\(\int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) [184]

Optimal result

Integrand size = 32, antiderivative size = 464 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {4 a b i (f h-e i)^2 x}{d f^3}+\frac {6 b^2 i (f h-e i)^2 x}{d f^3}+\frac {3 b^2 i^2 (f h-e i) (e+f x)^2}{4 d f^4}+\frac {2 b^2 i^3 (e+f x)^3}{27 d f^4}+\frac {b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}-\frac {4 b^2 i (f h-e i)^2 (e+f x) \log (c (e+f x))}{d f^4}-\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {3 b i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}-\frac {2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^4}-\frac {2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac {2 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac {i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4} \]

[Out]

-4*a*b*i*(-e*i+f*h)^2*x/d/f^3+6*b^2*i*(-e*i+f*h)^2*x/d/f^3+3/4*b^2*i^2*(-e*i+f*h)*(f*x+e)^2/d/f^4+2/27*b^2*i^3
*(f*x+e)^3/d/f^4+1/3*b^2*(-e*i+f*h)^3*ln(f*x+e)^2/d/f^4-4*b^2*i*(-e*i+f*h)^2*(f*x+e)*ln(c*(f*x+e))/d/f^4-2*b*i
*(-e*i+f*h)^2*(f*x+e)*(a+b*ln(c*(f*x+e)))/d/f^4-3/2*b*i^2*(-e*i+f*h)*(f*x+e)^2*(a+b*ln(c*(f*x+e)))/d/f^4-2/9*b
*i^3*(f*x+e)^3*(a+b*ln(c*(f*x+e)))/d/f^4-2/3*b*(-e*i+f*h)^3*ln(f*x+e)*(a+b*ln(c*(f*x+e)))/d/f^4+2*i*(-e*i+f*h)
^2*(f*x+e)*(a+b*ln(c*(f*x+e)))^2/d/f^4+1/2*i^2*(-e*i+f*h)*(f*x+e)^2*(a+b*ln(c*(f*x+e)))^2/d/f^4+1/3*(i*x+h)^3*
(a+b*ln(c*(f*x+e)))^2/d/f+1/3*(-e*i+f*h)^3*(a+b*ln(c*(f*x+e)))^3/b/d/f^4

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2458, 12, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341, 2356, 45, 2372, 14, 2338} \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))^2}{2 d f^4}-\frac {3 b i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))}{2 d f^4}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}-\frac {2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac {2 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))^2}{d f^4}-\frac {2 b i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^4}-\frac {2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac {4 a b i x (f h-e i)^2}{d f^3}-\frac {4 b^2 i (e+f x) (f h-e i)^2 \log (c (e+f x))}{d f^4}+\frac {3 b^2 i^2 (e+f x)^2 (f h-e i)}{4 d f^4}+\frac {b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}+\frac {2 b^2 i^3 (e+f x)^3}{27 d f^4}+\frac {6 b^2 i x (f h-e i)^2}{d f^3} \]

[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) - (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)

Rule 12

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

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 30

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

Rule 45

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 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 2338

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]

Rule 2339

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 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 2356

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 2367

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 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 2388

