3.184 \(\int \frac{(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\)

Optimal. Leaf size=464 \[ \frac{i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))^2}{2 d f^4}-\frac{3 b i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))}{2 d f^4}+\frac{(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}-\frac{2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac{2 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac{2 b i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac{2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^4}+\frac{(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac{4 a b i x (f h-e i)^2}{d f^3}-\frac{4 b^2 i (e+f x) (f h-e i)^2 \log (c (e+f x))}{d f^4}+\frac{3 b^2 i^2 (e+f x)^2 (f h-e i)}{4 d f^4}+\frac{6 b^2 i x (f h-e i)^2}{d f^3}+\frac{b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}+\frac{2 b^2 i^3 (e+f x)^3}{27 d f^4} \]

[Out]

(-4*a*b*i*(f*h - e*i)^2*x)/(d*f^3) + (6*b^2*i*(f*h - e*i)^2*x)/(d*f^3) + (3*b^2*i^2*(f*h - e*i)*(e + f*x)^2)/(
4*d*f^4) + (2*b^2*i^3*(e + f*x)^3)/(27*d*f^4) + (b^2*(f*h - e*i)^3*Log[e + f*x]^2)/(3*d*f^4) - (4*b^2*i*(f*h -
e*i)^2*(e + f*x)*Log[c*(e + f*x)])/(d*f^4) - (2*b*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^4)
- (3*b*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^4) - (2*b*i^3*(e + f*x)^3*(a + b*Log[c*(e
+ f*x)]))/(9*d*f^4) - (2*b*(f*h - e*i)^3*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(3*d*f^4) + (2*i*(f*h - e*i)^
2*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^4) + (i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(2*
d*f^4) + ((h + i*x)^3*(a + b*Log[c*(e + f*x)])^2)/(3*d*f) + ((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*
f^4)

________________________________________________________________________________________

Rubi [A]  time = 0.981334, antiderivative size = 459, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 15, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.469, Rules used = {2411, 12, 2346, 2302, 30, 2296, 2295, 2330, 2305, 2304, 2319, 43, 2334, 14, 2301} \[ \frac{i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))^2}{2 d f^4}-\frac{b i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))}{2 d f^4}-\frac{b \left (\frac{9 i^2 (e+f x)^2 (f h-e i)}{f^2}+\frac{18 i (e+f x) (f h-e i)^2}{f^2}+\frac{6 (f h-e i)^3 \log (e+f x)}{f^2}+\frac{2 i^3 (e+f x)^3}{f^2}\right ) (a+b \log (c (e+f x)))}{9 d f^2}+\frac{(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac{2 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))^2}{d f^4}+\frac{(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac{4 a b i x (f h-e i)^2}{d f^3}-\frac{4 b^2 i (e+f x) (f h-e i)^2 \log (c (e+f x))}{d f^4}+\frac{3 b^2 i^2 (e+f x)^2 (f h-e i)}{4 d f^4}+\frac{6 b^2 i x (f h-e i)^2}{d f^3}+\frac{b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}+\frac{2 b^2 i^3 (e+f x)^3}{27 d f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*i*(f*h - e*i)^2*x)/(d*f^3) + (6*b^2*i*(f*h - e*i)^2*x)/(d*f^3) + (3*b^2*i^2*(f*h - e*i)*(e + f*x)^2)/(
