\(\int \frac {(i+j x)^2 (a+b \log (c (d (e+f x)^p)^q))^3}{g+h x} \, dx\) [535]

Optimal result

Integrand size = 35, antiderivative size = 742 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\frac {6 a b^2 j (f i-e j) p^2 q^2 x}{f h}+\frac {6 a b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {6 b^3 j (f i-e j) p^3 q^3 x}{f h}-\frac {6 b^3 j (h i-g j) p^3 q^3 x}{h^2}-\frac {3 b^3 j^2 p^3 q^3 (e+f x)^2}{8 f^2 h}+\frac {6 b^3 j (f i-e j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}+\frac {6 b^3 j (h i-g j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {3 b^2 j^2 p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2 h}-\frac {3 b j (f i-e j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac {3 b j (h i-g j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}-\frac {3 b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2 h}+\frac {j (f i-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2 h}+\frac {j (h i-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h^2}+\frac {j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2 h}+\frac {(h i-g j)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^2 (h i-g j)^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\frac {6 b^3 (h i-g j)^2 p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h^3} \]

[Out]

6*a*b^2*j*(-e*j+f*i)*p^2*q^2*x/f/h+6*a*b^2*j*(-g*j+h*i)*p^2*q^2*x/h^2-6*b^3*j*(-e*j+f*i)*p^3*q^3*x/f/h-6*b^3*j
*(-g*j+h*i)*p^3*q^3*x/h^2-3/8*b^3*j^2*p^3*q^3*(f*x+e)^2/f^2/h+6*b^3*j*(-e*j+f*i)*p^2*q^2*(f*x+e)*ln(c*(d*(f*x+
e)^p)^q)/f^2/h+6*b^3*j*(-g*j+h*i)*p^2*q^2*(f*x+e)*ln(c*(d*(f*x+e)^p)^q)/f/h^2+3/4*b^2*j^2*p^2*q^2*(f*x+e)^2*(a
+b*ln(c*(d*(f*x+e)^p)^q))/f^2/h-3*b*j*(-e*j+f*i)*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2/h-3*b*j*(-g*j+h
*i)*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f/h^2-3/4*b*j^2*p*q*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2/
h+j*(-e*j+f*i)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f^2/h+j*(-g*j+h*i)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/
f/h^2+1/2*j^2*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f^2/h+(-g*j+h*i)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3*ln(f*(h
*x+g)/(-e*h+f*g))/h^3+3*b*(-g*j+h*i)^2*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h^3-
6*b^2*(-g*j+h*i)^2*p^2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(3,-h*(f*x+e)/(-e*h+f*g))/h^3+6*b^3*(-g*j+h*i)^2
*p^3*q^3*polylog(4,-h*(f*x+e)/(-e*h+f*g))/h^3

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2465, 2436, 2333, 2332, 2443, 2481, 2421, 2430, 6724, 2448, 2437, 2342, 2341, 2495} \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\frac {3 b^2 j^2 p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2 h}-\frac {6 b^2 p^2 q^2 (h i-g j)^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^3}+\frac {6 a b^2 j p^2 q^2 x (f i-e j)}{f h}+\frac {6 a b^2 j p^2 q^2 x (h i-g j)}{h^2}-\frac {3 b j p q (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}+\frac {j (e+f x) (f i-e j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2 h}-\frac {3 b j^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2 h}+\frac {j^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2 h}+\frac {3 b p q (h i-g j)^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^3}+\frac {(h i-g j)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^3}-\frac {3 b j p q (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}+\frac {j (e+f x) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h^2}+\frac {6 b^3 j p^2 q^2 (e+f x) (f i-e j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}+\frac {6 b^3 j p^2 q^2 (e+f x) (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac {3 b^3 j^2 p^3 q^3 (e+f x)^2}{8 f^2 h}+\frac {6 b^3 p^3 q^3 (h i-g j)^2 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^3 j p^3 q^3 x (f i-e j)}{f h}-\frac {6 b^3 j p^3 q^3 x (h i-g j)}{h^2} \]

[In]

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

[Out]