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 2458

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f} \\ & = \frac {i \text {Subst}\left (\int \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}+\frac {(f h-e i) \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2} \\ & = \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}-\frac {(2 b) \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{3 d f}+\frac {(i (f h-e i)) \text {Subst}\left (\int \left (\frac {f h-e i}{f}+\frac {i x}{f}\right ) (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac {(f h-e i)^2 \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^3} \\ & = -\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {b i^2 (f h-e i) (e+f x)^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 {2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {i x \left (18 f^2 h^2+9 f h i (-4 e+x)+i^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f h-e i)^3 \log (x)}{6 f^3 x} \, dx,x,e+f x\right )}{3 d f}+\frac {(i (f h-e i)) \text {Subst}\left (\int \left (\frac {(f h-e i) (a+b \log (c x))^2}{f}+\frac {i x (a+b \log (c x))^2}{f}\right ) \, dx,x,e+f x\right )}{d f^3}+\frac {\left (i (f h-e i)^2\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}+\frac {(f h-e i)^3 \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^4} \\ & = -\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {b i^2 (f h-e i) (e+f x)^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 {2 b (f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{3 d f^4}+\frac {i (f h-e i)^2 (e+f x) (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 {b^2 \text {Subst}\left (\int \frac {i x \left (18 f^2 h^2+9 f h i (-4 e+x)+i^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f h-e i)^3 \log (x)}{x} \, dx,x,e+f x\right )}{9 d f^4}+\frac {\left (i^2 (f h-e i)\right ) \text {Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}+\frac {\left (i (f h-e i)^2\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}-\frac {\left (2 b i (f h-e i)^2\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}+\frac {(f h-e i)^3 \text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^4} \\ & = -\frac {2 a b i (f h-e i)^2 x}{d f^3}-\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {b i^2 (f h-e i) (e+f x)^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 {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 (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac {i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac {b^2 \text {Subst}\left (\int \left (i \left (18 (f h-e i)^2+9 i (f h-e i) x+2 i^2 x^2\right )+\frac {6 (f h-e i)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{9 d f^4}-\frac {\left (b i^2 (f h-e i)\right ) \text {Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac {\left (2 b i (f h-e i)^2\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^4}-\frac {\left (2 b^2 i (f h-e i)^2\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^4} \\ & = -\frac {4 a b i (f h-e i)^2 x}{d f^3}+\frac {2 b^2 i (f h-e i)^2 x}{d f^3}+\frac {b^2 i^2 (f h-e i) (e+f x)^2}{4 d f^4}-\frac {2 b^2 i (f h-e i)^2 (e+f x) \log (c (e+f x))}{d f^4}-\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {3 b i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}-\frac {2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 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 (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac {i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4}+\frac {\left (b^2 i\right ) \text {Subst}\left (\int \left (18 (f h-e i)^2+9 i (f h-e i) x+2 i^2 x^2\right ) \, dx,x,e+f x\right )}{9 d f^4}-\frac {\left (2 b^2 i (f h-e i)^2\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^4}+\frac {\left (2 b^2 (f h-e i)^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{3 d f^4} \\ & = -\frac {4 a b i (f h-e i)^2 x}{d f^3}+\frac {6 b^2 i (f h-e i)^2 x}{d f^3}+\frac {3 b^2 i^2 (f h-e i) (e+f x)^2}{4 d f^4}+\frac {2 b^2 i^3 (e+f x)^3}{27 d f^4}+\frac {b^2 (f h-e i)^3 \log ^2(e+f x)}{3 d f^4}-\frac {4 b^2 i (f h-e i)^2 (e+f x) \log (c (e+f x))}{d f^4}-\frac {2 b i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}-\frac {3 b i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}-\frac {2 b i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{9 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 (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))^2}{d f^4}+\frac {i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^4}+\frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f}+\frac {(f h-e i)^3 (a+b \log (c (e+f x)))^3}{3 b d f^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.58 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {324 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))^2+162 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))^2+36 i^3 (e+f x)^3 (a+b \log (c (e+f x)))^2+\frac {36 (f h-e i)^3 (a+b \log (c (e+f x)))^3}{b}-648 b i (f h-e i)^2 ((a-b) f x+b (e+f x) \log (c (e+f x)))+81 b i^2 (f h-e i) \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )+8 b i^3 \left (b f x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{108 d f^4} \]

[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)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.55

