3.2.83 \(\int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) [183]

Optimal. Leaf size=579 \[ -\frac {4 a b i (f h-e i)^3 x}{d f^4}+\frac {8 b^2 i (f h-e i)^3 x}{d f^4}+\frac {3 b^2 i^2 (f h-e i)^2 (e+f x)^2}{2 d f^5}+\frac {8 b^2 i^3 (f h-e i) (e+f x)^3}{27 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}-\frac {4 b^2 i (f h-e i)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {3 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5} \]

[Out]

-4*a*b*i*(-e*i+f*h)^3*x/d/f^4+8*b^2*i*(-e*i+f*h)^3*x/d/f^4+3/2*b^2*i^2*(-e*i+f*h)^2*(f*x+e)^2/d/f^5+8/27*b^2*i
^3*(-e*i+f*h)*(f*x+e)^3/d/f^5+1/32*b^2*i^4*(f*x+e)^4/d/f^5+7/12*b^2*(-e*i+f*h)^4*ln(f*x+e)^2/d/f^5-4*b^2*i*(-e
*i+f*h)^3*(f*x+e)*ln(c*(f*x+e))/d/f^5-4*b*i*(-e*i+f*h)^3*(f*x+e)*(a+b*ln(c*(f*x+e)))/d/f^5-3*b*i^2*(-e*i+f*h)^
2*(f*x+e)^2*(a+b*ln(c*(f*x+e)))/d/f^5-8/9*b*i^3*(-e*i+f*h)*(f*x+e)^3*(a+b*ln(c*(f*x+e)))/d/f^5-1/8*b*i^4*(f*x+
e)^4*(a+b*ln(c*(f*x+e)))/d/f^5-7/6*b*(-e*i+f*h)^4*ln(f*x+e)*(a+b*ln(c*(f*x+e)))/d/f^5+2*i*(-e*i+f*h)^3*(f*x+e)
*(a+b*ln(c*(f*x+e)))^2/d/f^5+1/2*i^2*(-e*i+f*h)^2*(f*x+e)^2*(a+b*ln(c*(f*x+e)))^2/d/f^5+1/3*(-e*i+f*h)*(i*x+h)
^3*(a+b*ln(c*(f*x+e)))^2/d/f^2+1/4*(i*x+h)^4*(a+b*ln(c*(f*x+e)))^2/d/f+1/3*(-e*i+f*h)^4*(a+b*ln(c*(f*x+e)))^3/
b/d/f^5

________________________________________________________________________________________

Rubi [A]
time = 1.07, antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 15, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2458, 12, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341, 2356, 45, 2372, 14, 2338} \begin {gather*} -\frac {8 b i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))}{9 d f^5}+\frac {i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}-\frac {3 b i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^5}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))^2}{d f^5}-\frac {4 b i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))}{d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}+\frac {(h+i x)^3 (f h-e i) (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {4 a b i x (f h-e i)^3}{d f^4}-\frac {4 b^2 i (e+f x) (f h-e i)^3 \log (c (e+f x))}{d f^5}+\frac {8 b^2 i^3 (e+f x)^3 (f h-e i)}{27 d f^5}+\frac {3 b^2 i^2 (e+f x)^2 (f h-e i)^2}{2 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {8 b^2 i x (f h-e i)^3}{d f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((h + i*x)^4*(a + b*Log[c*(e + f*x)])^2)/(d*e + d*f*x),x]

[Out]

(-4*a*b*i*(f*h - e*i)^3*x)/(d*f^4) + (8*b^2*i*(f*h - e*i)^3*x)/(d*f^4) + (3*b^2*i^2*(f*h - e*i)^2*(e + f*x)^2)
/(2*d*f^5) + (8*b^2*i^3*(f*h - e*i)*(e + f*x)^3)/(27*d*f^5) + (b^2*i^4*(e + f*x)^4)/(32*d*f^5) + (7*b^2*(f*h -
e*i)^4*Log[e + f*x]^2)/(12*d*f^5) - (4*b^2*i*(f*h - e*i)^3*(e + f*x)*Log[c*(e + f*x)])/(d*f^5) - (4*b*i*(f*h
- e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^5) - (3*b*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*
x)]))/(d*f^5) - (8*b*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/(9*d*f^5) - (b*i^4*(e + f*x)^4*(a +
b*Log[c*(e + f*x)]))/(8*d*f^5) - (7*b*(f*h - e*i)^4*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(6*d*f^5) + (2*i*(
f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^5) + (i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e +
f*x)])^2)/(2*d*f^5) + ((f*h - e*i)*(h + i*x)^3*(a + b*Log[c*(e + f*x)])^2)/(3*d*f^2) + ((h + i*x)^4*(a + b*Log
[c*(e + f*x)])^2)/(4*d*f) + ((f*h - e*i)^4*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^5)