4*d*f^4) + (2*b^2*i^3*(e + f*x)^3)/(27*d*f^4) + (b^2*(f*h - e*i)^3*Log[e + f*x]^2)/(3*d*f^4) - (4*b^2*i*(f*h -
e*i)^2*(e + f*x)*Log[c*(e + f*x)])/(d*f^4) - (b*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^
4) - (b*((18*i*(f*h - e*i)^2*(e + f*x))/f^2 + (9*i^2*(f*h - e*i)*(e + f*x)^2)/f^2 + (2*i^3*(e + f*x)^3)/f^2 +
(6*(f*h - e*i)^3*Log[e + f*x])/f^2)*(a + b*Log[c*(e + f*x)]))/(9*d*f^2) + (2*i*(f*h - e*i)^2*(e + f*x)*(a + b*
Log[c*(e + f*x)])^2)/(d*f^4) + (i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(2*d*f^4) + ((h + i*x)
^3*(a + b*Log[c*(e + f*x)])^2)/(3*d*f) + ((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^4)

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right )^3 (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right )^3 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{184 \operatorname{Subst}\left (\int \left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right )^2 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}-\frac{(184 e-f h) \operatorname{Subst}\left (\int \frac{\left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right )^2 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{3 d f}-\frac{(184 (184 e-f h)) \operatorname{Subst}\left (\int \left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right ) (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac{(184 e-f h)^2 \operatorname{Subst}\left (\int \frac{\left (\frac{-184 e+f h}{f}+\frac{184 x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^3}\\ &=-\frac{2 b \left (\frac{1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac{152352 (184 e-f h) (e+f x)^2}{f^3}+\frac{6229504 (e+f x)^3}{f^3}-\frac{3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{184 x \left (304704 e^2+9 f^2 h^2+828 f h x+33856 x^2-3312 e (f h+46 x)\right )-3 (184 e-f h)^3 \log (x)}{3 f^3 x} \, dx,x,e+f x\right )}{3 d f}-\frac{(184 (184 e-f h)) \operatorname{Subst}\left (\int \left (\frac{(-184 e+f h) (a+b \log (c x))^2}{f}+\frac{184 x (a+b \log (c x))^2}{f}\right ) \, dx,x,e+f x\right )}{d f^3}+\frac{\left (184 (184 e-f h)^2\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}-\frac{(184 e-f h)^3 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^4}\\ &=-\frac{2 b \left (\frac{1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac{152352 (184 e-f h) (e+f x)^2}{f^3}+\frac{6229504 (e+f x)^3}{f^3}-\frac{3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac{184 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{184 x \left (304704 e^2+9 f^2 h^2+828 f h x+33856 x^2-3312 