method	result	size
norman	\(\frac {b^{2} i \left (e^{2} i^{2}-3 e f h i +3 f^{2} h^{2}\right ) x \ln \left (c \left (f x +e \right )\right )^{2}}{d \,f^{3}}-\frac {\left (18 a^{2} e^{3} i^{3}-54 a^{2} e^{2} f h \,i^{2}+54 a^{2} e \,f^{2} h^{2} i -18 a^{2} f^{3} h^{3}-66 a b \,e^{3} i^{3}+162 a b \,e^{2} f h \,i^{2}-108 a b e \,f^{2} h^{2} i +85 b^{2} e^{3} i^{3}-189 b^{2} e^{2} f h \,i^{2}+108 b^{2} e \,f^{2} h^{2} i \right ) \ln \left (c \left (f x +e \right )\right )}{18 d \,f^{4}}-\frac {b \left (6 a \,e^{3} i^{3}-18 a \,e^{2} f h \,i^{2}+18 a e \,f^{2} h^{2} i -6 a \,f^{3} h^{3}-11 b \,e^{3} i^{3}+27 b \,e^{2} f h \,i^{2}-18 b e \,f^{2} h^{2} i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{6 d \,f^{4}}-\frac {b^{2} \left (e^{3} i^{3}-3 e^{2} f h \,i^{2}+3 e \,f^{2} h^{2} i -f^{3} h^{3}\right ) \ln \left (c \left (f x +e \right )\right )^{3}}{3 d \,f^{4}}+\frac {i \left (18 a^{2} e^{2} i^{2}-54 a^{2} e f h i +54 a^{2} f^{2} h^{2}-66 a b \,e^{2} i^{2}+162 a b e f h i -108 a b \,f^{2} h^{2}+85 b^{2} e^{2} i^{2}-189 b^{2} e f h i +108 b^{2} f^{2} h^{2}\right ) x}{18 d \,f^{3}}-\frac {i^{2} \left (18 a^{2} e i -54 a^{2} f h -30 a b e i +54 a b f h +19 b^{2} e i -27 b^{2} f h \right ) x^{2}}{36 d \,f^{2}}+\frac {i^{3} \left (9 a^{2}-6 a b +2 b^{2}\right ) x^{3}}{27 d f}+\frac {b^{2} i^{3} x^{3} \ln \left (c \left (f x +e \right )\right )^{2}}{3 d f}+\frac {b i \left (6 a \,e^{2} i^{2}-18 a e f h i +18 a \,f^{2} h^{2}-11 b \,e^{2} i^{2}+27 b e f h i -18 b \,f^{2} h^{2}\right ) x \ln \left (c \left (f x +e \right )\right )}{3 d \,f^{3}}-\frac {b \,i^{2} \left (6 a e i -18 a f h -5 b e i +9 b f h \right ) x^{2} \ln \left (c \left (f x +e \right )\right )}{6 d \,f^{2}}+\frac {2 b \,i^{3} \left (3 a -b \right ) x^{3} \ln \left (c \left (f x +e \right )\right )}{9 d f}-\frac {b^{2} i^{2} \left (e i -3 f h \right ) x^{2} \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{2}}\)	\(721\)
risch	\(-\frac {b \left (-2 b \,f^{3} i^{3} x^{3}+3 b e \,f^{2} i^{3} x^{2}-9 b \,f^{3} h \,i^{2} x^{2}-6 b \,e^{2} f \,i^{3} x +18 b e \,f^{2} h \,i^{2} x -18 b \,f^{3} h^{2} i x +6 a \,e^{3} i^{3}-18 a \,e^{2} f h \,i^{2}+18 a e \,f^{2} h^{2} i -6 a \,f^{3} h^{3}-11 b \,e^{3} i^{3}+27 b \,e^{2} f h \,i^{2}-18 b e \,f^{2} h^{2} i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{6 d \,f^{4}}+\frac {b i x \left (12 a \,f^{2} i^{2} x^{2}-4 b \,f^{2} i^{2} x^{2}-18 a e f \,i^{2} x +54 a \,f^{2} h i x +15 b e f \,i^{2} x -27 b \,f^{2} h i x +36 a \,e^{2} i^{2}-108 a e f h i +108 a \,f^{2} h^{2}-66 b \,e^{2} i^{2}+162 b e f h i -108 b \,f^{2} h^{2}\right ) \ln \left (c \left (f x +e \right )\right )}{18 d \,f^{3}}+\frac {9 a b e h \,i^{2} x}{d \,f^{2}}-\frac {9 \ln \left (f x +e \right ) a b \,e^{2} h \,i^{2}}{d \,f^{3}}+\frac {6 \ln \left (f x +e \right ) a b e \,h^{2} i}{d \,f^{2}}+\frac {3 \ln \left (f x +e \right ) a^{2} e^{2} h \,i^{2}}{d \,f^{3}}-\frac {3 \ln \left (f x +e \right ) a^{2} e \,h^{2} i}{d \,f^{2}}+\frac {11 \ln \left (f x +e \right ) a b \,e^{3} i^{3}}{3 d \,f^{4}}+\frac {21 \ln \left (f x +e \right ) b^{2} e^{2} h \,i^{2}}{2 d \,f^{3}}-\frac {6 \ln \left (f x +e \right ) b^{2} e \,h^{2} i}{d \,f^{2}}+\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} h^{3}}{3 d f}+\frac {\ln \left (f x +e \right ) a^{2} h^{3}}{d f}+\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e^{2} h \,i^{2}}{d \,f^{3}}-\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e \,h^{2} i}{d \,f^{2}}+\frac {5 a b e \,i^{3} x^{2}}{6 d \,f^{2}}-\frac {3 a b h \,i^{2} x^{2}}{2 d f}-\frac {3 a^{2} e h \,i^{2} x}{d \,f^{2}}-\frac {11 a b \,e^{2} i^{3} x}{3 d \,f^{3}}-\frac {6 a b \,h^{2} i x}{d f}-\frac {21 b^{2} e h \,i^{2} x}{2 d \,f^{2}}-\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e^{3} i^{3}}{3 d \,f^{4}}+\frac {a^{2} i^{3} x^{3}}{3 d f}-\frac {2 a b \,i^{3} x^{3}}{9 d f}-\frac {a^{2} e \,i^{3} x^{2}}{2 d \,f^{2}}+\frac {3 a^{2} h \,i^{2} x^{2}}{2 d f}-\frac {19 b^{2} e \,i^{3} x^{2}}{36 d \,f^{2}}+\frac {3 b^{2} h \,i^{2} x^{2}}{4 d f}+\frac {a^{2} e^{2} i^{3} x}{d \,f^{3}}+\frac {3 a^{2} h^{2} i x}{d f}+\frac {85 b^{2} e^{2} i^{3} x}{18 d \,f^{3}}+\frac {6 b^{2} h^{2} i x}{d f}-\frac {\ln \left (f x +e \right ) a^{2} e^{3} i^{3}}{d \,f^{4}}-\frac {85 \ln \left (f x +e \right ) b^{2} e^{3} i^{3}}{18 d \,f^{4}}+\frac {2 b^{2} i^{3} x^{3}}{27 d f}\)	\(927\)
parts	\(\frac {a^{2} \left (\frac {i \left (\frac {1}{3} f^{2} i^{2} x^{3}-\frac {1}{2} e f \,i^{2} x^{2}+\frac {3}{2} f^{2} h i \,x^{2}+x \,e^{2} i^{2}-3 x e f h i +3 x \,f^{2} h^{2}\right )}{f^{3}}+\frac {\left (-e^{3} i^{3}+3 e^{2} f h \,i^{2}-3 e \,f^{2} h^{2} i +f^{3} h^{3}\right ) \ln \left (f x +e \right )}{f^{4}}\right )}{d}+\frac {b^{2} \left (-\frac {c \,e^{3} i^{3} \ln \left (c f x +c e \right )^{3}}{3 f^{3}}+\frac {c \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{3}}{f^{2}}-\frac {c e \,h^{2} i \ln \left (c f x +c e \right )^{3}}{f}+\frac {c \,h^{3} \ln \left (c f x +c e \right )^{3}}{3}+\frac {3 e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f^{3}}-\frac {6 e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f^{2}}+\frac {3 h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f}-\frac {3 e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}+\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3}}+\frac {3 h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}+\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}+\frac {i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )^{2}}{3}-\frac {2 \left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{9}+\frac {2 \left (c f x +c e \right )^{3}}{27}\right )}{c^{2} f^{3}}\right )}{d c f}+\frac {2 a b \left (-\frac {c \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3}}+\frac {3 c \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2}}-\frac {3 c e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f}+\frac {c \,h^{3} \ln \left (c f x +c e \right )^{2}}{2}+\frac {3 e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3}}-\frac {6 e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2}}+\frac {3 h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f}-\frac {3 e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3}}+\frac {3 h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}+\frac {i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3}}\right )}{d c f}\)	\(948\)
derivativedivides	\(\text {Expression too large to display}\)	\(1178\)
default	\(\text {Expression too large to display}\)	\(1178\)
parallelrisch	\(\text {Expression too large to display}\)	\(1204\)