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*} \int \frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^4 (a+b \log (c x))^2}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^4 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {183 \text {Subst}\left (\int \left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^3 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^2}-\frac {(183 e-f h) \text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^3 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^4 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{2 d f}-\frac {(183 (183 e-f h)) \text {Subst}\left (\int \left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^2 (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^3}+\frac {(183 e-f h)^2 \text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^2 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^3}\\ &=\frac {b \left (\frac {2928 (183 e-f h)^3 (e+f x)}{f^4}-\frac {401868 (183 e-f h)^2 (e+f x)^2}{f^4}+\frac {32685264 (183 e-f h) (e+f x)^3}{f^4}-\frac {1121513121 (e+f x)^4}{f^4}-\frac {4 (183 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{8 d f}-\frac {(183 e-f h) (h+183 x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {b^2 \text {Subst}\left (\int \frac {-2928 (183 e-f h)^3+401868 (-183 e+f h)^2 x-32685264 (183 e-f h) x^2+1121513121 x^3+\frac {4 (-183 e+f h)^4 \log (x)}{x}}{4 f^4} \, dx,x,e+f x\right )}{2 d f}+\frac {(2 b (183 e-f h)) \text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right )^3 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{3 d f^2}+\frac {\left (183 (183 e-f h)^2\right ) \text {Subst}\left (\int \left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right ) (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^4}-\frac {(183 e-f h)^3 \text {Subst}\left (\int \frac {\left (\frac {-183 e+f h}{f}+\frac {183 x}{f}\right ) (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^4}\\ &=\frac {b (183 e-f h) \left (\frac {1098 (183 e-f h)^2 (e+f x)}{f^3}-\frac {100467 (183 e-f h) (e+f x)^2}{f^3}+\frac {4085658 (e+f x)^3}{f^3}-\frac {2 (183 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f^2}+\frac {b \left (\frac {2928 (183 e-f h)^3 (e+f x)}{f^4}-\frac {401868 (183 e-f h)^2 (e+f x)^2}{f^4}+\frac {32685264 (183 e-f h) (e+f x)^3}{f^4}-\frac {1121513121 (e+f x)^4}{f^4}-\frac {4 (183 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{8 d f}-\frac {(183 e-f h) (h+183 x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {b^2 \text {Subst}\left (\int \left (-2928 (183 e-f h)^3+401868 (-183 e+f h)^2 x-32685264 (183 e-f h) x^2+1121513121 x^3+\frac {4 (-183 e+f h)^4 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{8 d f^5}-\frac {\left (2 b^2 (183 e-f h)\right ) \text {Subst}\left (\int \frac {549 x \left (66978 e^2+2 f^2 h^2+183 f h x+7442 x^2-183 e (4 f h+183 x)\right )-2 (183 e-f h)^3 \log (x)}{2 f^3 x} \, dx,x,e+f x\right )}{3 d f^2}+\frac {\left (183 (183 e-f h)^2\right ) \text {Subst}\left (\int \left (\frac {(-183 e+f h) (a+b \log (c x))^2}{f}+\frac {183 x (a+b \log (c x))^2}{f}\right ) \, dx,x,e+f x\right )}{d f^4}-\frac {\left (183 (183 e-f h)^3\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}+\frac {(183 e-f h)^4 \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^5}\\ &=-\frac {366 b^2 (183 e-f h)^3 x}{d f^4}+\frac {100467 b^2 (183 e-f h)^2 (e+f x)^2}{4 d f^5}-\frac {1361886 b^2 (183 e-f h) (e+f x)^3}{d f^5}+\frac {1121513121 b^2 (e+f x)^4}{32 d f^5}+\frac {b (183 e-f h) \left (\frac {1098 (183 e-f h)^2 (e+f x)}{f^3}-\frac {100467 (183 e-f h) (e+f x)^2}{f^3}+\frac {4085658 (e+f x)^3}{f^3}-\frac {2 (183 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f^2}+\frac {b \left (\frac {2928 (183 e-f h)^3 (e+f x)}{f^4}-\frac {401868 (183 e-f h)^2 (e+f x)^2}{f^4}+\frac {32685264 (183 e-f h) (e+f x)^3}{f^4}-\frac {1121513121 (e+f x)^4}{f^4}-\frac {4 (183 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{8 d f}-\frac {(183 e-f h) (h+183 x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {183 (183 e-f h)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}-\frac {\left (b^2 (183 e-f h)\right ) \text {Subst}\left (\int \frac {549 x \left (66978 e^2+2 f^2 h^2+183 f h x+7442 x^2-183 e (4 f h+183 x)\right )-2 (183 e-f h)^3 \log (x)}{x} \, dx,x,e+f x\right )}{3 d f^5}+\frac {\left (33489 (183 e-f h)^2\right ) \text {Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}-\frac {\left (183 (183 e-f h)^3\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}+\frac {\left (366 b (183 e-f h)^3\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}+\frac {(183 e-f h)^4 \text {Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d f^5}+\frac {\left (b^2 (183 e-f h)^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{2 d f^5}\\ &=\frac {366 a b (183 e-f h)^3 x}{d f^4}-\frac {366 b^2 (183 e-f h)^3 x}{d f^4}+\frac {100467 b^2 (183 e-f h)^2 (e+f x)^2}{4 d f^5}-\frac {1361886 b^2 (183 e-f h) (e+f x)^3}{d f^5}+\frac {1121513121 b^2 (e+f x)^4}{32 d f^5}+\frac {b^2 (183 e-f h)^4 \log ^2(e+f x)}{4 d f^5}+\frac {b (183 e-f h) \left (\frac {1098 (183 e-f h)^2 (e+f x)}{f^3}-\frac {100467 (183 e-f h) (e+f x)^2}{f^3}+\frac {4085658 (e+f x)^3}{f^3}-\frac {2 (183 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f^2}+\frac {b \left (\frac {2928 (183 e-f h)^3 (e+f x)}{f^4}-\frac {401868 (183 e-f h)^2 (e+f x)^2}{f^4}+\frac {32685264 (183 e-f h) (e+f x)^3}{f^4}-\frac {1121513121 (e+f x)^4}{f^4}-\frac {4 (183 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{8 d f}-\frac {(183 e-f h) (h+183 x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {366 (183 e-f h)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {33489 (183 e-f h)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(183 e-f h)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}-\frac {\left (b^2 (183 e-f h)\right ) \text {Subst}\left (\int \left (549 \left (2 (183 e-f h)^2-183 (183 e-f h) x+7442 x^2\right )-\frac {2 (183 e-f h)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{3 d f^5}-\frac {\left (33489 b (183 e-f h)^2\right ) \text {Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}+\frac {\left (366 b (183 e-f h)^3\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}+\frac {\left (366 b^2 (183 e-f h)^3\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^5}\\ &=\frac {732 a b (183 e-f h)^3 x}{d f^4}-\frac {732 b^2 (183 e-f h)^3 x}{d f^4}+\frac {33489 b^2 (183 e-f h)^2 (e+f x)^2}{d f^5}-\frac {1361886 b^2 (183 e-f h) (e+f x)^3}{d f^5}+\frac {1121513121 b^2 (e+f x)^4}{32 d f^5}+\frac {b^2 (183 e-f h)^4 \log ^2(e+f x)}{4 d f^5}+\frac {366 b^2 (183 e-f h)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {33489 b (183 e-f h)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^5}+\frac {b (183 e-f h) \left (\frac {1098 (183 e-f h)^2 (e+f x)}{f^3}-\frac {100467 (183 e-f h) (e+f x)^2}{f^3}+\frac {4085658 (e+f x)^3}{f^3}-\frac {2 (183 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f^2}+\frac {b \left (\frac {2928 (183 e-f h)^3 (e+f x)}{f^4}-\frac {401868 (183 e-f h)^2 (e+f x)^2}{f^4}+\frac {32685264 (183 e-f h) (e+f x)^3}{f^4}-\frac {1121513121 (e+f x)^4}{f^4}-\frac {4 (183 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{8 d f}-\frac {(183 e-f h) (h+183 x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {366 (183 e-f h)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {33489 (183 e-f h)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(183 e-f h)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}-\frac {\left (183 b^2 (183 e-f h)\right ) \text {Subst}\left (\int \left (2 (183 e-f h)^2-183 (183 e-f h) x+7442 x^2\right ) \, dx,x,e+f x\right )}{d f^5}+\frac {\left (366 b^2 (183 e-f h)^3\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^5}+\frac {\left (2 b^2 (183 e-f h)^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{3 d f^5}\\ &=\frac {732 a b (183 e-f h)^3 x}{d f^4}-\frac {1464 b^2 (183 e-f h)^3 x}{d f^4}+\frac {100467 b^2 (183 e-f h)^2 (e+f x)^2}{2 d f^5}-\frac {1815848 b^2 (183 e-f h) (e+f x)^3}{d f^5}+\frac {1121513121 b^2 (e+f x)^4}{32 d f^5}+\frac {7 b^2 (183 e-f h)^4 \log ^2(e+f x)}{12 d f^5}+\frac {732 b^2 (183 e-f h)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {33489 b (183 e-f h)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^5}+\frac {b (183 e-f h) \left (\frac {1098 (183 e-f h)^2 (e+f x)}{f^3}-\frac {100467 (183 e-f h) (e+f x)^2}{f^3}+\frac {4085658 (e+f x)^3}{f^3}-\frac {2 (183 e-f h)^3 \log (e+f x)}{f^3}\right ) (a+b \log (c (e+f x)))}{3 d f^2}+\frac {b \left (\frac {2928 (183 e-f h)^3 (e+f x)}{f^4}-\frac {401868 (183 e-f h)^2 (e+f x)^2}{f^4}+\frac {32685264 (183 e-f h) (e+f x)^3}{f^4}-\frac {1121513121 (e+f x)^4}{f^4}-\frac {4 (183 e-f h)^4 \log (e+f x)}{f^4}\right ) (a+b \log (c (e+f x)))}{8 d f}-\frac {(183 e-f h) (h+183 x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+183 x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {366 (183 e-f h)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {33489 (183 e-f h)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(183 e-f h)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}\\ \end {align*}

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Mathematica [A]
time = 0.64, size = 733, normalized size = 1.27 \begin {gather*} \frac {f i x \left (72 a^2 \left (-12 e^3 i^3+6 e^2 f i^2 (8 h+i x)-4 e f^2 i \left (18 h^2+6 h i x+i^2 x^2\right )+f^3 \left (48 h^3+36 h^2 i x+16 h i^2 x^2+3 i^3 x^3\right )\right )-12 a b \left (-300 e^3 i^3+6 e^2 f i^2 (176 h+13 i x)-4 e f^2 i \left (324 h^2+60 h i x+7 i^2 x^2\right )+f^3 \left (576 h^3+216 h^2 i x+64 h i^2 x^2+9 i^3 x^3\right )\right )+b^2 \left (-4980 e^3 i^3+30 e^2 f i^2 (544 h+23 i x)-4 e f^2 i \left (4536 h^2+456 h i x+37 i^2 x^2\right )+f^3 \left (6912 h^3+1296 h^2 i x+256 h i^2 x^2+27 i^3 x^3\right )\right )\right )+12 \left (72 a^2 (f h-e i)^4-12 a b e i \left (-48 f^3 h^3+108 e f^2 h^2 i-88 e^2 f h i^2+25 e^3 i^3\right )+b^2 e i \left (-576 f^3 h^3+1512 e f^2 h^2 i-1360 e^2 f h i^2+415 e^3 i^3\right )\right ) \log (e+f x)+12 b f i x \left (12 a \left (-12 e^3 i^3+6 e^2 f i^2 (8 h+i x)-4 e f^2 i \left (18 h^2+6 h i x+i^2 x^2\right )+f^3 \left (48 h^3+36 h^2 i x+16 h i^2 x^2+3 i^3 x^3\right )\right )-b \left (-300 e^3 i^3+6 e^2 f i^2 (176 h+13 i x)-4 e f^2 i \left (324 h^2+60 h i x+7 i^2 x^2\right )+f^3 \left (576 h^3+216 h^2 i x+64 h i^2 x^2+9 i^3 x^3\right )\right )\right ) \log (c (e+f x))+72 b \left (12 a (f h-e i)^4-b i (e+f x) \left (25 e^3 i^3-e^2 f i^2 (88 h+13 i x)+e f^2 i \left (108 h^2+40 h i x+7 i^2 x^2\right )-f^3 \left (48 h^3+36 h^2 i x+16 h i^2 x^2+3 i^3 x^3\right )\right )\right ) \log ^2(c (e+f x))+288 b^2 (f h-e i)^4 \log ^3(c (e+f x))}{864 d f^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((h + i*x)^4*(a + b*Log[c*(e + f*x)])^2)/(d*e + d*f*x),x]