(6*a*b^2*j*(f*i - e*j)*p^2*q^2*x)/(f*h) + (6*a*b^2*j*(h*i - g*j)*p^2*q^2*x)/h^2 - (6*b^3*j*(f*i - e*j)*p^3*q^3
*x)/(f*h) - (6*b^3*j*(h*i - g*j)*p^3*q^3*x)/h^2 - (3*b^3*j^2*p^3*q^3*(e + f*x)^2)/(8*f^2*h) + (6*b^3*j*(f*i -
e*j)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f^2*h) + (6*b^3*j*(h*i - g*j)*p^2*q^2*(e + f*x)*Log[c*(d*(e
+ f*x)^p)^q])/(f*h^2) + (3*b^2*j^2*p^2*q^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(4*f^2*h) - (3*b*j*(f
*i - e*j)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f^2*h) - (3*b*j*(h*i - g*j)*p*q*(e + f*x)*(a + b*
Log[c*(d*(e + f*x)^p)^q])^2)/(f*h^2) - (3*b*j^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*f^2*h)
+ (j*(f*i - e*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(f^2*h) + (j*(h*i - g*j)*(e + f*x)*(a + b*Log[c
*(d*(e + f*x)^p)^q])^3)/(f*h^2) + (j^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(2*f^2*h) + ((h*i - g*j
)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h^3 + (3*b*(h*i - g*j)^2*p*q*(a + b*Log
[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^3 - (6*b^2*(h*i - g*j)^2*p^2*q^2*(a + b*L
og[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^3 + (6*b^3*(h*i - g*j)^2*p^3*q^3*PolyLog[
4, -((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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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 )^3}{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 )^3}{h^2}+\frac {(h i-g j)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{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 )^3}{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 )^3 \, 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 )^3 \, 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 )^3}{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 )^3 \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 )^3}{f}+\frac {j (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{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 )^3 \, 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 (3 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 )^2 \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 )^3}{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 )^3 \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 )^3 \, 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 )^3 \, 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 {(3 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 )^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 (3 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 )^2 \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 {3 b j (h i-g j) p q (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 (h i-g j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \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 )^3 \, 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 )^3 \, 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 (6 b^2 j (h i-g j) p^2 q^2\right ) \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 (6 b^2 (h i-g j)^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \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 {6 a b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {3 b j (h i-g j) p q (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 (f i-e j) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{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 )^3}{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 )^3}{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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^2 (h i-g j)^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\text {Subst}\left (\frac {\left (3 b j^2 p q\right ) \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 )}{2 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 {(3 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 )^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 (6 b^3 j (h i-g j) p^2 q^2\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 (6 b^3 (h i-g j)^2 p^3 q^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\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 {6 a b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {6 b^3 j (h i-g j) p^3 q^3 x}{h^2}+\frac {6 b^3 j (h i-g j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}-\frac {3 b j (f i-e j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac {3 b j (h i-g j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}-\frac {3 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}{4 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 )^3}{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 )^3}{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 )^3}{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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^2 (h i-g j)^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\frac {6 b^3 (h i-g j)^2 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\text {Subst}\left (\frac {\left (3 b^2 j^2 p^2 q^2\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 )}{2 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 (6 b^2 j (f i-e j) p^2 q^2\right ) \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 {6 a b^2 j (f i-e j) p^2 q^2 x}{f h}+\frac {6 a b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {6 b^3 j (h i-g j) p^3 q^3 x}{h^2}-\frac {3 b^3 j^2 p^3 q^3 (e+f x)^2}{8 f^2 h}+\frac {6 b^3 j (h i-g j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {3 b^2 j^2 p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2 h}-\frac {3 b j (f i-e j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac {3 b j (h i-g j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}-\frac {3 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}{4 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 )^3}{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 )^3}{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 )^3}{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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^2 (h i-g j)^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\frac {6 b^3 (h i-g j)^2 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\text {Subst}\left (\frac {\left (6 b^3 j (f i-e j) p^2 q^2\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 {6 a b^2 j (f i-e j) p^2 q^2 x}{f h}+\frac {6 a b^2 j (h i-g j) p^2 q^2 x}{h^2}-\frac {6 b^3 j (f i-e j) p^3 q^3 x}{f h}-\frac {6 b^3 j (h i-g j) p^3 q^3 x}{h^2}-\frac {3 b^3 j^2 p^3 q^3 (e+f x)^2}{8 f^2 h}+\frac {6 b^3 j (f i-e j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2 h}+\frac {6 b^3 j (h i-g j) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^2}+\frac {3 b^2 j^2 p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2 h}-\frac {3 b j (f i-e j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2 h}-\frac {3 b j (h i-g j) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h^2}-\frac {3 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}{4 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 )^3}{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 )^3}{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 )^3}{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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^3}+\frac {3 b (h i-g j)^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}-\frac {6 b^2 (h i-g j)^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3}+\frac {6 b^3 (h i-g j)^2 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4056\) vs. \(2(742)=1484\).