[In]

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

[Out]

b^2*i*(e^2*i^2-3*e*f*h*i+3*f^2*h^2)/d/f^3*x*ln(c*(f*x+e))^2-1/18*(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)/d/f^4*ln(c*(f*x+e))-1/6*b*(6*a*e^3*i^3-18*a*e^2*f*h*i^2+18*a*e*f^2*h^2*i-6*a*f^3*h^3-
11*b*e^3*i^3+27*b*e^2*f*h*i^2-18*b*e*f^2*h^2*i)/d/f^4*ln(c*(f*x+e))^2-1/3*b^2*(e^3*i^3-3*e^2*f*h*i^2+3*e*f^2*h
^2*i-f^3*h^3)/d/f^4*ln(c*(f*x+e))^3+1/18*i*(18*a^2*e^2*i^2-54*a^2*e*f*h*i+54*a^2*f^2*h^2-66*a*b*e^2*i^2+162*a*
b*e*f*h*i-108*a*b*f^2*h^2+85*b^2*e^2*i^2-189*b^2*e*f*h*i+108*b^2*f^2*h^2)/d/f^3*x-1/36*i^2*(18*a^2*e*i-54*a^2*
f*h-30*a*b*e*i+54*a*b*f*h+19*b^2*e*i-27*b^2*f*h)/d/f^2*x^2+1/27*i^3*(9*a^2-6*a*b+2*b^2)/d/f*x^3+1/3*b^2*i^3/d/
f*x^3*ln(c*(f*x+e))^2+1/3*b*i*(6*a*e^2*i^2-18*a*e*f*h*i+18*a*f^2*h^2-11*b*e^2*i^2+27*b*e*f*h*i-18*b*f^2*h^2)/d
/f^3*x*ln(c*(f*x+e))-1/6*b*i^2*(6*a*e*i-18*a*f*h-5*b*e*i+9*b*f*h)/d/f^2*x^2*ln(c*(f*x+e))+2/9*b*i^3*(3*a-b)/d/
f*x^3*ln(c*(f*x+e))-1/2*b^2*i^2*(e*i-3*f*h)/d/f^2*x^2*ln(c*(f*x+e))^2