[Out]

(f*i*x*(72*a^2*(-12*e^3*i^3 + 6*e^2*f*i^2*(8*h + i*x) - 4*e*f^2*i*(18*h^2 + 6*h*i*x + i^2*x^2) + f^3*(48*h^3 +
36*h^2*i*x + 16*h*i^2*x^2 + 3*i^3*x^3)) - 12*a*b*(-300*e^3*i^3 + 6*e^2*f*i^2*(176*h + 13*i*x) - 4*e*f^2*i*(32
4*h^2 + 60*h*i*x + 7*i^2*x^2) + f^3*(576*h^3 + 216*h^2*i*x + 64*h*i^2*x^2 + 9*i^3*x^3)) + b^2*(-4980*e^3*i^3 +
30*e^2*f*i^2*(544*h + 23*i*x) - 4*e*f^2*i*(4536*h^2 + 456*h*i*x + 37*i^2*x^2) + f^3*(6912*h^3 + 1296*h^2*i*x
+ 256*h*i^2*x^2 + 27*i^3*x^3))) + 12*(72*a^2*(f*h - e*i)^4 - 12*a*b*e*i*(-48*f^3*h^3 + 108*e*f^2*h^2*i - 88*e^
2*f*h*i^2 + 25*e^3*i^3) + b^2*e*i*(-576*f^3*h^3 + 1512*e*f^2*h^2*i - 1360*e^2*f*h*i^2 + 415*e^3*i^3))*Log[e +
f*x] + 12*b*f*i*x*(12*a*(-12*e^3*i^3 + 6*e^2*f*i^2*(8*h + i*x) - 4*e*f^2*i*(18*h^2 + 6*h*i*x + i^2*x^2) + f^3*
(48*h^3 + 36*h^2*i*x + 16*h*i^2*x^2 + 3*i^3*x^3)) - b*(-300*e^3*i^3 + 6*e^2*f*i^2*(176*h + 13*i*x) - 4*e*f^2*i
*(324*h^2 + 60*h*i*x + 7*i^2*x^2) + f^3*(576*h^3 + 216*h^2*i*x + 64*h*i^2*x^2 + 9*i^3*x^3)))*Log[c*(e + f*x)]
+ 72*b*(12*a*(f*h - e*i)^4 - b*i*(e + f*x)*(25*e^3*i^3 - e^2*f*i^2*(88*h + 13*i*x) + e*f^2*i*(108*h^2 + 40*h*i
*x + 7*i^2*x^2) - f^3*(48*h^3 + 36*h^2*i*x + 16*h*i^2*x^2 + 3*i^3*x^3)))*Log[c*(e + f*x)]^2 + 288*b^2*(f*h - e
*i)^4*Log[c*(e + f*x)]^3)/(864*d*f^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1890\) vs. \(2(557)=1114\).
time = 0.75, size = 1891, normalized size = 3.27

method	result	size
norman	\(\frac {\left (72 a^{2} e^{4} i^{4}-288 a^{2} e^{3} f h \,i^{3}+432 a^{2} e^{2} f^{2} h^{2} i^{2}-288 a^{2} e \,f^{3} h^{3} i +72 a^{2} h^{4} f^{4}-300 a b \,e^{4} i^{4}+1056 a b \,e^{3} f h \,i^{3}-1296 a b \,e^{2} f^{2} h^{2} i^{2}+576 a b e \,f^{3} h^{3} i +415 b^{2} e^{4} i^{4}-1360 b^{2} e^{3} f h \,i^{3}+1512 b^{2} e^{2} f^{2} h^{2} i^{2}-576 b^{2} e \,f^{3} h^{3} i \right ) \ln \left (c \left (f x +e \right )\right )}{72 d \,f^{5}}+\frac {b \left (12 a \,e^{4} i^{4}-48 a \,e^{3} f h \,i^{3}+72 a \,e^{2} f^{2} h^{2} i^{2}-48 a e \,f^{3} h^{3} i +12 a \,h^{4} f^{4}-25 b \,e^{4} i^{4}+88 b \,e^{3} f h \,i^{3}-108 b \,e^{2} f^{2} h^{2} i^{2}+48 b e \,f^{3} h^{3} i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{12 d \,f^{5}}+\frac {b^{2} \left (e^{4} i^{4}-4 e^{3} f h \,i^{3}+6 e^{2} f^{2} h^{2} i^{2}-4 e \,f^{3} h^{3} i +f^{4} h^{4}\right ) \ln \left (c \left (f x +e \right )\right )^{3}}{3 d \,f^{5}}-\frac {i \left (72 a^{2} e^{3} i^{3}-288 a^{2} e^{2} f h \,i^{2}+432 a^{2} e \,f^{2} h^{2} i -288 a^{2} f^{3} h^{3}-300 a b \,e^{3} i^{3}+1056 a b \,e^{2} f h \,i^{2}-1296 a b e \,f^{2} h^{2} i +576 a b \,f^{3} h^{3}+415 b^{2} e^{3} i^{3}-1360 b^{2} e^{2} f h \,i^{2}+1512 b^{2} e \,f^{2} h^{2} i -576 b^{2} f^{3} h^{3}\right ) x}{72 d \,f^{4}}+\frac {i^{2} \left (72 a^{2} e^{2} i^{2}-288 a^{2} e f h i +432 a^{2} f^{2} h^{2}-156 a b \,e^{2} i^{2}+480 a b e f h i -432 a b \,f^{2} h^{2}+115 b^{2} e^{2} i^{2}-304 b^{2} e f h i +216 b^{2} f^{2} h^{2}\right ) x^{2}}{144 d \,f^{3}}-\frac {i^{3} \left (72 a^{2} e i -288 a^{2} f h -84 a b e i +192 a b f h +37 b^{2} e i -64 b^{2} f h \right ) x^{3}}{216 f^{2} d}+\frac {i^{4} \left (8 a^{2}-4 b a +b^{2}\right ) x^{4}}{32 d f}+\frac {b^{2} i^{4} x^{4} \ln \left (c \left (f x +e \right )\right )^{2}}{4 d f}-\frac {b i \left (12 a \,e^{3} i^{3}-48 a \,e^{2} f h \,i^{2}+72 a e \,f^{2} h^{2} i -48 a \,f^{3} h^{3}-25 b \,e^{3} i^{3}+88 b \,e^{2} f h \,i^{2}-108 b e \,f^{2} h^{2} i +48 b \,f^{3} h^{3}\right ) x \ln \left (c \left (f x +e \right )\right )}{6 d \,f^{4}}+\frac {b \,i^{2} \left (12 a \,e^{2} i^{2}-48 a e f h i +72 a \,f^{2} h^{2}-13 b \,e^{2} i^{2}+40 b e f h i -36 b \,f^{2} h^{2}\right ) x^{2} \ln \left (c \left (f x +e \right )\right )}{12 d \,f^{3}}-\frac {b \,i^{3} \left (12 a e i -48 a f h -7 b e i +16 b f h \right ) x^{3} \ln \left (c \left (f x +e \right )\right )}{18 d \,f^{2}}+\frac {b \,i^{4} \left (4 a -b \right ) x^{4} \ln \left (c \left (f x +e \right )\right )}{8 d f}-\frac {b^{2} i \left (e^{3} i^{3}-4 e^{2} f h \,i^{2}+6 e \,f^{2} h^{2} i -4 f^{3} h^{3}\right ) x \ln \left (c \left (f x +e \right )\right )^{2}}{d \,f^{4}}+\frac {b^{2} i^{2} \left (e^{2} i^{2}-4 e f h i +6 f^{2} h^{2}\right ) x^{2} \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{3}}-\frac {b^{2} i^{3} \left (e i -4 f h \right ) x^{3} \ln \left (c \left (f x +e \right )\right )^{2}}{3 d \,f^{2}}\)	\(1135\)
risch	\(\text {Expression too large to display}\)	\(1475\)
derivativedivides	\(\text {Expression too large to display}\)	\(1891\)
default	\(\text {Expression too large to display}\)	\(1891\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x+h)^4*(a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e),x,method=_RETURNVERBOSE)

[Out]