Time = 0.89 (sec) , antiderivative size = 4056, normalized size of antiderivative = 5.47 \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-48*a^2*b*e*f*h^2*i*j*p*q + 24*a^2*b*e*f*g*h*j^2*p*q + 16*a^3*f^2*h^2*i*j*x - 8*a^3*f^2*g*h*j^2*x - 48*a^2*b*
f^2*h^2*i*j*p*q*x + 24*a^2*b*f^2*g*h*j^2*p*q*x + 12*a^2*b*e*f*h^2*j^2*p*q*x + 96*a*b^2*f^2*h^2*i*j*p^2*q^2*x -
48*a*b^2*f^2*g*h*j^2*p^2*q^2*x - 36*a*b^2*e*f*h^2*j^2*p^2*q^2*x - 96*b^3*f^2*h^2*i*j*p^3*q^3*x + 48*b^3*f^2*g
*h*j^2*p^3*q^3*x + 42*b^3*e*f*h^2*j^2*p^3*q^3*x + 4*a^3*f^2*h^2*j^2*x^2 - 6*a^2*b*f^2*h^2*j^2*p*q*x^2 + 6*a*b^
2*f^2*h^2*j^2*p^2*q^2*x^2 - 3*b^3*f^2*h^2*j^2*p^3*q^3*x^2 + 48*a^2*b*e*f*h^2*i*j*p*q*Log[e + f*x] - 24*a^2*b*e
*f*g*h*j^2*p*q*Log[e + f*x] - 12*a^2*b*e^2*h^2*j^2*p*q*Log[e + f*x] + 36*a*b^2*e^2*h^2*j^2*p^2*q^2*Log[e + f*x
] + 96*b^3*e*f*h^2*i*j*p^3*q^3*Log[e + f*x] - 48*b^3*e*f*g*h*j^2*p^3*q^3*Log[e + f*x] - 42*b^3*e^2*h^2*j^2*p^3
*q^3*Log[e + f*x] - 48*a*b^2*e*f*h^2*i*j*p^2*q^2*Log[e + f*x]^2 + 24*a*b^2*e*f*g*h*j^2*p^2*q^2*Log[e + f*x]^2
+ 12*a*b^2*e^2*h^2*j^2*p^2*q^2*Log[e + f*x]^2 - 18*b^3*e^2*h^2*j^2*p^3*q^3*Log[e + f*x]^2 + 16*b^3*e*f*h^2*i*j
*p^3*q^3*Log[e + f*x]^3 - 8*b^3*e*f*g*h*j^2*p^3*q^3*Log[e + f*x]^3 - 4*b^3*e^2*h^2*j^2*p^3*q^3*Log[e + f*x]^3
- 96*a*b^2*e*f*h^2*i*j*p*q*Log[c*(d*(e + f*x)^p)^q] + 48*a*b^2*e*f*g*h*j^2*p*q*Log[c*(d*(e + f*x)^p)^q] + 48*a
^2*b*f^2*h^2*i*j*x*Log[c*(d*(e + f*x)^p)^q] - 24*a^2*b*f^2*g*h*j^2*x*Log[c*(d*(e + f*x)^p)^q] - 96*a*b^2*f^2*h
^2*i*j*p*q*x*Log[c*(d*(e + f*x)^p)^q] + 48*a*b^2*f^2*g*h*j^2*p*q*x*Log[c*(d*(e + f*x)^p)^q] + 24*a*b^2*e*f*h^2
*j^2*p*q*x*Log[c*(d*(e + f*x)^p)^q] + 96*b^3*f^2*h^2*i*j*p^2*q^2*x*Log[c*(d*(e + f*x)^p)^q] - 48*b^3*f^2*g*h*j
^2*p^2*q^2*x*Log[c*(d*(e + f*x)^p)^q] - 36*b^3*e*f*h^2*j^2*p^2*q^2*x*Log[c*(d*(e + f*x)^p)^q] + 12*a^2*b*f^2*h
^2*j^2*x^2*Log[c*(d*(e + f*x)^p)^q] - 12*a*b^2*f^2*h^2*j^2*p*q*x^2*Log[c*(d*(e + f*x)^p)^q] + 6*b^3*f^2*h^2*j^
2*p^2*q^2*x^2*Log[c*(d*(e + f*x)^p)^q] + 96*a*b^2*e*f*h^2*i*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] - 48*a
*b^2*e*f*g*h*j^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] - 24*a*b^2*e^2*h^2*j^2*p*q*Log[e + f*x]*Log[c*(d*(e
+ f*x)^p)^q] + 36*b^3*e^2*h^2*j^2*p^2*q^2*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] - 48*b^3*e*f*h^2*i*j*p^2*q^2*
Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q] + 24*b^3*e*f*g*h*j^2*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q] +
12*b^3*e^2*h^2*j^2*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q] - 48*b^3*e*f*h^2*i*j*p*q*Log[c*(d*(e + f*x
)^p)^q]^2 + 24*b^3*e*f*g*h*j^2*p*q*Log[c*(d*(e + f*x)^p)^q]^2 + 48*a*b^2*f^2*h^2*i*j*x*Log[c*(d*(e + f*x)^p)^q
]^2 - 24*a*b^2*f^2*g*h*j^2*x*Log[c*(d*(e + f*x)^p)^q]^2 - 48*b^3*f^2*h^2*i*j*p*q*x*Log[c*(d*(e + f*x)^p)^q]^2
+ 24*b^3*f^2*g*h*j^2*p*q*x*Log[c*(d*(e + f*x)^p)^q]^2 + 12*b^3*e*f*h^2*j^2*p*q*x*Log[c*(d*(e + f*x)^p)^q]^2 +