e (f h+46 x)\right )-3 (184 e-f h)^3 \log (x)}{x} \, dx,x,e+f x\right )}{9 d f^4}-\frac{(33856 (184 e-f h)) \operatorname{Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}+\frac{\left (184 (184 e-f h)^2\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}-\frac{\left (368 b (184 e-f h)^2\right ) \operatorname{Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac{(184 e-f h)^3 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^4}\\ &=-\frac{368 a b (184 e-f h)^2 x}{d f^3}-\frac{2 b \left (\frac{1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac{152352 (184 e-f h) (e+f x)^2}{f^3}+\frac{6229504 (e+f x)^3}{f^3}-\frac{3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac{368 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}-\frac{16928 (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac{(184 e-f h)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \left (184 \left (9 (184 e-f h)^2-828 (184 e-f h) x+33856 x^2\right )-\frac{3 (184 e-f h)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{9 d f^4}+\frac{(33856 b (184 e-f h)) \operatorname{Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac{\left (368 b (184 e-f h)^2\right ) \operatorname{Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac{\left (368 b^2 (184 e-f h)^2\right ) \operatorname{Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^4}\\ &=-\frac{736 a b (184 e-f h)^2 x}{d f^3}+\frac{368 b^2 (184 e-f h)^2 x}{d f^3}-\frac{8464 b^2 (184 e-f h) (e+f x)^2}{d f^4}-\frac{368 b^2 (184 e-f h)^2 (e+f x) \log (c (e+f x))}{d f^4}+\frac{16928 b (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac{2 b \left (\frac{1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac{152352 (184 e-f h) (e+f x)^2}{f^3}+\frac{6229504 (e+f x)^3}{f^3}-\frac{3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac{368 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}-\frac{16928 (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac{(184 e-f h)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac{\left (368 b^2\right ) \operatorname{Subst}\left (\int \left (9 (184 e-f h)^2-828 (184 e-f h) x+33856 x^2\right ) \, dx,x,e+f x\right )}{9 d f^4}-\frac{\left (368 b^2 (184 e-f h)^2\right ) \operatorname{Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^4}-\frac{\left (2 b^2 (184 e-f h)^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,e+f x\right )}{3 d f^4}\\ &=-\frac{736 a b (184 e-f h)^2 x}{d f^3}+\frac{1104 b^2 (184 e-f h)^2 x}{d f^3}-\frac{25392 b^2 (184 e-f h) (e+f x)^2}{d f^4}+\frac{12459008 b^2 (e+f x)^3}{27 d f^4}-\frac{b^2 (184 e-f h)^3 \log ^2(e+f x)}{3 d f^4}-\frac{736 b^2 (184 e-f h)^2 (e+f x) \log (c (e+f x))}{d f^4}+\frac{16928 b (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac{2 b \left (\frac{1656 (184 e-f h)^2 (e+f