-g j)^2 \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^3}+\operatorname{Subst}\left (\frac{j \int \left (\frac{(530 f-e j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{f}+\frac{j (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{f}\right ) \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(j (530 h-g j)) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (2 b f (530 h-g j)^2 p q\right ) \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^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{j (530 h-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{(530 h-g j)^2 \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^3}+\operatorname{Subst}\left (\frac{j^2 \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(j (530 f-e j)) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(2 b j (530 h-g j) p q) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (2 b (530 h-g j)^2 p q\right ) \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^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 a b j (530 h-g j) p q x}{h^2}+\frac{j (530 h-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{(530 h-g j)^2 \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^3}+\frac{2 b (530 h-g j)^2 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^3}+\operatorname{Subst}\left (\frac{j^2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(j (530 f-e j)) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (2 b^2 j (530 h-g j) p q\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (2 b^2 (530 h-g j)^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^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 a b j (530 h-g j) p q x}{h^2}+\frac{2 b^2 j (530 h-g j) p^2 q^2 x}{h^2}-\frac{2 b^2 j (530 h-g j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac{j (530 f-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}+\frac{j (530 h-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac{(530 h-g j)^2 \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^3}+\frac{2 b (530 h-g j)^2 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^3}-\frac{2 b^2 (530 h-g j)^2 p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^3}-\operatorname{Subst}\left (\frac{\left (b j^2 p q\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(2 b j (530 f-e j) p q) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 a b j (530 f-e j) p q x}{f h}-\frac{2 a b j (530 h-g j) p q x}{h^2}+\frac{2 b^2 j (530 h-g j) p^2 q^2 x}{h^2}+\frac{b^2 j^2 p^2 q^2 (e+f x)^2}{4 f^2 h}-\frac{2 b^2 j (530 h-g j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac{b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2 h}+\frac{j (530 f-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}+\frac{j (530 h-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac{(530 h-g j)^2 \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^3}+\frac{2 b (530 h-g j)^2 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^3}-\frac{2 b^2 (530 h-g j)^2 p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^3}-\operatorname{Subst}\left (\frac{\left (2 b^2 j (530 f-e j) p q\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 a b j (530 f-e j) p q x}{f h}-\frac{2 a b j (530 h-g j) p q x}{h^2}+\frac{2 b^2 j (530 f-e j) p^2 q^2 x}{f h}+\frac{2 b^2 j (530 h-g j) p^2 q^2 x}{h^2}+\frac{b^2 j^2 p^2 q^2 (e+f x)^2}{4 f^2 h}-\frac{2 b^2 j (530 f-e j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}-\frac{2 b^2 j (530 h-g j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac{b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2 h}+\frac{j (530 f-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}+\frac{j (530 h-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac{j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac{(530 h-g j)^2 \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^3}+\frac{2 b (530 h-g j)^2 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^3}-\frac{2 b^2 (530 h-g j)^2 p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^3}\\ \end{align*}

Mathematica [A]  time = 0.692227, size = 927, normalized size = 1.79 \[ \frac{-8 b f^2 h^2 p q \left (-a+b p q \log (e+f x)-b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+\text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )\right ) i^2+4 b^2 f^2 h^2 p^2 q^2 \left (\log \left (\frac{f (g+h x)}{f g-e h}\right ) \log ^2(e+f x)+2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \log (e+f x)-2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )\right ) i^2-16 b f h j p q \left (-a+b p q \log (e+f x)-b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (-h (e+f x)+\log (e+f x) \left (e h+f x h-f g \log \left (\frac{f (g+h x)}{f g-e h}\right )\right )-f g \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )\right ) i+8 b^2 f h j p^2 q^2 \left (h \left ((e+f x) \log ^2(e+f x)-2 (e+f x) \log (e+f x)+2 f x\right )-f g \left (\log \left (\frac{f (g+h x)}{f g-e h}\right ) \log ^2(e+f x)+2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \log (e+f x)-2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )\right )\right ) i+2 f^2 h^2 j^2 x^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+4 f^2 h j (2 h i-g j) x \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+4 f^2 (h i-g j)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (g+h x)+2 b j^2 p q \left (-a+b p q \log (e+f x)-b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (-4 f^2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) g^2+f h (f x (h x-4 g)-2 e (2 g+h x))+2 \log (e+f x) \left (h (e+f x) (2 f g+e h-f h x)-2 f^2 g^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )\right )\right )-b^2 j^2 p^2 q^2 \left (-4 f^2 \left (\log \left (\frac{f (g+h x)}{f g-e h}\right ) \log ^2(e+f x)+2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \log (e+f x)-2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )\right ) g^2+4 f h \left ((e+f x) \log ^2(e+f x)-2 (e+f x) \log (e+f x)+2 f x\right ) g+h^2 \left (2 \left (e^2-f^2 x^2\right ) \log ^2(e+f x)+\left (-6 e^2-4 f x e+2 f^2 x^2\right ) \log (e+f x)+f x (6 e-f x)\right )\right )}{4 f^2 h^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*f^2*h*j*(2*h*i - g*j)*x*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2 + 2*f^2*h^2*j^2*x^2*(a - b*
p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2 + 4*f^2*(h*i - g*j)^2*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e
+ f*x)^p)^q])^2*Log[g + h*x] - 8*b*f^2*h^2*i^2*p*q*(-a + b*p*q*Log[e + f*x] - b*Log[c*(d*(e + f*x)^p)^q])*(Lo
g[e + f*x]*Log[(f*(g + h*x))/(f*g - e*h)] + PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)]) - 16*b*f*h*i*j*p*q*(-a +
b*p*q*Log[e + f*x] - b*Log[c*(d*(e + f*x)^p)^q])*(-(h*(e + f*x)) + Log[e + f*x]*(e*h + f*h*x - f*g*Log[(f*(g
+ h*x))/(f*g - e*h)]) - f*g*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)]) + 2*b*j^2*p*q*(-a + b*p*q*Log[e + f*x] -
b*Log[c*(d*(e + f*x)^p)^q])*(f*h*(f*x*(-4*g + h*x) - 2*e*(2*g + h*x)) + 2*Log[e + f*x]*(h*(e + f*x)*(2*f*g +
e*h - f*h*x) - 2*f^2*g^2*Log[(f*(g + h*x))/(f*g - e*h)]) - 4*f^2*g^2*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)])
+ 8*b^2*f*h*i*j*p^2*q^2*(h*(2*f*x - 2*(e + f*x)*Log[e + f*x] + (e + f*x)*Log[e + f*x]^2) - f*g*(Log[e + f*x]^
2*Log[(f*(g + h*x))/(f*g - e*h)] + 2*Log[e + f*x]*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - 2*PolyLog[3, (h*(
e + f*x))/(-(f*g) + e*h)])) - b^2*j^2*p^2*q^2*(4*f*g*h*(2*f*x - 2*(e + f*x)*Log[e + f*x] + (e + f*x)*Log[e + f
*x]^2) + h^2*(f*x*(6*e - f*x) + (-6*e^2 - 4*e*f*x + 2*f^2*x^2)*Log[e + f*x] + 2*(e^2 - f^2*x^2)*Log[e + f*x]^2
) - 4*f^2*g^2*(Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 2*Log[e + f*x]*PolyLog[2, (h*(e + f*x))/(-(f*g)
+ e*h)] - 2*PolyLog[3, (h*(e + f*x))/(-(f*g) + e*h)])) + 4*b^2*f^2*h^2*i^2*p^2*q^2*(Log[e + f*x]^2*Log[(f*(g
+ h*x))/(f*g - e*h)] + 2*Log[e + f*x]*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - 2*PolyLog[3, (h*(e + f*x))/(-
(f*g) + e*h)]))/(4*f^2*h^3)

