\(\int \frac {(i+j x)^2 (a+b \log (c (d (e+f x)^p)^q))^2}{g+h x} \, dx\) [530]

Optimal result

Integrand size = 35, antiderivative size = 519 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{g+h x} \, dx=-\frac {2 a b j (f i-e j) p q x}{f h}-\frac {2 a b j (h i-g j) p q x}{h^2}+\frac {2 b^2 j (f i-e j) p^2 q^2 x}{f h}+\frac {2 b^2 j (h i-g j) p^2 q^2 x}{h^2}+\frac {b^2 j^2 p^2 q^2 (e+f x)^2}{4 f^2 h}-\frac {2 b^2 j (f i-e j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}-\frac {2 b^2 j (h i-g j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac {b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2 h}+\frac {j (f i-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}+\frac {j (h i-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac {j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {2 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {2 b^2 (h i-g j)^2 p^2 q^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h^3} \]

[Out]

-2*a*b*j*(-e*j+f*i)*p*q*x/f/h-2*a*b*j*(-g*j+h*i)*p*q*x/h^2+2*b^2*j*(-e*j+f*i)*p^2*q^2*x/f/h+2*b^2*j*(-g*j+h*i)
*p^2*q^2*x/h^2+1/4*b^2*j^2*p^2*q^2*(f*x+e)^2/f^2/h-2*b^2*j*(-e*j+f*i)*p*q*(f*x+e)*ln(c*(d*(f*x+e)^p)^q)/f^2/h-
2*b^2*j*(-g*j+h*i)*p*q*(f*x+e)*ln(c*(d*(f*x+e)^p)^q)/f/h^2-1/2*b*j^2*p*q*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))
/f^2/h+j*(-e*j+f*i)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2/h+j*(-g*j+h*i)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q
))^2/f/h^2+1/2*j^2*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2/h+(-g*j+h*i)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*ln
(f*(h*x+g)/(-e*h+f*g))/h^3+2*b*(-g*j+h*i)^2*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h
^3-2*b^2*(-g*j+h*i)^2*p^2*q^2*polylog(3,-h*(f*x+e)/(-e*h+f*g))/h^3

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2465, 2436, 2333, 2332, 2443, 2481, 2421, 6724, 2448, 2437, 2342, 2341, 2495} \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{g+h x} \, dx=\frac {j (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac {b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2 h}+\frac {j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2 h}+\frac {2 b p q (h i-g j)^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^3}+\frac {(h i-g j)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^3}+\frac {j (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}-\frac {2 a b j p q x (f i-e j)}{f h}-\frac {2 a b j p q x (h i-g j)}{h^2}-\frac {2 b^2 j p q (e+f x) (f i-e j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}-\frac {2 b^2 j p q (e+f x) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {b^2 j^2 p^2 q^2 (e+f x)^2}{4 f^2 h}-\frac {2 b^2 p^2 q^2 (h i-g j)^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\frac {2 b^2 j p^2 q^2 x (f i-e j)}{f h}+\frac {2 b^2 j p^2 q^2 x (h i-g j)}{h^2} \]

[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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

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 2341

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 2342

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2436

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 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 2443

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 2448

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 2465

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 2481

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 2495

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]]

Rule 6724

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {(i+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 ) \\ & = \text {Subst}\left (\int \left (\frac {j (h i-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{h^2}+\frac {(h i-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 (i+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 ) \\ & = \text {Subst}\left (\frac {j \int (i+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 )+\text {Subst}\left (\frac {(j (h i-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 )+\text {Subst}\left (\frac {(h i-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 {(h i-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}+\text {Subst}\left (\frac {j \int \left (\frac {(f i-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 )+\text {Subst}\left (\frac {(j (h i-g j)) \text {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 )-\text {Subst}\left (\frac {\left (2 b f (h i-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 (h i-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 {(h i-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}+\text {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 )+\text {Subst}\left (\frac {(j (f i-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 )-\text {Subst}\left (\frac {(2 b j (h i-g j) p q) \text {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 )-\text {Subst}\left (\frac {\left (2 b (h i-g j)^2 p q\right ) \text {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 (h i-g j) p q x}{h^2}+\frac {j (h i-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 {(h i-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 (h i-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}+\text {Subst}\left (\frac {j^2 \text {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 )+\text {Subst}\left (\frac {(j (f i-e j)) \text {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 )-\text {Subst}\left (\frac {\left (2 b^2 j (h i-g j) p q\right ) \text {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 )-\text {Subst}\left (\frac {\left (2 b^2 (h i-g j)^2 p^2 q^2\right ) \text {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 (h i-g j) p q x}{h^2}+\frac {2 b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {2 b^2 j (h i-g j) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {j (f i-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 (h i-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 {(h i-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 (h i-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 (h i-g j)^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\text {Subst}\left (\frac {\left (b j^2 p q\right ) \text {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 )-\text {Subst}\left (\frac {(2 b j (f i-e j) p q) \text {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 (f i-e j) p q x}{f h}-\frac {2 a b j (h i-g j) p q x}{h^2}+\frac {2 b^2 j (h i-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 (h i-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 (f i-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 (h i-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 {(h i-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 (h i-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 (h i-g j)^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\text {Subst}\left (\frac {\left (2 b^2 j (f i-e j) p q\right ) \text {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 (f i-e j) p q x}{f h}-\frac {2 a b j (h i-g j) p q x}{h^2}+\frac {2 b^2 j (f i-e j) p^2 q^2 x}{f h}+\frac {2 b^2 j (h i-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 (f i-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 (h i-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 (f i-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 (h i-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 {(h i-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 (h i-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 (h i-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] (verified)

Time = 0.41 (sec) , antiderivative size = 927, normalized size of antiderivative = 1.79 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{g+h x} \, dx=\frac {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+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 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)-8 b f^2 h^2 i^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 )+\operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )-16 b f h i 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 h x-f g \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )-f g \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )+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 (f h (f x (-4 g+h x)-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 )-4 f^2 g^2 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )+8 b^2 f h i j p^2 q^2 \left (h \left (2 f x-2 (e+f x) \log (e+f x)+(e+f x) \log ^2(e+f x)\right )-f g \left (\log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 \log (e+f x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )\right )\right )-b^2 j^2 p^2 q^2 \left (4 f g h \left (2 f x-2 (e+f x) \log (e+f x)+(e+f x) \log ^2(e+f x)\right )+h^2 \left (f x (6 e-f x)+\left (-6 e^2-4 e f x+2 f^2 x^2\right ) \log (e+f x)+2 \left (e^2-f^2 x^2\right ) \log ^2(e+f x)\right )-4 f^2 g^2 \left (\log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 \log (e+f x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )\right )\right )+4 b^2 f^2 h^2 i^2 p^2 q^2 \left (\log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 \log (e+f x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )\right )}{4 f^2 h^3} \]

[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)

Maple [F]

\[\int \frac {\left (j x +i \right )^{2} {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{h x +g}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[In]

integrate((j*x+i)**2*(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*(i + j*x)**2/(g + h*x), x)

Maxima [F]

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

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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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