x)}{f^3}-\frac{152352 (184 e-f h) (e+f x)^2}{f^3}+\frac{6229504 (e+f x)^3}{f^3}-\frac{3 (184 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{9 d f}+\frac{(h+184 x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac{368 (184 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}-\frac{16928 (184 e-f h) (e+f x)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac{(184 e-f h)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}\\ \end{align*}

Mathematica [A]  time = 0.305459, size = 267, normalized size = 0.58 \[ \frac{8 b i^3 \left (b f x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 (a+b \log (c (e+f x)))\right )+162 i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))^2+81 b i^2 (f h-e i) \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )+324 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))^2-648 b i (f h-e i)^2 (f x (a-b)+b (e+f x) \log (c (e+f x)))+\frac{36 (f h-e i)^3 (a+b \log (c (e+f x)))^3}{b}+36 i^3 (e+f x)^3 (a+b \log (c (e+f x)))^2}{108 d f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(324*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)])^2 + 162*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e +
f*x)])^2 + 36*i^3*(e + f*x)^3*(a + b*Log[c*(e + f*x)])^2 + (36*(f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^3)/b - 6
48*b*i*(f*h - e*i)^2*((a - b)*f*x + b*(e + f*x)*Log[c*(e + f*x)]) + 81*b*i^2*(f*h - e*i)*(b*f*x*(2*e + f*x) -
2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])) + 8*b*i^3*(b*f*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 3*(e + f*x)^3*(a + b*Lo
g[c*(e + f*x)])))/(108*d*f^4)

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Maple [B]  time = 0.064, size = 1485, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

575/108/f^4/d*b^2*e^3*i^3+11/6/f^4/d*a^2*e^3*i^3-1/3/f^4/d*b^2*e^3*i^3*ln(c*f*x+c*e)^3+11/6/f^4/d*b^2*e^3*i^3*
ln(c*f*x+c*e)^2-85/18/f^4/d*b^2*e^3*i^3*ln(c*f*x+c*e)-1/f^4/d*a^2*e^3*i^3*ln(c*f*x+c*e)+1/f/d*a*b*h^3*ln(c*f*x
+c*e)^2+1/3/f/d*b^2*i^3*ln(c*f*x+c*e)^2*x^3-6/f^2/d*a*b*e*h*i^2*ln(c*f*x+c*e)*x+2/3/f/d*a*b*i^3*ln(c*f*x+c*e)*
x^3+1/3/f/d*a^2*i^3*x^3+1/3/f/d*b^2*h^3*ln(c*f*x+c*e)^3+1/f/d*a^2*h^3*ln(c*f*x+c*e)+2/27/f/d*b^2*i^3*x^3-2/9/f
/d*b^2*i^3*ln(c*f*x+c*e)*x^3+1/f^3/d*a^2*e^2*i^3*x+3/f/d*a^2*h^2*i*x+6/f/d*b^2*h^2*i*x+3/2/f/d*a^2*h*i^2*x^2+3
/4/f/d*b^2*h*i^2*x^2-19/36/f^2/d*b^2*e*i^3*x^2+85/18/f^3/d*b^2*e^2*i^3*x-2/9/f/d*a*b*i^3*x^3-1/2/f^2/d*a^2*e*i
^3*x^2-11/3/f^3/d*b^2*e^2*i^3*ln(c*f*x+c*e)*x+5/6/f^2/d*b^2*e*i^3*ln(c*f*x+c*e)*x^2-3/f^2/d*a^2*e*h^2*i*ln(c*f
*x+c*e)+3/f/d*b^2*h^2*i*ln(c*f*x+c*e)^2*x+11/3/f^4/d*a*b*i^3*ln(c*f*x+c*e)*e^3+3/2/f/d*b^2*h*i^2*ln(c*f*x+c*e)
^2*x^2-1/2/f^2/d*b^2*e*i^3*ln(c*f*x+c*e)^2*x^2+1/f^3/d*b^2*e^2*i^3*ln(c*f*x+c*e)^2*x-6/f^2/d*b^2*h^2*i*ln(c*f*