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.31 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {4 \, {\left (9 \, a^{2} - 6 \, a b + 2 \, b^{2}\right )} f^{3} i^{3} x^{3} + 36 \, {\left (b^{2} f^{3} h^{3} - 3 \, b^{2} e f^{2} h^{2} i + 3 \, b^{2} e^{2} f h i^{2} - b^{2} e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{3} + 3 \, {\left (27 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{3} h i^{2} - {\left (18 \, a^{2} - 30 \, a b + 19 \, b^{2}\right )} e f^{2} i^{3}\right )} x^{2} + 18 \, {\left (2 \, b^{2} f^{3} i^{3} x^{3} + 6 \, a b f^{3} h^{3} - 18 \, {\left (a b - b^{2}\right )} e f^{2} h^{2} i + 9 \, {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} f h i^{2} - {\left (6 \, a b - 11 \, b^{2}\right )} e^{3} i^{3} + 3 \, {\left (3 \, b^{2} f^{3} h i^{2} - b^{2} e f^{2} i^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{2} f^{3} h^{2} i - 3 \, b^{2} e f^{2} h i^{2} + b^{2} e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 6 \, {\left (54 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{3} h^{2} i - 27 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f^{2} h i^{2} + {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{2} f i^{3}\right )} x + 6 \, {\left (4 \, {\left (3 \, a b - b^{2}\right )} f^{3} i^{3} x^{3} + 18 \, a^{2} f^{3} h^{3} - 54 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f^{2} h^{2} i + 27 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} f h i^{2} - {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{3} i^{3} + 3 \, {\left (9 \, {\left (2 \, a b - b^{2}\right )} f^{3} h i^{2} - {\left (6 \, a b - 5 \, b^{2}\right )} e f^{2} i^{3}\right )} x^{2} + 6 \, {\left (18 \, {\left (a b - b^{2}\right )} f^{3} h^{2} i - 9 \, {\left (2 \, a b - 3 \, b^{2}\right )} e f^{2} h i^{2} + {\left (6 \, a b - 11 \, b^{2}\right )} e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )}{108 \, d f^{4}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (428) = 856\).

Time = 1.10 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.98 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=x^{3} \left (\frac {a^{2} i^{3}}{3 d f} - \frac {2 a b i^{3}}{9 d f} + \frac {2 b^{2} i^{3}}{27 d f}\right ) + x^{2} \left (- \frac {a^{2} e i^{3}}{2 d f^{2}} + \frac {3 a^{2} h i^{2}}{2 d f} + \frac {5 a b e i^{3}}{6 d f^{2}} - \frac {3 a b h i^{2}}{2 d f} - \frac {19 b^{2} e i^{3}}{36 d f^{2}} + \frac {3 b^{2} h i^{2}}{4 d f}\right ) + x \left (\frac {a^{2} e^{2} i^{3}}{d f^{3}} - \frac {3 a^{2} e h i^{2}}{d f^{2}} + \frac {3 a^{2} h^{2} i}{d f} - \frac {11 a b e^{2} i^{3}}{3 d f^{3}} + \frac {9 a b e h i^{2}}{d f^{2}} - \frac {6 a b h^{2} i}{d f} + \frac {85 b^{2} e^{2} i^{3}}{18 d f^{3}} - \frac {21 b^{2} e h i^{2}}{2 d f^{2}} + \frac {6 b^{2} h^{2} i}{d f}\right ) + \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}} \]

[In]

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

[Out]

x**3*(a**2*i**3/(3*d*f) - 2*a*b*i**3/(9*d*f) + 2*b**2*i**3/(27*d*f)) + x**2*(-a**2*e*i**3/(2*d*f**2) + 3*a**2*
h*i**2/(2*d*f) + 5*a*b*e*i**3/(6*d*f**2) - 3*a*b*h*i**2/(2*d*f) - 19*b**2*e*i**3/(36*d*f**2) + 3*b**2*h*i**2/(
4*d*f)) + x*(a**2*e**2*i**3/(d*f**3) - 3*a**2*e*h*i**2/(d*f**2) + 3*a**2*h**2*i/(d*f) - 11*a*b*e**2*i**3/(3*d*
f**3) + 9*a*b*e*h*i**2/(d*f**2) - 6*a*b*h**2*i/(d*f) + 85*b**2*e**2*i**3/(18*d*f**3) - 21*b**2*e*h*i**2/(2*d*f
**2) + 6*b**2*h**2*i/(d*f)) + (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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 964 vs. \(2 (446) = 892\).