1/c/f*(24/f^3/d*a*b*e^2*h*i^3*((c*f*x+c*e)*ln(c*f*x+c*e)-c*f*x-c*e)+c/d*a*b*h^4*ln(c*f*x+c*e)^2+12/f^3/d*b^2*e
^2*h*i^3*((c*f*x+c*e)*ln(c*f*x+c*e)^2-2*(c*f*x+c*e)*ln(c*f*x+c*e)+2*c*f*x+2*c*e)+8/f/d*a*b*h^3*i*((c*f*x+c*e)*
ln(c*f*x+c*e)-c*f*x-c*e)+3/c/f^2/d*a^2*h^2*i^2*(c*f*x+c*e)^2+4/c^2/f^3/d*b^2*h*i^3*(1/3*(c*f*x+c*e)^3*ln(c*f*x
+c*e)^2-2/9*(c*f*x+c*e)^3*ln(c*f*x+c*e)+2/27*(c*f*x+c*e)^3)+12/f^3/d*a^2*e^2*h*i^3*(c*f*x+c*e)-12/f^2/d*a^2*e*
h^2*i^2*(c*f*x+c*e)-12/f^2/d*b^2*e*h^2*i^2*((c*f*x+c*e)*ln(c*f*x+c*e)^2-2*(c*f*x+c*e)*ln(c*f*x+c*e)+2*c*f*x+2*
c*e)+c/f^4/d*a^2*e^4*i^4*ln(c*f*x+c*e)-4/3/c^2/f^4/d*a^2*e*i^4*(c*f*x+c*e)^3-8/f^4/d*a*b*e^3*i^4*((c*f*x+c*e)*
ln(c*f*x+c*e)-c*f*x-c*e)+3/c/f^4/d*a^2*e^2*i^4*(c*f*x+c*e)^2-4/c^2/f^4/d*b^2*e*i^4*(1/3*(c*f*x+c*e)^3*ln(c*f*x
+c*e)^2-2/9*(c*f*x+c*e)^3*ln(c*f*x+c*e)+2/27*(c*f*x+c*e)^3)+6/c/f^4/d*b^2*e^2*i^4*(1/2*(c*f*x+c*e)^2*ln(c*f*x+
c*e)^2-1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)+1/4*(c*f*x+c*e)^2)+4/3/c^2/f^3/d*a^2*h*i^3*(c*f*x+c*e)^3+2/c^3/f^4/d*a*
b*i^4*(1/4*(c*f*x+c*e)^4*ln(c*f*x+c*e)-1/16*(c*f*x+c*e)^4)+1/3*c/d*b^2*h^4*ln(c*f*x+c*e)^3+c/d*a^2*h^4*ln(c*f*
x+c*e)-4/f^4/d*b^2*e^3*i^4*((c*f*x+c*e)*ln(c*f*x+c*e)^2-2*(c*f*x+c*e)*ln(c*f*x+c*e)+2*c*f*x+2*c*e)+4/f/d*b^2*h
^3*i*((c*f*x+c*e)*ln(c*f*x+c*e)^2-2*(c*f*x+c*e)*ln(c*f*x+c*e)+2*c*f*x+2*c*e)+1/4/c^3/f^4/d*a^2*i^4*(c*f*x+c*e)
^4+12/c/f^4/d*a*b*e^2*i^4*(1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)-1/4*(c*f*x+c*e)^2)+8/c^2/f^3/d*a*b*h*i^3*(1/3*(c*f*
x+c*e)^3*ln(c*f*x+c*e)-1/9*(c*f*x+c*e)^3)-4*c/f/d*a^2*e*h^3*i*ln(c*f*x+c*e)-24/f^2/d*a*b*e*h^2*i^2*((c*f*x+c*e
)*ln(c*f*x+c*e)-c*f*x-c*e)-4*c/f^3/d*a^2*e^3*h*i^3*ln(c*f*x+c*e)+c/f^4/d*a*b*e^4*i^4*ln(c*f*x+c*e)^2-8/c^2/f^4
/d*a*b*e*i^4*(1/3*(c*f*x+c*e)^3*ln(c*f*x+c*e)-1/9*(c*f*x+c*e)^3)-4/3*c/f^3/d*b^2*e^3*h*i^3*ln(c*f*x+c*e)^3+2*c
/f^2/d*b^2*e^2*h^2*i^2*ln(c*f*x+c*e)^3-12/c/f^3/d*b^2*e*h*i^3*(1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)^2-1/2*(c*f*x+c*
e)^2*ln(c*f*x+c*e)+1/4*(c*f*x+c*e)^2)-6/c/f^3/d*a^2*e*h*i^3*(c*f*x+c*e)^2+6*c/f^2/d*a^2*e^2*h^2*i^2*ln(c*f*x+c
*e)-4/3*c/f/d*b^2*e*h^3*i*ln(c*f*x+c*e)^3+12/c/f^2/d*a*b*h^2*i^2*(1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)-1/4*(c*f*x+c
*e)^2)+6*c/f^2/d*a*b*e^2*h^2*i^2*ln(c*f*x+c*e)^2-4*c/f/d*a*b*e*h^3*i*ln(c*f*x+c*e)^2-4*c/f^3/d*a*b*e^3*h*i^3*l
n(c*f*x+c*e)^2-24/c/f^3/d*a*b*e*h*i^3*(1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)-1/4*(c*f*x+c*e)^2)-4/f^4/d*a^2*e^3*i^4*
(c*f*x+c*e)+4/f/d*a^2*h^3*i*(c*f*x+c*e)+1/c^3/f^4/d*b^2*i^4*(1/4*(c*f*x+c*e)^4*ln(c*f*x+c*e)^2-1/8*(c*f*x+c*e)
^4*ln(c*f*x+c*e)+1/32*(c*f*x+c*e)^4)+1/3*c/f^4/d*b^2*e^4*i^4*ln(c*f*x+c*e)^3+6/c/f^2/d*b^2*h^2*i^2*(1/2*(c*f*x
+c*e)^2*ln(c*f*x+c*e)^2-1/2*(c*f*x+c*e)^2*ln(c*f*x+c*e)+1/4*(c*f*x+c*e)^2))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1437 vs. \(2 (554) = 1108\).
time = 0.32, size = 1437, normalized size = 2.48 \begin {gather*} -a b h^{4} {\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 )} + 8 i \, a b h^{3} {\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 {b^{2} h^{4} \log \left (c f x + c e\right )^{3}}{3 \, d f} + 4 i \, a^{2} h^{3} {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - 6 \, a b h^{2} {\left (\frac {f x^{2} - 2 \, x e}{d f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac {2 \, a b h^{4} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - 3 \, a^{2} h^{2} {\left (\frac {f x^{2} - 2 \, x e}{d f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}}\right )} - \frac {4}{3} i \, a b h {\left (\frac {2 \, f^{2} x^{3} - 3 \, f x^{2} e + 6 \, x e^{2}}{d f^{3}} - \frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}}\right )} \log \left (c f x + c e\right ) + \frac {a^{2} h^{4} \log \left (d f x + d e\right )}{d f} - \frac {2}{3} i \, a^{2} h {\left (\frac {2 \, f^{2} x^{3} - 3 \, f x^{2} e + 6 \, x e^{2}}{d f^{3}} - \frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}}\right )} + \frac {1}{6} \, a b {\left (\frac {3 \, f^{3} x^{4} - 4 \, f^{2} x^{3} e + 6 \, f x^{2} e^{2} - 12 \, x e^{3}}{d f^{4}} + \frac {12 \, e^{4} \log \left (f x + e\right )}{d f^{5}}\right )} \log \left (c f x + c e\right ) + \frac {4 i \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b h^{3}}{d f^{2}} + \frac {1}{12} \, a^{2} {\left (\frac {3 \, f^{3} x^{4} - 4 \, f^{2} x^{3} e + 6 \, f x^{2} e^{2} - 12 \, x e^{3}}{d f^{4}} + \frac {12 \, e^{4} \log \left (f x + e\right )}{d f^{5}}\right )} + \frac {3 \, {\left (f^{2} x^{2} - 6 \, f x e + 2 \, e^{2} \log \left (f x + e\right )^{2} + 6 \, e^{2} \log \left (f x + e\right )\right )} a b h^{2}}{d f^{3}} - \frac {4 i \, {\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^{3}}{3 \, c^{2} d f^{2}} + \frac {2 i \, {\left (4 \, f^{3} x^{3} - 15 \, f^{2} x^{2} e + 66 \, f x e^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} - 66 \, e^{3} \log \left (f x + e\right )\right )} a b h}{9 \, 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^{2}}{2 \, c^{3} d f^{3}} - \frac {{\left (9 \, f^{4} x^{4} - 28 \, f^{3} x^{3} e + 78 \, f^{2} x^{2} e^{2} - 300 \, f x e^{3} + 72 \, e^{4} \log \left (f x + e\right )^{2} + 300 \, e^{4} \log \left (f x + e\right )\right )} a b}{72 \, d f^{5}} + \frac {i \, {\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} h}{27 \, c^{4} d f^{4}} + \frac {{\left (288 \, c^{5} e^{4} \log \left (c f x + c e\right )^{3} + 27 \, {\left (c f x + c e\right )}^{4} {\left (8 \, c \log \left (c f x + c e\right )^{2} - 4 \, c \log \left (c f x + c e\right ) + c\right )} - 128 \, {\left (9 \, c^{2} e \log \left (c f x + c e\right )^{2} - 6 \, c^{2} e \log \left (c f x + c e\right ) + 2 \, c^{2} e\right )} {\left (c f x + c e\right )}^{3} + 1296 \, {\left (2 \, 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 ) + c^{3} e^{2}\right )} {\left (c f x + c e\right )}^{2} - 3456 \, {\left (c^{4} e^{3} \log \left (c f x + c e\right )^{2} - 2 \, c^{4} e^{3} \log \left (c f x + c e\right ) + 2 \, c^{4} e^{3}\right )} {\left (c f x + c e\right )}\right )} b^{2}}{864 \, c^{5} d f^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^4*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="maxima")