x+c*e)*e-1/f^4/d*a*b*e^3*i^3*ln(c*f*x+c*e)^2+21/2/f^3/d*a*b*e^2*h*i^2+3/f^2/d*b^2*h^2*i*ln(c*f*x+c*e)^2*e-6/f/
d*b^2*h^2*i*ln(c*f*x+c*e)*x+1/f^3/d*b^2*e^2*h*i^2*ln(c*f*x+c*e)^3+3/f^3/d*a^2*e^2*h*i^2*ln(c*f*x+c*e)-1/f^2/d*
a*b*i^3*ln(c*f*x+c*e)*x^2*e+9/f^2/d*b^2*e*h*i^2*ln(c*f*x+c*e)*x+6/f/d*a*b*h^2*i*ln(c*f*x+c*e)*x-9/f^3/d*a*b*e^
2*h*i^2*ln(c*f*x+c*e)+9/f^2/d*a*b*e*h*i^2*x+2/f^3/d*a*b*i^3*ln(c*f*x+c*e)*x*e^2-3/f^2/d*a*b*e*h^2*i*ln(c*f*x+c
*e)^2+6/f^2/d*a*b*h^2*i*ln(c*f*x+c*e)*e+3/f/d*a*b*h*i^2*ln(c*f*x+c*e)*x^2+3/f^3/d*a*b*e^2*h*i^2*ln(c*f*x+c*e)^
2-3/f^2/d*b^2*e*h*i^2*ln(c*f*x+c*e)^2*x-6/f^2/d*a*b*e*h^2*i-3/f^2/d*a^2*e*h*i^2*x-11/3/f^3/d*a*b*i^3*x*e^2-21/
2/f^2/d*b^2*e*h*i^2*x-3/2/f/d*a*b*h*i^2*x^2-6/f/d*a*b*h^2*i*x+5/6/f^2/d*a*b*i^3*x^2*e-3/2/f/d*b^2*h*i^2*ln(c*f
*x+c*e)*x^2-9/2/f^3/d*b^2*e^2*h*i^2*ln(c*f*x+c*e)^2+3/f^2/d*a^2*e*h^2*i-9/2/f^3/d*a^2*e^2*h*i^2-45/4/f^3/d*b^2
*e^2*h*i^2-85/18/f^4/d*a*b*e^3*i^3+6/f^2/d*b^2*e*h^2*i+21/2/f^3/d*b^2*e^2*h*i^2*ln(c*f*x+c*e)-1/f^2/d*b^2*e*h^
2*i*ln(c*f*x+c*e)^3

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Maxima [B]  time = 1.35131, size = 1301, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*a*b*h^2*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) - 1/3*a*b*i^3*(6*e^3*log(f*x + e)/(d*f^4) - (2
*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3))*log(c*f*x + c*e) + 3*a*b*h*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 -
2*e*x)/(d*f^2))*log(c*f*x + c*e) - a*b*h^3*(2*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - (log(f*x + e)^2 + 2*l
og(f*x + e)*log(c))/(d*f)) + 3*a^2*h^2*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) - 1/6*a^2*i^3*(6*e^3*log(f*x + e)/
(d*f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3)) + 3/2*a^2*h*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2*
e*x)/(d*f^2)) + 1/3*b^2*h^3*log(c*f*x + c*e)^3/(d*f) + 2*a*b*h^3*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) + a^2
*h^3*log(d*f*x + d*e)/(d*f) + 3*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*a*b*h^2*i/(d*f^2) - 3/2*(f^2*x^2
+ 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*a*b*h*i^2/(d*f^3) - (c^2*e*log(c*f*x + c*e)^3 - 3*(c*f
*x + c*e)*(c*log(c*f*x + c*e)^2 - 2*c*log(c*f*x + c*e) + 2*c))*b^2*h^2*i/(c^2*d*f^2) - 1/18*(4*f^3*x^3 - 15*e*
f^2*x^2 - 18*e^3*log(f*x + e)^2 + 66*e^2*f*x - 66*e^3*log(f*x + e))*a*b*i^3/(d*f^4) + 1/4*(4*c^3*e^2*log(c*f*x
+ c*e)^3 + 3*(c*f*x + c*e)^2*(2*c*log(c*f*x + c*e)^2 - 2*c*log(c*f*x + c*e) + c) - 24*(c^2*e*log(c*f*x + c*e)
^2 - 2*c^2*e*log(c*f*x + c*e) + 2*c^2*e)*(c*f*x + c*e))*b^2*h*i^2/(c^3*d*f^3) - 1/108*(36*c^4*e^3*log(c*f*x +
c*e)^3 - 4*(c*f*x + c*e)^3*(9*c*log(c*f*x + c*e)^2 - 6*c*log(c*f*x + c*e) + 2*c) + 81*(2*c^2*e*log(c*f*x + c*e
)^2 - 2*c^2*e*log(c*f*x + c*e) + c^2*e)*(c*f*x + c*e)^2 - 324*(c^3*e^2*log(c*f*x + c*e)^2 - 2*c^3*e^2*log(c*f*