Time = 0.28 (sec) , antiderivative size = 964, normalized size of antiderivative = 2.08 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=6 \, a b h^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) - \frac {1}{3} \, a b i^{3} {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + 3 \, a b h i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} \log \left (c f x + c e\right ) - a b h^{3} {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + 3 \, a^{2} h^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - \frac {1}{6} \, a^{2} i^{3} {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} + \frac {3}{2} \, a^{2} h i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} + \frac {b^{2} h^{3} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac {2 \, a b h^{3} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a^{2} h^{3} \log \left (d f x + d e\right )}{d f} + \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b h^{2} i}{d f^{2}} - \frac {3 \, {\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} a b h i^{2}}{2 \, d f^{3}} - \frac {{\left (c^{2} e \log \left (c f x + c e\right )^{3} - 3 \, {\left (c f x + c e\right )} {\left (c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + 2 \, c\right )}\right )} b^{2} h^{2} i}{c^{2} d f^{2}} - \frac {{\left (4 \, f^{3} x^{3} - 15 \, e f^{2} x^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} + 66 \, e^{2} f x - 66 \, e^{3} \log \left (f x + e\right )\right )} a b i^{3}}{18 \, d f^{4}} + \frac {{\left (4 \, c^{3} e^{2} \log \left (c f x + c e\right )^{3} + 3 \, {\left (c f x + c e\right )}^{2} {\left (2 \, c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + c\right )} - 24 \, {\left (c^{2} e \log \left (c f x + c e\right )^{2} - 2 \, c^{2} e \log \left (c f x + c e\right ) + 2 \, c^{2} e\right )} {\left (c f x + c e\right )}\right )} b^{2} h i^{2}}{4 \, c^{3} d f^{3}} - \frac {{\left (36 \, c^{4} e^{3} \log \left (c f x + c e\right )^{3} - 4 \, {\left (c f x + c e\right )}^{3} {\left (9 \, c \log \left (c f x + c e\right )^{2} - 6 \, c \log \left (c f x + c e\right ) + 2 \, c\right )} + 81 \, {\left (2 \, c^{2} e \log \left (c f x + c e\right )^{2} - 2 \, c^{2} e \log \left (c f x + c e\right ) + c^{2} e\right )} {\left (c f x + c e\right )}^{2} - 324 \, {\left (c^{3} e^{2} \log \left (c f x + c e\right )^{2} - 2 \, c^{3} e^{2} \log \left (c f x + c e\right ) + 2 \, c^{3} e^{2}\right )} {\left (c f x + c e\right )}\right )} b^{2} i^{3}}{108 \, c^{4} d f^{4}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.66 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {1}{6} \, {\left (\frac {2 \, b^{2} i^{3} x^{3}}{d f} + \frac {3 \, {\left (3 \, b^{2} f h i^{2} - b^{2} e i^{3}\right )} x^{2}}{d f^{2}} + \frac {6 \, {\left (3 \, b^{2} f^{2} h^{2} i - 3 \, b^{2} e f h i^{2} + b^{2} e^{2} i^{3}\right )} x}{d f^{3}} + \frac {6 \, a b f^{3} h^{3} - 18 \, a b e f^{2} h^{2} i + 18 \, b^{2} e f^{2} h^{2} i + 18 \, a b e^{2} f h i^{2} - 27 \, b^{2} e^{2} f h i^{2} - 6 \, a b e^{3} i^{3} + 11 \, b^{2} e^{3} i^{3}}{d f^{4}}\right )} \log \left (c f x + c e\right )^{2} + \frac {{\left (9 \, a^{2} i^{3} - 6 \, a b i^{3} + 2 \, b^{2} i^{3}\right )} x^{3}}{27 \, d f} + \frac {1}{18} \, {\left (\frac {4 \, {\left (3 \, a b i^{3} - b^{2} i^{3}\right )} x^{3}}{d f} + \frac {3 \, {\left (18 \, a b f h i^{2} - 9 \, b^{2} f h i^{2} - 6 \, a b e i^{3} + 5 \, b^{2} e i^{3}\right )} x^{2}}{d f^{2}} + \frac {6 \, {\left (18 \, a b f^{2} h^{2} i - 18 \, b^{2} f^{2} h^{2} i - 18 \, a b e f h i^{2} + 27 \, b^{2} e f h i^{2} + 6 \, a b e^{2} i^{3} - 11 \, b^{2} e^{2} i^{3}\right )} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac {{\left (54 \, a^{2} f h i^{2} - 54 \, a b f h i^{2} + 27 \, b^{2} f h i^{2} - 18 \, a^{2} e i^{3} + 30 \, a b e i^{3} - 19 \, b^{2} e i^{3}\right )} x^{2}}{36 \, d f^{2}} + \frac {{\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 \, d f^{4}} + \frac {{\left (54 \, a^{2} f^{2} h^{2} i - 108 \, a b f^{2} h^{2} i + 108 \, b^{2} f^{2} h^{2} i - 54 \, a^{2} e f h i^{2} + 162 \, a b e f h i^{2} - 189 \, b^{2} e f h i^{2} + 18 \, a^{2} e^{2} i^{3} - 66 \, a b e^{2} i^{3} + 85 \, b^{2} e^{2} i^{3}\right )} x}{18 \, d f^{3}} + \frac {{\left (18 \, a^{2} f^{3} h^{3} - 54 \, a^{2} e f^{2} h^{2} i + 108 \, a b e f^{2} h^{2} i - 108 \, b^{2} e f^{2} h^{2} i + 54 \, a^{2} e^{2} f h i^{2} - 162 \, a b e^{2} f h i^{2} + 189 \, b^{2} e^{2} f h i^{2} - 18 \, a^{2} e^{3} i^{3} + 66 \, a b e^{3} i^{3} - 85 \, b^{2} e^{3} i^{3}\right )} \log \left (f x + e\right )}{18 \, d f^{4}} \]