[Out]

-a*b*h^4*(2*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - (log(f*x + e)^2 + 2*log(f*x + e)*log(c))/(d*f)) + 8*I*a*
b*h^3*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + 1/3*b^2*h^4*log(c*f*x + c*e)^3/(d*f) + 4*I*a^2*h^3
*(x/(d*f) - e*log(f*x + e)/(d*f^2)) - 6*a*b*h^2*((f*x^2 - 2*x*e)/(d*f^2) + 2*e^2*log(f*x + e)/(d*f^3))*log(c*f
*x + c*e) + 2*a*b*h^4*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - 3*a^2*h^2*((f*x^2 - 2*x*e)/(d*f^2) + 2*e^2*log
(f*x + e)/(d*f^3)) - 4/3*I*a*b*h*((2*f^2*x^3 - 3*f*x^2*e + 6*x*e^2)/(d*f^3) - 6*e^3*log(f*x + e)/(d*f^4))*log(
c*f*x + c*e) + a^2*h^4*log(d*f*x + d*e)/(d*f) - 2/3*I*a^2*h*((2*f^2*x^3 - 3*f*x^2*e + 6*x*e^2)/(d*f^3) - 6*e^3
*log(f*x + e)/(d*f^4)) + 1/6*a*b*((3*f^3*x^4 - 4*f^2*x^3*e + 6*f*x^2*e^2 - 12*x*e^3)/(d*f^4) + 12*e^4*log(f*x
+ e)/(d*f^5))*log(c*f*x + c*e) + 4*I*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*a*b*h^3/(d*f^2) + 1/12*a^2*
((3*f^3*x^4 - 4*f^2*x^3*e + 6*f*x^2*e^2 - 12*x*e^3)/(d*f^4) + 12*e^4*log(f*x + e)/(d*f^5)) + 3*(f^2*x^2 - 6*f*
x*e + 2*e^2*log(f*x + e)^2 + 6*e^2*log(f*x + e))*a*b*h^2/(d*f^3) - 4/3*I*(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^3/(c^2*d*f^2) + 2/9*I*(4*f^3*x^3 - 15*f^2*x^
2*e + 66*f*x*e^2 - 18*e^3*log(f*x + e)^2 - 66*e^3*log(f*x + e))*a*b*h/(d*f^4) - 1/2*(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^2/(c^3*d*f^3) - 1/72*(9*f^4*x^4 - 28*f^3*x^3*e + 78*f^
2*x^2*e^2 - 300*f*x*e^3 + 72*e^4*log(f*x + e)^2 + 300*e^4*log(f*x + e))*a*b/(d*f^5) + 1/27*I*(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*l
og(c*f*x + c*e) + 2*c^3*e^2)*(c*f*x + c*e))*b^2*h/(c^4*d*f^4) + 1/864*(288*c^5*e^4*log(c*f*x + c*e)^3 + 27*(c*
f*x + c*e)^4*(8*c*log(c*f*x + c*e)^2 - 4*c*log(c*f*x + c*e) + c) - 128*(9*c^2*e*log(c*f*x + c*e)^2 - 6*c^2*e*l
og(c*f*x + c*e) + 2*c^2*e)*(c*f*x + c*e)^3 + 1296*(2*c^3*e^2*log(c*f*x + c*e)^2 - 2*c^3*e^2*log(c*f*x + c*e) +
c^3*e^2)*(c*f*x + c*e)^2 - 3456*(c^4*e^3*log(c*f*x + c*e)^2 - 2*c^4*e^3*log(c*f*x + c*e) + 2*c^4*e^3)*(c*f*x
+ c*e))*b^2/(c^5*d*f^5)

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Fricas [A]
time = 0.38, size = 833, normalized size = 1.44 \begin {gather*} -\frac {3456 \, {\left (-i \, a^{2} + 2 i \, a b - 2 i \, b^{2}\right )} f^{4} h^{3} x + 1296 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{4} h^{2} x^{2} + 128 \, {\left (9 i \, a^{2} - 6 i \, a b + 2 i \, b^{2}\right )} f^{4} h x^{3} - 27 \, {\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} f^{4} x^{4} + 12 \, {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} f x e^{3} - 288 \, {\left (b^{2} f^{4} h^{4} - 4 i \, b^{2} f^{3} h^{3} e - 6 \, b^{2} f^{2} h^{2} e^{2} + 4 i \, b^{2} f h e^{3} + b^{2} e^{4}\right )} \log \left (c f x + c e\right )^{3} - 72 \, {\left (12 \, a b f^{4} h^{4} + 48 i \, b^{2} f^{4} h^{3} x - 36 \, b^{2} f^{4} h^{2} x^{2} - 16 i \, b^{2} f^{4} h x^{3} + 3 \, b^{2} f^{4} x^{4} + {\left (12 \, a b - 25 \, b^{2}\right )} e^{4} - 4 \, {\left (3 \, b^{2} f x + 2 \, {\left (-6 i \, a b + 11 i \, b^{2}\right )} f h\right )} e^{3} - 6 \, {\left (8 i \, b^{2} f^{2} h x - b^{2} f^{2} x^{2} + 6 \, {\left (2 \, a b - 3 \, b^{2}\right )} f^{2} h^{2}\right )} e^{2} + 4 \, {\left (18 \, b^{2} f^{3} h^{2} x + 6 i \, b^{2} f^{3} h x^{2} - b^{2} f^{3} x^{3} - 12 \, {\left (i \, a b - i \, b^{2}\right )} f^{3} h^{3}\right )} e\right )} \log \left (c f x + c e\right )^{2} + 6 \, {\left (32 \, {\left (18 i \, a^{2} - 66 i \, a b + 85 i \, b^{2}\right )} f^{2} h x - {\left (72 \, a^{2} - 156 \, a b + 115 \, b^{2}\right )} f^{2} x^{2}\right )} e^{2} - 4 \, {\left (648 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} f^{3} h^{2} x - 24 \, {\left (-18 i \, a^{2} + 30 i \, a b - 19 i \, b^{2}\right )} f^{3} h x^{2} - {\left (72 \, a^{2} - 84 \, a b + 37 \, b^{2}\right )} f^{3} x^{3}\right )} e - 12 \, {\left (72 \, a^{2} f^{4} h^{4} - 576 \, {\left (-i \, a b + i \, b^{2}\right )} f^{4} h^{3} x - 216 \, {\left (2 \, a b - b^{2}\right )} f^{4} h^{2} x^{2} - 64 \, {\left (3 i \, a b - i \, b^{2}\right )} f^{4} h x^{3} + 9 \, {\left (4 \, a b - b^{2}\right )} f^{4} x^{4} + {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{4} - 4 \, {\left (4 \, {\left (-18 i \, a^{2} + 66 i \, a b - 85 i \, b^{2}\right )} f h + 3 \, {\left (12 \, a b - 25 \, b^{2}\right )} f x\right )} e^{3} - 6 \, {\left (36 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} f^{2} h^{2} + 16 \, {\left (6 i \, a b - 11 i \, b^{2}\right )} f^{2} h x - {\left (12 \, a b - 13 \, b^{2}\right )} f^{2} x^{2}\right )} e^{2} - 4 \, {\left (72 \, {\left (i \, a^{2} - 2 i \, a b + 2 i \, b^{2}\right )} f^{3} h^{3} - 108 \, {\left (2 \, a b - 3 \, b^{2}\right )} f^{3} h^{2} x + 12 \, {\left (-6 i \, a b + 5 i \, b^{2}\right )} f^{3} h x^{2} + {\left (12 \, a b - 7 \, b^{2}\right )} f^{3} x^{3}\right )} e\right )} \log \left (c f x + c e\right )}{864 \, d f^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^4*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="fricas")

[Out]