x + c*e) + 2*c^3*e^2)*(c*f*x + c*e))*b^2*i^3/(c^4*d*f^4)

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Fricas [A]  time = 1.80296, size = 1278, normalized size = 2.75 \begin{align*} \frac{4 \,{\left (9 \, a^{2} - 6 \, a b + 2 \, b^{2}\right )} f^{3} i^{3} x^{3} + 36 \,{\left (b^{2} f^{3} h^{3} - 3 \, b^{2} e f^{2} h^{2} i + 3 \, b^{2} e^{2} f h i^{2} - b^{2} e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{3} + 3 \,{\left (27 \,{\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{3} h i^{2} -{\left (18 \, a^{2} - 30 \, a b + 19 \, b^{2}\right )} e f^{2} i^{3}\right )} x^{2} + 18 \,{\left (2 \, b^{2} f^{3} i^{3} x^{3} + 6 \, a b f^{3} h^{3} - 18 \,{\left (a b - b^{2}\right )} e f^{2} h^{2} i + 9 \,{\left (2 \, a b - 3 \, b^{2}\right )} e^{2} f h i^{2} -{\left (6 \, a b - 11 \, b^{2}\right )} e^{3} i^{3} + 3 \,{\left (3 \, b^{2} f^{3} h i^{2} - b^{2} e f^{2} i^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} f^{3} h^{2} i - 3 \, b^{2} e f^{2} h i^{2} + b^{2} e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 6 \,{\left (54 \,{\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{3} h^{2} i - 27 \,{\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f^{2} h i^{2} +{\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{2} f i^{3}\right )} x + 6 \,{\left (4 \,{\left (3 \, a b - b^{2}\right )} f^{3} i^{3} x^{3} + 18 \, a^{2} f^{3} h^{3} - 54 \,{\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f^{2} h^{2} i + 27 \,{\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} f h i^{2} -{\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{3} i^{3} + 3 \,{\left (9 \,{\left (2 \, a b - b^{2}\right )} f^{3} h i^{2} -{\left (6 \, a b - 5 \, b^{2}\right )} e f^{2} i^{3}\right )} x^{2} + 6 \,{\left (18 \,{\left (a b - b^{2}\right )} f^{3} h^{2} i - 9 \,{\left (2 \, a b - 3 \, b^{2}\right )} e f^{2} h i^{2} +{\left (6 \, a b - 11 \, b^{2}\right )} e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )}{108 \, d f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(4*(9*a^2 - 6*a*b + 2*b^2)*f^3*i^3*x^3 + 36*(b^2*f^3*h^3 - 3*b^2*e*f^2*h^2*i + 3*b^2*e^2*f*h*i^2 - b^2*e
^3*i^3)*log(c*f*x + c*e)^3 + 3*(27*(2*a^2 - 2*a*b + b^2)*f^3*h*i^2 - (18*a^2 - 30*a*b + 19*b^2)*e*f^2*i^3)*x^2
+ 18*(2*b^2*f^3*i^3*x^3 + 6*a*b*f^3*h^3 - 18*(a*b - b^2)*e*f^2*h^2*i + 9*(2*a*b - 3*b^2)*e^2*f*h*i^2 - (6*a*b
- 11*b^2)*e^3*i^3 + 3*(3*b^2*f^3*h*i^2 - b^2*e*f^2*i^3)*x^2 + 6*(3*b^2*f^3*h^2*i - 3*b^2*e*f^2*h*i^2 + b^2*e^
2*f*i^3)*x)*log(c*f*x + c*e)^2 + 6*(54*(a^2 - 2*a*b + 2*b^2)*f^3*h^2*i - 27*(2*a^2 - 6*a*b + 7*b^2)*e*f^2*h*i^
2 + (18*a^2 - 66*a*b + 85*b^2)*e^2*f*i^3)*x + 6*(4*(3*a*b - b^2)*f^3*i^3*x^3 + 18*a^2*f^3*h^3 - 54*(a^2 - 2*a*
b + 2*b^2)*e*f^2*h^2*i + 27*(2*a^2 - 6*a*b + 7*b^2)*e^2*f*h*i^2 - (18*a^2 - 66*a*b + 85*b^2)*e^3*i^3 + 3*(9*(2