[In]

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

[Out]

1/6*(2*b^2*i^3*x^3/(d*f) + 3*(3*b^2*f*h*i^2 - b^2*e*i^3)*x^2/(d*f^2) + 6*(3*b^2*f^2*h^2*i - 3*b^2*e*f*h*i^2 +
b^2*e^2*i^3)*x/(d*f^3) + (6*a*b*f^3*h^3 - 18*a*b*e*f^2*h^2*i + 18*b^2*e*f^2*h^2*i + 18*a*b*e^2*f*h*i^2 - 27*b^
2*e^2*f*h*i^2 - 6*a*b*e^3*i^3 + 11*b^2*e^3*i^3)/(d*f^4))*log(c*f*x + c*e)^2 + 1/27*(9*a^2*i^3 - 6*a*b*i^3 + 2*
b^2*i^3)*x^3/(d*f) + 1/18*(4*(3*a*b*i^3 - b^2*i^3)*x^3/(d*f) + 3*(18*a*b*f*h*i^2 - 9*b^2*f*h*i^2 - 6*a*b*e*i^3
+ 5*b^2*e*i^3)*x^2/(d*f^2) + 6*(18*a*b*f^2*h^2*i - 18*b^2*f^2*h^2*i - 18*a*b*e*f*h*i^2 + 27*b^2*e*f*h*i^2 + 6
*a*b*e^2*i^3 - 11*b^2*e^2*i^3)*x/(d*f^3))*log(c*f*x + c*e) + 1/36*(54*a^2*f*h*i^2 - 54*a*b*f*h*i^2 + 27*b^2*f*
h*i^2 - 18*a^2*e*i^3 + 30*a*b*e*i^3 - 19*b^2*e*i^3)*x^2/(d*f^2) + 1/3*(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/(d*f^4) + 1/18*(54*a^2*f^2*h^2*i - 108*a*b*f^2*h^2*i + 108*b^2*
f^2*h^2*i - 54*a^2*e*f*h*i^2 + 162*a*b*e*f*h*i^2 - 189*b^2*e*f*h*i^2 + 18*a^2*e^2*i^3 - 66*a*b*e^2*i^3 + 85*b^
2*e^2*i^3)*x/(d*f^3) + 1/18*(18*a^2*f^3*h^3 - 54*a^2*e*f^2*h^2*i + 108*a*b*e*f^2*h^2*i - 108*b^2*e*f^2*h^2*i +
54*a^2*e^2*f*h*i^2 - 162*a*b*e^2*f*h*i^2 + 189*b^2*e^2*f*h*i^2 - 18*a^2*e^3*i^3 + 66*a*b*e^3*i^3 - 85*b^2*e^3
*i^3)*log(f*x + e)/(d*f^4)

Mupad [B] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.73 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=x^2\,\left (\frac {i^2\,\left (18\,a^2\,f\,h-5\,b^2\,e\,i+9\,b^2\,f\,h+6\,a\,b\,e\,i-18\,a\,b\,f\,h\right )}{12\,d\,f^2}-\frac {e\,i^3\,\left (9\,a^2-6\,a\,b+2\,b^2\right )}{18\,d\,f^2}\right )+{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^3\,x^3}{3\,d\,f^2}-\frac {b^2\,i^2\,x^2\,\left (e\,i-3\,f\,h\right )}{2\,d\,f^3}+\frac {b^2\,i\,x\,\left (e^2\,i^2-3\,e\,f\,h\,i+3\,f^2\,h^2\right )}{d\,f^4}\right )+\frac {11\,b^2\,e^3\,i^3-27\,b^2\,e^2\,f\,h\,i^2+18\,b^2\,e\,f^2\,h^2\,i-6\,a\,b\,e^3\,i^3+18\,a\,b\,e^2\,f\,h\,i^2-18\,a\,b\,e\,f^2\,h^2\,i+6\,a\,b\,f^3\,h^3}{6\,d\,f^4}\right )+x\,\left (\frac {54\,a^2\,f^2\,h^2\,i-36\,a\,b\,e^2\,i^3+108\,a\,b\,e\,f\,h\,i^2-108\,a\,b\,f^2\,h^2\,i+66\,b^2\,e^2\,i^3-162\,b^2\,e\,f\,h\,i^2+108\,b^2\,f^2\,h^2\,i}{18\,d\,f^3}-\frac {e\,\left (\frac {i^2\,\left (18\,a^2\,f\,h-5\,b^2\,e\,i+9\,b^2\,f\,h+6\,a\,b\,e\,i-18\,a\,b\,f\,h\right )}{6\,d\,f^2}-\frac {e\,i^3\,\left (9\,a^2-6\,a\,b+2\,b^2\right )}{9\,d\,f^2}\right )}{f}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x^2\,\left (5\,e\,b^2\,i^3-9\,f\,h\,b^2\,i^2-6\,a\,e\,b\,i^3+18\,a\,f\,h\,b\,i^2\right )}{6\,d\,f^3}-\frac {x\,\left (11\,b^2\,e^2\,i^3-27\,b^2\,e\,f\,h\,i^2+18\,b^2\,f^2\,h^2\,i-6\,a\,b\,e^2\,i^3+18\,a\,b\,e\,f\,h\,i^2-18\,a\,b\,f^2\,h^2\,i\right )}{3\,d\,f^4}+\frac {2\,b\,i^3\,x^3\,\left (3\,a-b\right )}{9\,d\,f^2}\right )-\frac {\ln \left (e+f\,x\right )\,\left (18\,a^2\,e^3\,i^3-54\,a^2\,e^2\,f\,h\,i^2+54\,a^2\,e\,f^2\,h^2\,i-18\,a^2\,f^3\,h^3-66\,a\,b\,e^3\,i^3+162\,a\,b\,e^2\,f\,h\,i^2-108\,a\,b\,e\,f^2\,h^2\,i+85\,b^2\,e^3\,i^3-189\,b^2\,e^2\,f\,h\,i^2+108\,b^2\,e\,f^2\,h^2\,i\right )}{18\,d\,f^4}+\frac {i^3\,x^3\,\left (9\,a^2-6\,a\,b+2\,b^2\right )}{27\,d\,f}-\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^3\,i^3-3\,e^2\,f\,h\,i^2+3\,e\,f^2\,h^2\,i-f^3\,h^3\right )}{3\,d\,f^4} \]