-1/864*(3456*(-I*a^2 + 2*I*a*b - 2*I*b^2)*f^4*h^3*x + 1296*(2*a^2 - 2*a*b + b^2)*f^4*h^2*x^2 + 128*(9*I*a^2 -
6*I*a*b + 2*I*b^2)*f^4*h*x^3 - 27*(8*a^2 - 4*a*b + b^2)*f^4*x^4 + 12*(72*a^2 - 300*a*b + 415*b^2)*f*x*e^3 - 28
8*(b^2*f^4*h^4 - 4*I*b^2*f^3*h^3*e - 6*b^2*f^2*h^2*e^2 + 4*I*b^2*f*h*e^3 + b^2*e^4)*log(c*f*x + c*e)^3 - 72*(1
2*a*b*f^4*h^4 + 48*I*b^2*f^4*h^3*x - 36*b^2*f^4*h^2*x^2 - 16*I*b^2*f^4*h*x^3 + 3*b^2*f^4*x^4 + (12*a*b - 25*b^
2)*e^4 - 4*(3*b^2*f*x + 2*(-6*I*a*b + 11*I*b^2)*f*h)*e^3 - 6*(8*I*b^2*f^2*h*x - b^2*f^2*x^2 + 6*(2*a*b - 3*b^2
)*f^2*h^2)*e^2 + 4*(18*b^2*f^3*h^2*x + 6*I*b^2*f^3*h*x^2 - b^2*f^3*x^3 - 12*(I*a*b - I*b^2)*f^3*h^3)*e)*log(c*
f*x + c*e)^2 + 6*(32*(18*I*a^2 - 66*I*a*b + 85*I*b^2)*f^2*h*x - (72*a^2 - 156*a*b + 115*b^2)*f^2*x^2)*e^2 - 4*
(648*(2*a^2 - 6*a*b + 7*b^2)*f^3*h^2*x - 24*(-18*I*a^2 + 30*I*a*b - 19*I*b^2)*f^3*h*x^2 - (72*a^2 - 84*a*b + 3
7*b^2)*f^3*x^3)*e - 12*(72*a^2*f^4*h^4 - 576*(-I*a*b + I*b^2)*f^4*h^3*x - 216*(2*a*b - b^2)*f^4*h^2*x^2 - 64*(
3*I*a*b - I*b^2)*f^4*h*x^3 + 9*(4*a*b - b^2)*f^4*x^4 + (72*a^2 - 300*a*b + 415*b^2)*e^4 - 4*(4*(-18*I*a^2 + 66
*I*a*b - 85*I*b^2)*f*h + 3*(12*a*b - 25*b^2)*f*x)*e^3 - 6*(36*(2*a^2 - 6*a*b + 7*b^2)*f^2*h^2 + 16*(6*I*a*b -
11*I*b^2)*f^2*h*x - (12*a*b - 13*b^2)*f^2*x^2)*e^2 - 4*(72*(I*a^2 - 2*I*a*b + 2*I*b^2)*f^3*h^3 - 108*(2*a*b -
3*b^2)*f^3*h^2*x + 12*(-6*I*a*b + 5*I*b^2)*f^3*h*x^2 + (12*a*b - 7*b^2)*f^3*x^3)*e)*log(c*f*x + c*e))/(d*f^5)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (534) = 1068\).
time = 2.27, size = 1479, normalized size = 2.55 \begin {gather*} x^{4} \left (\frac {a^{2} i^{4}}{4 d f} - \frac {a b i^{4}}{8 d f} + \frac {b^{2} i^{4}}{32 d f}\right ) + x^{3} \left (- \frac {a^{2} e i^{4}}{3 d f^{2}} + \frac {4 a^{2} h i^{3}}{3 d f} + \frac {7 a b e i^{4}}{18 d f^{2}} - \frac {8 a b h i^{3}}{9 d f} - \frac {37 b^{2} e i^{4}}{216 d f^{2}} + \frac {8 b^{2} h i^{3}}{27 d f}\right ) + x^{2} \left (\frac {a^{2} e^{2} i^{4}}{2 d f^{3}} - \frac {2 a^{2} e h i^{3}}{d f^{2}} + \frac {3 a^{2} h^{2} i^{2}}{d f} - \frac {13 a b e^{2} i^{4}}{12 d f^{3}} + \frac {10 a b e h i^{3}}{3 d f^{2}} - \frac {3 a b h^{2} i^{2}}{d f} + \frac {115 b^{2} e^{2} i^{4}}{144 d f^{3}} - \frac {19 b^{2} e h i^{3}}{9 d f^{2}} + \frac {3 b^{2} h^{2} i^{2}}{2 d f}\right ) + x \left (- \frac {a^{2} e^{3} i^{4}}{d f^{4}} + \frac {4 a^{2} e^{2} h i^{3}}{d f^{3}} - \frac {6 a^{2} e h^{2} i^{2}}{d f^{2}} + \frac {4 a^{2} h^{3} i}{d f} + \frac {25 a b e^{3} i^{4}}{6 d f^{4}} - \frac {44 a b e^{2} h i^{3}}{3 d f^{3}} + \frac {18 a b e h^{2} i^{2}}{d f^{2}} - \frac {8 a b h^{3} i}{d f} - \frac {415 b^{2} e^{3} i^{4}}{72 d f^{4}} + \frac {170 b^{2} e^{2} h i^{3}}{9 d f^{3}} - \frac {21 b^{2} e h^{2} i^{2}}{d f^{2}} + \frac {8 b^{2} h^{3} i}{d f}\right ) + \frac {\left (- 144 a b e^{3} i^{4} x + 576 a b e^{2} f h i^{3} x + 72 a b e^{2} f i^{4} x^{2} - 864 a b e f^{2} h^{2} i^{2} x - 288 a b e f^{2} h i^{3} x^{2} - 48 a b e f^{2} i^{4} x^{3} + 576 a b f^{3} h^{3} i x + 432 a b f^{3} h^{2} i^{2} x^{2} + 192 a b f^{3} h i^{3} x^{3} + 36 a b f^{3} i^{4} x^{4} + 300 b^{2} e^{3} i^{4} x - 1056 b^{2} e^{2} f h i^{3} x - 78 b^{2} e^{2} f i^{4} x^{2} + 1296 b^{2} e f^{2} h^{2} i^{2} x + 240 b^{2} e f^{2} h i^{3} x^{2} + 28 b^{2} e f^{2} i^{4} x^{3} - 576 b^{2} f^{3} h^{3} i x - 216 b^{2} f^{3} h^{2} i^{2} x^{2} - 64 b^{2} f^{3} h i^{3} x^{3} - 9 b^{2} f^{3} i^{4} x^{4}\right ) \log {\left (c \left (e + f x\right ) \right )}}{72 d f^{4}} + \frac {\left (b^{2} e^{4} i^{4} - 4 b^{2} e^{3} f h i^{3} + 6 b^{2} e^{2} f^{2} h^{2} i^{2} - 4 b^{2} e f^{3} h^{3} i + b^{2} f^{4} h^{4}\right ) \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{5}} + \frac {\left (72 a^{2} e^{4} i^{4} - 288 a^{2} e^{3} f h i^{3} + 432 a^{2} e^{2} f^{2} h^{2} i^{2} - 288 a^{2} e f^{3} h^{3} i + 72 a^{2} f^{4} h^{4} - 300 a b e^{4} i^{4} + 1056 a b e^{3} f h i^{3} - 1296 a b e^{2} f^{2} h^{2} i^{2} + 576 a b e f^{3} h^{3} i + 415 b^{2} e^{4} i^{4} - 1360 b^{2} e^{3} f h i^{3} + 1512 b^{2} e^{2} f^{2} h^{2} i^{2} - 576 b^{2} e f^{3} h^{3} i\right ) \log {\left (e + f x \right )}}{72 d f^{5}} + \frac {\left (12 a b e^{4} i^{4} - 48 a b e^{3} f h i^{3} + 72 a b e^{2} f^{2} h^{2} i^{2} - 48 a b e f^{3} h^{3} i + 12 a b f^{4} h^{4} - 25 b^{2} e^{4} i^{4} + 88 b^{2} e^{3} f h i^{3} - 12 b^{2} e^{3} f i^{4} x - 108 b^{2} e^{2} f^{2} h^{2} i^{2} + 48 b^{2} e^{2} f^{2} h i^{3} x + 6 b^{2} e^{2} f^{2} i^{4} x^{2} + 48 b^{2} e f^{3} h^{3} i - 72 b^{2} e f^{3} h^{2} i^{2} x - 24 b^{2} e f^{3} h i^{3} x^{2} - 4 b^{2} e f^{3} i^{4} x^{3} + 48 b^{2} f^{4} h^{3} i x + 36 b^{2} f^{4} h^{2} i^{2} x^{2} + 16 b^{2} f^{4} h i^{3} x^{3} + 3 b^{2} f^{4} i^{4} x^{4}\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{12 d f^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)**4*(a+b*ln(c*(f*x+e)))**2/(d*f*x+d*e),x)