*a*b - b^2)*f^3*h*i^2 - (6*a*b - 5*b^2)*e*f^2*i^3)*x^2 + 6*(18*(a*b - b^2)*f^3*h^2*i - 9*(2*a*b - 3*b^2)*e*f^2
*h*i^2 + (6*a*b - 11*b^2)*e^2*f*i^3)*x)*log(c*f*x + c*e))/(d*f^4)

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Sympy [B]  time = 4.22983, size = 865, normalized size = 1.86 \begin{align*} \frac{x^{3} \left (9 a^{2} i^{3} - 6 a b i^{3} + 2 b^{2} i^{3}\right )}{27 d f} - \frac{x^{2} \left (18 a^{2} e i^{3} - 54 a^{2} f h i^{2} - 30 a b e i^{3} + 54 a b f h i^{2} + 19 b^{2} e i^{3} - 27 b^{2} f h i^{2}\right )}{36 d f^{2}} + \frac{x \left (18 a^{2} e^{2} i^{3} - 54 a^{2} e f h i^{2} + 54 a^{2} f^{2} h^{2} i - 66 a b e^{2} i^{3} + 162 a b e f h i^{2} - 108 a b f^{2} h^{2} i + 85 b^{2} e^{2} i^{3} - 189 b^{2} e f h i^{2} + 108 b^{2} f^{2} h^{2} i\right )}{18 d f^{3}} + \frac{\left (36 a b e^{2} i^{3} x - 108 a b e f h i^{2} x - 18 a b e f i^{3} x^{2} + 108 a b f^{2} h^{2} i x + 54 a b f^{2} h i^{2} x^{2} + 12 a b f^{2} i^{3} x^{3} - 66 b^{2} e^{2} i^{3} x + 162 b^{2} e f h i^{2} x + 15 b^{2} e f i^{3} x^{2} - 108 b^{2} f^{2} h^{2} i x - 27 b^{2} f^{2} h i^{2} x^{2} - 4 b^{2} f^{2} i^{3} x^{3}\right ) \log{\left (c \left (e + f x\right ) \right )}}{18 d f^{3}} + \frac{\left (- b^{2} e^{3} i^{3} + 3 b^{2} e^{2} f h i^{2} - 3 b^{2} e f^{2} h^{2} i + b^{2} f^{3} h^{3}\right ) \log{\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{4}} - \frac{\left (18 a^{2} e^{3} i^{3} - 54 a^{2} e^{2} f h i^{2} + 54 a^{2} e f^{2} h^{2} i - 18 a^{2} f^{3} h^{3} - 66 a b e^{3} i^{3} + 162 a b e^{2} f h i^{2} - 108 a b e f^{2} h^{2} i + 85 b^{2} e^{3} i^{3} - 189 b^{2} e^{2} f h i^{2} + 108 b^{2} e f^{2} h^{2} i\right ) \log{\left (e + f x \right )}}{18 d f^{4}} + \frac{\left (- 6 a b e^{3} i^{3} + 18 a b e^{2} f h i^{2} - 18 a b e f^{2} h^{2} i + 6 a b f^{3} h^{3} + 11 b^{2} e^{3} i^{3} - 27 b^{2} e^{2} f h i^{2} + 6 b^{2} e^{2} f i^{3} x + 18 b^{2} e f^{2} h^{2} i - 18 b^{2} e f^{2} h i^{2} x - 3 b^{2} e f^{2} i^{3} x^{2} + 18 b^{2} f^{3} h^{2} i x + 9 b^{2} f^{3} h i^{2} x^{2} + 2 b^{2} f^{3} i^{3} x^{3}\right ) \log{\left (c \left (e + f x\right ) \right )}^{2}}{6 d f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(9*a**2*i**3 - 6*a*b*i**3 + 2*b**2*i**3)/(27*d*f) - x**2*(18*a**2*e*i**3 - 54*a**2*f*h*i**2 - 30*a*b*e*i*
*3 + 54*a*b*f*h*i**2 + 19*b**2*e*i**3 - 27*b**2*f*h*i**2)/(36*d*f**2) + x*(18*a**2*e**2*i**3 - 54*a**2*e*f*h*i
**2 + 54*a**2*f**2*h**2*i - 66*a*b*e**2*i**3 + 162*a*b*e*f*h*i**2 - 108*a*b*f**2*h**2*i + 85*b**2*e**2*i**3 -
189*b**2*e*f*h*i**2 + 108*b**2*f**2*h**2*i)/(18*d*f**3) + (36*a*b*e**2*i**3*x - 108*a*b*e*f*h*i**2*x - 18*a*b*
e*f*i**3*x**2 + 108*a*b*f**2*h**2*i*x + 54*a*b*f**2*h*i**2*x**2 + 12*a*b*f**2*i**3*x**3 - 66*b**2*e**2*i**3*x