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Maple [F]  time = 0.856, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( jx+i \right ) ^{2} \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((j*x+i)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, a^{2} i j{\left (\frac{x}{h} - \frac{g \log \left (h x + g\right )}{h^{2}}\right )} + \frac{1}{2} \, a^{2} j^{2}{\left (\frac{2 \, g^{2} \log \left (h x + g\right )}{h^{3}} + \frac{h x^{2} - 2 \, g x}{h^{2}}\right )} + \frac{a^{2} i^{2} \log \left (h x + g\right )}{h} + \int \frac{2 \,{\left (i^{2} \log \left (c\right ) + i^{2} \log \left (d^{q}\right )\right )} a b +{\left (i^{2} \log \left (c\right )^{2} + 2 \, i^{2} \log \left (c\right ) \log \left (d^{q}\right ) + i^{2} \log \left (d^{q}\right )^{2}\right )} b^{2} +{\left (2 \,{\left (j^{2} \log \left (c\right ) + j^{2} \log \left (d^{q}\right )\right )} a b +{\left (j^{2} \log \left (c\right )^{2} + 2 \, j^{2} \log \left (c\right ) \log \left (d^{q}\right ) + j^{2} \log \left (d^{q}\right )^{2}\right )} b^{2}\right )} x^{2} +{\left (b^{2} j^{2} x^{2} + 2 \, b^{2} i j x + b^{2} i^{2}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 2 \,{\left (2 \,{\left (i j \log \left (c\right ) + i j \log \left (d^{q}\right )\right )} a b +{\left (i j \log \left (c\right )^{2} + 2 \, i j \log \left (c\right ) \log \left (d^{q}\right ) + i j \log \left (d^{q}\right )^{2}\right )} b^{2}\right )} x + 2 \,{\left (a b i^{2} +{\left (i^{2} \log \left (c\right ) + i^{2} \log \left (d^{q}\right )\right )} b^{2} +{\left (a b j^{2} +{\left (j^{2} \log \left (c\right ) + j^{2} \log \left (d^{q}\right )\right )} b^{2}\right )} x^{2} + 2 \,{\left (a b i j +{\left (i j \log \left (c\right ) + i j \log \left (d^{q}\right )\right )} b^{2}\right )} x\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((j*x+i)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g),x, algorithm="maxima")

[Out]

2*a^2*i*j*(x/h - g*log(h*x + g)/h^2) + 1/2*a^2*j^2*(2*g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + a^2*i^2*lo
g(h*x + g)/h + integrate((2*(i^2*log(c) + i^2*log(d^q))*a*b + (i^2*log(c)^2 + 2*i^2*log(c)*log(d^q) + i^2*log(
d^q)^2)*b^2 + (2*(j^2*log(c) + j^2*log(d^q))*a*b + (j^2*log(c)^2 + 2*j^2*log(c)*log(d^q) + j^2*log(d^q)^2)*b^2
)*x^2 + (b^2*j^2*x^2 + 2*b^2*i*j*x + b^2*i^2)*log(((f*x + e)^p)^q)^2 + 2*(2*(i*j*log(c) + i*j*log(d^q))*a*b +
(i*j*log(c)^2 + 2*i*j*log(c)*log(d^q) + i*j*log(d^q)^2)*b^2)*x + 2*(a*b*i^2 + (i^2*log(c) + i^2*log(d^q))*b^2
+ (a*b*j^2 + (j^2*log(c) + j^2*log(d^q))*b^2)*x^2 + 2*(a*b*i*j + (i*j*log(c) + i*j*log(d^q))*b^2)*x)*log(((f*x
+ e)^p)^q))/(h*x + g), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

[Out]

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