[Out]

x**4*(a**2*i**4/(4*d*f) - a*b*i**4/(8*d*f) + b**2*i**4/(32*d*f)) + x**3*(-a**2*e*i**4/(3*d*f**2) + 4*a**2*h*i*
*3/(3*d*f) + 7*a*b*e*i**4/(18*d*f**2) - 8*a*b*h*i**3/(9*d*f) - 37*b**2*e*i**4/(216*d*f**2) + 8*b**2*h*i**3/(27
*d*f)) + x**2*(a**2*e**2*i**4/(2*d*f**3) - 2*a**2*e*h*i**3/(d*f**2) + 3*a**2*h**2*i**2/(d*f) - 13*a*b*e**2*i**
4/(12*d*f**3) + 10*a*b*e*h*i**3/(3*d*f**2) - 3*a*b*h**2*i**2/(d*f) + 115*b**2*e**2*i**4/(144*d*f**3) - 19*b**2
*e*h*i**3/(9*d*f**2) + 3*b**2*h**2*i**2/(2*d*f)) + x*(-a**2*e**3*i**4/(d*f**4) + 4*a**2*e**2*h*i**3/(d*f**3) -
6*a**2*e*h**2*i**2/(d*f**2) + 4*a**2*h**3*i/(d*f) + 25*a*b*e**3*i**4/(6*d*f**4) - 44*a*b*e**2*h*i**3/(3*d*f**
3) + 18*a*b*e*h**2*i**2/(d*f**2) - 8*a*b*h**3*i/(d*f) - 415*b**2*e**3*i**4/(72*d*f**4) + 170*b**2*e**2*h*i**3/
(9*d*f**3) - 21*b**2*e*h**2*i**2/(d*f**2) + 8*b**2*h**3*i/(d*f)) + (-144*a*b*e**3*i**4*x + 576*a*b*e**2*f*h*i*
*3*x + 72*a*b*e**2*f*i**4*x**2 - 864*a*b*e*f**2*h**2*i**2*x - 288*a*b*e*f**2*h*i**3*x**2 - 48*a*b*e*f**2*i**4*
x**3 + 576*a*b*f**3*h**3*i*x + 432*a*b*f**3*h**2*i**2*x**2 + 192*a*b*f**3*h*i**3*x**3 + 36*a*b*f**3*i**4*x**4
+ 300*b**2*e**3*i**4*x - 1056*b**2*e**2*f*h*i**3*x - 78*b**2*e**2*f*i**4*x**2 + 1296*b**2*e*f**2*h**2*i**2*x +
240*b**2*e*f**2*h*i**3*x**2 + 28*b**2*e*f**2*i**4*x**3 - 576*b**2*f**3*h**3*i*x - 216*b**2*f**3*h**2*i**2*x**
2 - 64*b**2*f**3*h*i**3*x**3 - 9*b**2*f**3*i**4*x**4)*log(c*(e + f*x))/(72*d*f**4) + (b**2*e**4*i**4 - 4*b**2*
e**3*f*h*i**3 + 6*b**2*e**2*f**2*h**2*i**2 - 4*b**2*e*f**3*h**3*i + b**2*f**4*h**4)*log(c*(e + f*x))**3/(3*d*f
**5) + (72*a**2*e**4*i**4 - 288*a**2*e**3*f*h*i**3 + 432*a**2*e**2*f**2*h**2*i**2 - 288*a**2*e*f**3*h**3*i + 7
2*a**2*f**4*h**4 - 300*a*b*e**4*i**4 + 1056*a*b*e**3*f*h*i**3 - 1296*a*b*e**2*f**2*h**2*i**2 + 576*a*b*e*f**3*
h**3*i + 415*b**2*e**4*i**4 - 1360*b**2*e**3*f*h*i**3 + 1512*b**2*e**2*f**2*h**2*i**2 - 576*b**2*e*f**3*h**3*i
)*log(e + f*x)/(72*d*f**5) + (12*a*b*e**4*i**4 - 48*a*b*e**3*f*h*i**3 + 72*a*b*e**2*f**2*h**2*i**2 - 48*a*b*e*
f**3*h**3*i + 12*a*b*f**4*h**4 - 25*b**2*e**4*i**4 + 88*b**2*e**3*f*h*i**3 - 12*b**2*e**3*f*i**4*x - 108*b**2*
e**2*f**2*h**2*i**2 + 48*b**2*e**2*f**2*h*i**3*x + 6*b**2*e**2*f**2*i**4*x**2 + 48*b**2*e*f**3*h**3*i - 72*b**
2*e*f**3*h**2*i**2*x - 24*b**2*e*f**3*h*i**3*x**2 - 4*b**2*e*f**3*i**4*x**3 + 48*b**2*f**4*h**3*i*x + 36*b**2*
f**4*h**2*i**2*x**2 + 16*b**2*f**4*h*i**3*x**3 + 3*b**2*f**4*i**4*x**4)*log(c*(e + f*x))**2/(12*d*f**5)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (554) = 1108\).
time = 4.17, size = 1588, normalized size = 2.74 \begin {gather*} \frac {288 \, b^{2} f^{4} h^{4} \log \left (c f x + c e\right )^{3} + 864 \, a b f^{4} h^{4} \log \left (c f x + c e\right )^{2} + 3456 i \, b^{2} f^{4} h^{3} x \log \left (c f x + c e\right )^{2} - 2592 \, b^{2} f^{4} h^{2} x^{2} \log \left (c f x + c e\right )^{2} - 1152 i \, b^{2} f^{4} h x^{3} \log \left (c f x + c e\right )^{2} + 216 \, b^{2} f^{4} x^{4} \log \left (c f x + c e\right )^{2} - 1152 i \, b^{2} f^{3} h^{3} e \log \left (c f x + c e\right )^{3} + 6912 i \, a b f^{4} h^{3} x \log \left (c f x + c e\right ) - 6912 i \, b^{2} f^{4} h^{3} x \log \left (c f x + c e\right ) - 5184 \, a b f^{4} h^{2} x^{2} \log \left (c f x + c e\right ) + 2592 \, b^{2} f^{4} h^{2} x^{2} \log \left (c f x + c e\right ) - 2304 i \, a b f^{4} h x^{3} \log \left (c f x + c e\right ) + 768 i \, b^{2} f^{4} h x^{3} \log \left (c f x + c e\right ) + 432 \, a b f^{4} x^{4} \log \left (c f x + c e\right ) - 108 \, b^{2} f^{4} x^{4} \log \left (c f x + c e\right ) - 3456 i \, a b f^{3} h^{3} e \log \left (c f x + c e\right )^{2} + 3456 i \, b^{2} f^{3} h^{3} e \log \left (c f x + c e\right )^{2} + 5184 \, b^{2} f^{3} h^{2} x e \log \left (c f x + c e\right )^{2} + 1728 i \, b^{2} f^{3} h x^{2} e \log \left (c f x + c e\right )^{2} - 288 \, b^{2} f^{3} x^{3} e \log \left (c f x + c e\right )^{2} + 864 \, a^{2} f^{4} h^{4} \log \left (f x + e\right ) + 3456 i \, a^{2} f^{4} h^{3} x - 6912 i \, a b f^{4} h^{3} x + 6912 i \, b^{2} f^{4} h^{3} x - 2592 \, a^{2} f^{4} h^{2} x^{2} + 2592 \, a b f^{4} h^{2} x^{2} - 1296 \, b^{2} f^{4} h^{2} x^{2} - 1152 i \, a^{2} f^{4} h x^{3} + 768 i \, a b f^{4} h x^{3} - 256 i \, b^{2} f^{4} h x^{3} + 216 \, a^{2} f^{4} x^{4} - 108 \, a b f^{4} x^{4} + 27 \, b^{2} f^{4} x^{4} + 10368 \, a b f^{3} h^{2} x e \log \left (c f x + c e\right ) - 15552 \, b^{2} f^{3} h^{2} x e \log \left (c f x + c e\right ) + 3456 i \, a b f^{3} h x^{2} e \log \left (c f x + c e\right ) - 2880 i \, b^{2} f^{3} h x^{2} e \log \left (c f x + c e\right ) - 576 \, a b f^{3} x^{3} e \log \left (c f x + c e\right ) + 336 \, b^{2} f^{3} x^{3} e \log \left (c f x + c e\right ) - 1728 \, b^{2} f^{2} h^{2} e^{2} \log \left (c f x + c e\right )^{3} - 3456 i \, a^{2} f^{3} h^{3} e \log \left (f x + e\right ) + 6912 i \, a b f^{3} h^{3} e \log \left (f x + e\right ) - 6912 i \, b^{2} f^{3} h^{3} e \log \left (f x + e\right ) + 5184 \, a^{2} f^{3} h^{2} x e - 15552 \, a b f^{3} h^{2} x e + 18144 \, b^{2} f^{3} h^{2} x e + 1728 i \, a^{2} f^{3} h x^{2} e - 2880 i \, a b f^{3} h x^{2} e + 1824 i \, b^{2} f^{3} h x^{2} e - 288 \, a^{2} f^{3} x^{3} e + 336 \, a b f^{3} x^{3} e - 148 \, b^{2} f^{3} x^{3} e - 5184 \, a b f^{2} h^{2} e^{2} \log \left (c f x + c e\right )^{2} + 7776 \, b^{2} f^{2} h^{2} e^{2} \log \left (c f x + c e\right )^{2} - 3456 i \, b^{2} f^{2} h x e^{2} \log \left (c f x + c e\right )^{2} + 432 \, b^{2} f^{2} x^{2} e^{2} \log \left (c f x + c e\right )^{2} - 6912 i \, a b f^{2} h x e^{2} \log \left (c f x + c e\right ) + 12672 i \, b^{2} f^{2} h x e^{2} \log \left (c f x + c e\right ) + 864 \, a b f^{2} x^{2} e^{2} \log \left (c f x + c e\right ) - 936 \, b^{2} f^{2} x^{2} e^{2} \log \left (c f x + c e\right ) + 1152 i \, b^{2} f h e^{3} \log \left (c f x + c e\right )^{3} - 5184 \, a^{2} f^{2} h^{2} e^{2} \log \left (f x + e\right ) + 15552 \, a b f^{2} h^{2} e^{2} \log \left (f x + e\right ) - 18144 \, b^{2} f^{2} h^{2} e^{2} \log \left (f x + e\right ) - 3456 i \, a^{2} f^{2} h x e^{2} + 12672 i \, a b f^{2} h x e^{2} - 16320 i \, b^{2} f^{2} h x e^{2} + 432 \, a^{2} f^{2} x^{2} e^{2} - 936 \, a b f^{2} x^{2} e^{2} + 690 \, b^{2} f^{2} x^{2} e^{2} + 3456 i \, a b f h e^{3} \log \left (c f x + c e\right )^{2} - 6336 i \, b^{2} f h e^{3} \log \left (c f x + c e\right )^{2} - 864 \, b^{2} f x e^{3} \log \left (c f x + c e\right )^{2} - 1728 \, a b f x e^{3} \log \left (c f x + c e\right ) + 3600 \, b^{2} f x e^{3} \log \left (c f x + c e\right ) + 288 \, b^{2} e^{4} \log \left (c f x + c e\right )^{3} + 3456 i \, a^{2} f h e^{3} \log \left (f x + e\right ) - 12672 i \, a b f h e^{3} \log \left (f x + e\right ) + 16320 i \, b^{2} f h e^{3} \log \left (f x + e\right ) - 864 \, a^{2} f x e^{3} + 3600 \, a b f x e^{3} - 4980 \, b^{2} f x e^{3} + 864 \, a b e^{4} \log \left (c f x + c e\right )^{2} - 1800 \, b^{2} e^{4} \log \left (c f x + c e\right )^{2} + 864 \, a^{2} e^{4} \log \left (f x + e\right ) - 3600 \, a b e^{4} \log \left (f x + e\right ) + 4980 \, b^{2} e^{4} \log \left (f x + e\right )}{864 \, d f^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)^4*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="giac")