[In]

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

[Out]

x^2*((i^2*(18*a^2*f*h - 5*b^2*e*i + 9*b^2*f*h + 6*a*b*e*i - 18*a*b*f*h))/(12*d*f^2) - (e*i^3*(9*a^2 - 6*a*b +
2*b^2))/(18*d*f^2)) + log(c*(e + f*x))^2*(f*((b^2*i^3*x^3)/(3*d*f^2) - (b^2*i^2*x^2*(e*i - 3*f*h))/(2*d*f^3) +
(b^2*i*x*(e^2*i^2 + 3*f^2*h^2 - 3*e*f*h*i))/(d*f^4)) + (11*b^2*e^3*i^3 - 6*a*b*e^3*i^3 + 6*a*b*f^3*h^3 + 18*b
^2*e*f^2*h^2*i - 27*b^2*e^2*f*h*i^2 - 18*a*b*e*f^2*h^2*i + 18*a*b*e^2*f*h*i^2)/(6*d*f^4)) + x*((66*b^2*e^2*i^3
+ 54*a^2*f^2*h^2*i + 108*b^2*f^2*h^2*i - 36*a*b*e^2*i^3 - 108*a*b*f^2*h^2*i - 162*b^2*e*f*h*i^2 + 108*a*b*e*f
*h*i^2)/(18*d*f^3) - (e*((i^2*(18*a^2*f*h - 5*b^2*e*i + 9*b^2*f*h + 6*a*b*e*i - 18*a*b*f*h))/(6*d*f^2) - (e*i^
3*(9*a^2 - 6*a*b + 2*b^2))/(9*d*f^2)))/f) + f*log(c*(e + f*x))*((x^2*(5*b^2*e*i^3 - 6*a*b*e*i^3 - 9*b^2*f*h*i^
2 + 18*a*b*f*h*i^2))/(6*d*f^3) - (x*(11*b^2*e^2*i^3 + 18*b^2*f^2*h^2*i - 6*a*b*e^2*i^3 - 18*a*b*f^2*h^2*i - 27
*b^2*e*f*h*i^2 + 18*a*b*e*f*h*i^2))/(3*d*f^4) + (2*b*i^3*x^3*(3*a - b))/(9*d*f^2)) - (log(e + f*x)*(18*a^2*e^3
*i^3 - 18*a^2*f^3*h^3 + 85*b^2*e^3*i^3 - 66*a*b*e^3*i^3 + 54*a^2*e*f^2*h^2*i - 54*a^2*e^2*f*h*i^2 + 108*b^2*e*
f^2*h^2*i - 189*b^2*e^2*f*h*i^2 - 108*a*b*e*f^2*h^2*i + 162*a*b*e^2*f*h*i^2))/(18*d*f^4) + (i^3*x^3*(9*a^2 - 6
*a*b + 2*b^2))/(27*d*f) - (b^2*log(c*(e + f*x))^3*(e^3*i^3 - f^3*h^3 + 3*e*f^2*h^2*i - 3*e^2*f*h*i^2))/(3*d*f^
4)