+ 162*b**2*e*f*h*i**2*x + 15*b**2*e*f*i**3*x**2 - 108*b**2*f**2*h**2*i*x - 27*b**2*f**2*h*i**2*x**2 - 4*b**2*f
**2*i**3*x**3)*log(c*(e + f*x))/(18*d*f**3) + (-b**2*e**3*i**3 + 3*b**2*e**2*f*h*i**2 - 3*b**2*e*f**2*h**2*i +
b**2*f**3*h**3)*log(c*(e + f*x))**3/(3*d*f**4) - (18*a**2*e**3*i**3 - 54*a**2*e**2*f*h*i**2 + 54*a**2*e*f**2*
h**2*i - 18*a**2*f**3*h**3 - 66*a*b*e**3*i**3 + 162*a*b*e**2*f*h*i**2 - 108*a*b*e*f**2*h**2*i + 85*b**2*e**3*i
**3 - 189*b**2*e**2*f*h*i**2 + 108*b**2*e*f**2*h**2*i)*log(e + f*x)/(18*d*f**4) + (-6*a*b*e**3*i**3 + 18*a*b*e
**2*f*h*i**2 - 18*a*b*e*f**2*h**2*i + 6*a*b*f**3*h**3 + 11*b**2*e**3*i**3 - 27*b**2*e**2*f*h*i**2 + 6*b**2*e**
2*f*i**3*x + 18*b**2*e*f**2*h**2*i - 18*b**2*e*f**2*h*i**2*x - 3*b**2*e*f**2*i**3*x**2 + 18*b**2*f**3*h**2*i*x
+ 9*b**2*f**3*h*i**2*x**2 + 2*b**2*f**3*i**3*x**3)*log(c*(e + f*x))**2/(6*d*f**4)

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Giac [B]  time = 1.18875, size = 1405, normalized size = 3.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(324*b^2*f^3*h^2*i*x*log(c*f*x + c*e)^2 - 36*b^2*f^3*i*x^3*log(c*f*x + c*e)^2 + 36*b^2*f^3*h^3*log(c*f*x
+ c*e)^3 - 108*b^2*f^2*h^2*i*e*log(c*f*x + c*e)^3 + 648*a*b*f^3*h^2*i*x*log(c*f*x + c*e) - 648*b^2*f^3*h^2*i*
x*log(c*f*x + c*e) - 72*a*b*f^3*i*x^3*log(c*f*x + c*e) + 24*b^2*f^3*i*x^3*log(c*f*x + c*e) + 108*a*b*f^3*h^3*l
og(c*f*x + c*e)^2 - 162*b^2*f^3*h*x^2*log(c*f*x + c*e)^2 - 324*a*b*f^2*h^2*i*e*log(c*f*x + c*e)^2 + 324*b^2*f^
2*h^2*i*e*log(c*f*x + c*e)^2 + 54*b^2*f^2*i*x^2*e*log(c*f*x + c*e)^2 + 324*a^2*f^3*h^2*i*x - 648*a*b*f^3*h^2*i
*x + 648*b^2*f^3*h^2*i*x - 36*a^2*f^3*i*x^3 + 24*a*b*f^3*i*x^3 - 8*b^2*f^3*i*x^3 - 324*a*b*f^3*h*x^2*log(c*f*x
+ c*e) + 162*b^2*f^3*h*x^2*log(c*f*x + c*e) + 108*a*b*f^2*i*x^2*e*log(c*f*x + c*e) - 90*b^2*f^2*i*x^2*e*log(c
*f*x + c*e) + 324*b^2*f^2*h*x*e*log(c*f*x + c*e)^2 + 108*a^2*f^3*h^3*log(f*x + e) - 324*a^2*f^2*h^2*i*e*log(f*
x + e) + 648*a*b*f^2*h^2*i*e*log(f*x + e) - 648*b^2*f^2*h^2*i*e*log(f*x + e) - 162*a^2*f^3*h*x^2 + 162*a*b*f^3
*h*x^2 - 81*b^2*f^3*h*x^2 + 54*a^2*f^2*i*x^2*e - 90*a*b*f^2*i*x^2*e + 57*b^2*f^2*i*x^2*e + 648*a*b*f^2*h*x*e*l
og(c*f*x + c*e) - 972*b^2*f^2*h*x*e*log(c*f*x + c*e) - 108*b^2*f*i*x*e^2*log(c*f*x + c*e)^2 - 108*b^2*f*h*e^2*
log(c*f*x + c*e)^3 + 324*a^2*f^2*h*x*e - 972*a*b*f^2*h*x*e + 1134*b^2*f^2*h*x*e - 216*a*b*f*i*x*e^2*log(c*f*x
+ c*e) + 396*b^2*f*i*x*e^2*log(c*f*x + c*e) - 324*a*b*f*h*e^2*log(c*f*x + c*e)^2 + 486*b^2*f*h*e^2*log(c*f*x +
c*e)^2 + 36*b^2*i*e^3*log(c*f*x + c*e)^3 - 108*a^2*f*i*x*e^2 + 396*a*b*f*i*x*e^2 - 510*b^2*f*i*x*e^2 + 108*a*
b*i*e^3*log(c*f*x + c*e)^2 - 198*b^2*i*e^3*log(c*f*x + c*e)^2 - 324*a^2*f*h*e^2*log(f*x + e) + 972*a*b*f*h*e^2
*log(f*x + e) - 1134*b^2*f*h*e^2*log(f*x + e) + 108*a^2*i*e^3*log(f*x + e) - 396*a*b*i*e^3*log(f*x + e) + 510*
b^2*i*e^3*log(f*x + e))/(d*f^4)