12*a*b^2*f^2*h^2*j^2*x^2*Log[c*(d*(e + f*x)^p)^q]^2 - 6*b^3*f^2*h^2*j^2*p*q*x^2*Log[c*(d*(e + f*x)^p)^q]^2 + 4
8*b^3*e*f*h^2*i*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2 - 24*b^3*e*f*g*h*j^2*p*q*Log[e + f*x]*Log[c*(d*(
e + f*x)^p)^q]^2 - 12*b^3*e^2*h^2*j^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2 + 16*b^3*f^2*h^2*i*j*x*Log[c
*(d*(e + f*x)^p)^q]^3 - 8*b^3*f^2*g*h*j^2*x*Log[c*(d*(e + f*x)^p)^q]^3 + 4*b^3*f^2*h^2*j^2*x^2*Log[c*(d*(e + f
*x)^p)^q]^3 + 8*a^3*f^2*h^2*i^2*Log[g + h*x] - 16*a^3*f^2*g*h*i*j*Log[g + h*x] + 8*a^3*f^2*g^2*j^2*Log[g + h*x
] - 24*a^2*b*f^2*h^2*i^2*p*q*Log[e + f*x]*Log[g + h*x] + 48*a^2*b*f^2*g*h*i*j*p*q*Log[e + f*x]*Log[g + h*x] -
24*a^2*b*f^2*g^2*j^2*p*q*Log[e + f*x]*Log[g + h*x] + 24*a*b^2*f^2*h^2*i^2*p^2*q^2*Log[e + f*x]^2*Log[g + h*x]
- 48*a*b^2*f^2*g*h*i*j*p^2*q^2*Log[e + f*x]^2*Log[g + h*x] + 24*a*b^2*f^2*g^2*j^2*p^2*q^2*Log[e + f*x]^2*Log[g
+ h*x] - 8*b^3*f^2*h^2*i^2*p^3*q^3*Log[e + f*x]^3*Log[g + h*x] + 16*b^3*f^2*g*h*i*j*p^3*q^3*Log[e + f*x]^3*Lo
g[g + h*x] - 8*b^3*f^2*g^2*j^2*p^3*q^3*Log[e + f*x]^3*Log[g + h*x] + 24*a^2*b*f^2*h^2*i^2*Log[c*(d*(e + f*x)^p
)^q]*Log[g + h*x] - 48*a^2*b*f^2*g*h*i*j*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 24*a^2*b*f^2*g^2*j^2*Log[c*(d
*(e + f*x)^p)^q]*Log[g + h*x] - 48*a*b^2*f^2*h^2*i^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] +
96*a*b^2*f^2*g*h*i*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] - 48*a*b^2*f^2*g^2*j^2*p*q*Log[e +
f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 24*b^3*f^2*h^2*i^2*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^
q]*Log[g + h*x] - 48*b^3*f^2*g*h*i*j*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 24*b^3*f^2
*g^2*j^2*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 24*a*b^2*f^2*h^2*i^2*Log[c*(d*(e + f*x
)^p)^q]^2*Log[g + h*x] - 48*a*b^2*f^2*g*h*i*j*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] + 24*a*b^2*f^2*g^2*j^2*L
og[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] - 24*b^3*f^2*h^2*i^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[g
+ h*x] + 48*b^3*f^2*g*h*i*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] - 24*b^3*f^2*g^2*j^2*p*q*
Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] + 8*b^3*f^2*h^2*i^2*Log[c*(d*(e + f*x)^p)^q]^3*Log[g + h*
x] - 16*b^3*f^2*g*h*i*j*Log[c*(d*(e + f*x)^p)^q]^3*Log[g + h*x] + 8*b^3*f^2*g^2*j^2*Log[c*(d*(e + f*x)^p)^q]^3
*Log[g + h*x] + 24*a^2*b*f^2*h^2*i^2*p*q*Log[e + f*x]*Log[(f*(g + h*x))/(f*g - e*h)] - 48*a^2*b*f^2*g*h*i*j*p*
q*Log[e + f*x]*Log[(f*(g + h*x))/(f*g - e*h)] + 24*a^2*b*f^2*g^2*j^2*p*q*Log[e + f*x]*Log[(f*(g + h*x))/(f*g -
e*h)] - 24*a*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)] + 48*a*b^2*f^2*g*h*i*j*p^2
*q^2*Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] - 24*a*b^2*f^2*g^2*j^2*p^2*q^2*Log[e + f*x]^2*Log[(f*(g + h
*x))/(f*g - e*h)] + 8*b^3*f^2*h^2*i^2*p^3*q^3*Log[e + f*x]^3*Log[(f*(g + h*x))/(f*g - e*h)] - 16*b^3*f^2*g*h*i
*j*p^3*q^3*Log[e + f*x]^3*Log[(f*(g + h*x))/(f*g - e*h)] + 8*b^3*f^2*g^2*j^2*p^3*q^3*Log[e + f*x]^3*Log[(f*(g
+ h*x))/(f*g - e*h)] + 48*a*b^2*f^2*h^2*i^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g -
e*h)] - 96*a*b^2*f^2*g*h*i*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 48*a*
b^2*f^2*g^2*j^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] - 24*b^3*f^2*h^2*i^2*
p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 48*b^3*f^2*g*h*i*j*p^2*q^2*Lo
g[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] - 24*b^3*f^2*g^2*j^2*p^2*q^2*Log[e + f*x]
^2*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 24*b^3*f^2*h^2*i^2*p*q*Log[e + f*x]*Log[c*(d*(e +
f*x)^p)^q]^2*Log[(f*(g + h*x))/(f*g - e*h)] - 48*b^3*f^2*g*h*i*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*
Log[(f*(g + h*x))/(f*g - e*h)] + 24*b^3*f^2*g^2*j^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*
x))/(f*g - e*h)] + 24*b*f^2*(h*i - g*j)^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, (h*(e + f*x))/(-(f
*g) + e*h)] - 48*b^2*f^2*(h*i - g*j)^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, (h*(e + f*x))/(-(f*
g) + e*h)] + 48*b^3*f^2*h^2*i^2*p^3*q^3*PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)] - 96*b^3*f^2*g*h*i*j*p^3*q^3*
PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)] + 48*b^3*f^2*g^2*j^2*p^3*q^3*PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)]
)/(8*f^2*h^3)

Maple [F]

\[\int \frac {\left (j x +i \right )^{2} {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}{h x +g}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g} \,d x } \]

[In]

integrate((j*x+i)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3/(h*x+g),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3} \left (i + j x\right )^{2}}{g + h x}\, dx \]

[In]

integrate((j*x+i)**2*(a+b*ln(c*(d*(f*x+e)**p)**q))**3/(h*x+g),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**3*(i + j*x)**2/(g + h*x), x)

Maxima [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g} \,d x } \]

[In]

integrate((j*x+i)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3/(h*x+g),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (j x + i\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g} \,d x } \]

[In]

integrate((j*x+i)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3/(h*x+g),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(i+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\int \frac {{\left (i+j\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{g+h\,x} \,d x \]

[In]

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

[Out]

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