[Out]

1/864*(288*b^2*f^4*h^4*log(c*f*x + c*e)^3 + 864*a*b*f^4*h^4*log(c*f*x + c*e)^2 + 3456*I*b^2*f^4*h^3*x*log(c*f*
x + c*e)^2 - 2592*b^2*f^4*h^2*x^2*log(c*f*x + c*e)^2 - 1152*I*b^2*f^4*h*x^3*log(c*f*x + c*e)^2 + 216*b^2*f^4*x
^4*log(c*f*x + c*e)^2 - 1152*I*b^2*f^3*h^3*e*log(c*f*x + c*e)^3 + 6912*I*a*b*f^4*h^3*x*log(c*f*x + c*e) - 6912
*I*b^2*f^4*h^3*x*log(c*f*x + c*e) - 5184*a*b*f^4*h^2*x^2*log(c*f*x + c*e) + 2592*b^2*f^4*h^2*x^2*log(c*f*x + c
*e) - 2304*I*a*b*f^4*h*x^3*log(c*f*x + c*e) + 768*I*b^2*f^4*h*x^3*log(c*f*x + c*e) + 432*a*b*f^4*x^4*log(c*f*x
+ c*e) - 108*b^2*f^4*x^4*log(c*f*x + c*e) - 3456*I*a*b*f^3*h^3*e*log(c*f*x + c*e)^2 + 3456*I*b^2*f^3*h^3*e*lo
g(c*f*x + c*e)^2 + 5184*b^2*f^3*h^2*x*e*log(c*f*x + c*e)^2 + 1728*I*b^2*f^3*h*x^2*e*log(c*f*x + c*e)^2 - 288*b
^2*f^3*x^3*e*log(c*f*x + c*e)^2 + 864*a^2*f^4*h^4*log(f*x + e) + 3456*I*a^2*f^4*h^3*x - 6912*I*a*b*f^4*h^3*x +
6912*I*b^2*f^4*h^3*x - 2592*a^2*f^4*h^2*x^2 + 2592*a*b*f^4*h^2*x^2 - 1296*b^2*f^4*h^2*x^2 - 1152*I*a^2*f^4*h*
x^3 + 768*I*a*b*f^4*h*x^3 - 256*I*b^2*f^4*h*x^3 + 216*a^2*f^4*x^4 - 108*a*b*f^4*x^4 + 27*b^2*f^4*x^4 + 10368*a
*b*f^3*h^2*x*e*log(c*f*x + c*e) - 15552*b^2*f^3*h^2*x*e*log(c*f*x + c*e) + 3456*I*a*b*f^3*h*x^2*e*log(c*f*x +
c*e) - 2880*I*b^2*f^3*h*x^2*e*log(c*f*x + c*e) - 576*a*b*f^3*x^3*e*log(c*f*x + c*e) + 336*b^2*f^3*x^3*e*log(c*
f*x + c*e) - 1728*b^2*f^2*h^2*e^2*log(c*f*x + c*e)^3 - 3456*I*a^2*f^3*h^3*e*log(f*x + e) + 6912*I*a*b*f^3*h^3*
e*log(f*x + e) - 6912*I*b^2*f^3*h^3*e*log(f*x + e) + 5184*a^2*f^3*h^2*x*e - 15552*a*b*f^3*h^2*x*e + 18144*b^2*
f^3*h^2*x*e + 1728*I*a^2*f^3*h*x^2*e - 2880*I*a*b*f^3*h*x^2*e + 1824*I*b^2*f^3*h*x^2*e - 288*a^2*f^3*x^3*e + 3
36*a*b*f^3*x^3*e - 148*b^2*f^3*x^3*e - 5184*a*b*f^2*h^2*e^2*log(c*f*x + c*e)^2 + 7776*b^2*f^2*h^2*e^2*log(c*f*
x + c*e)^2 - 3456*I*b^2*f^2*h*x*e^2*log(c*f*x + c*e)^2 + 432*b^2*f^2*x^2*e^2*log(c*f*x + c*e)^2 - 6912*I*a*b*f
^2*h*x*e^2*log(c*f*x + c*e) + 12672*I*b^2*f^2*h*x*e^2*log(c*f*x + c*e) + 864*a*b*f^2*x^2*e^2*log(c*f*x + c*e)
- 936*b^2*f^2*x^2*e^2*log(c*f*x + c*e) + 1152*I*b^2*f*h*e^3*log(c*f*x + c*e)^3 - 5184*a^2*f^2*h^2*e^2*log(f*x
+ e) + 15552*a*b*f^2*h^2*e^2*log(f*x + e) - 18144*b^2*f^2*h^2*e^2*log(f*x + e) - 3456*I*a^2*f^2*h*x*e^2 + 1267
2*I*a*b*f^2*h*x*e^2 - 16320*I*b^2*f^2*h*x*e^2 + 432*a^2*f^2*x^2*e^2 - 936*a*b*f^2*x^2*e^2 + 690*b^2*f^2*x^2*e^
2 + 3456*I*a*b*f*h*e^3*log(c*f*x + c*e)^2 - 6336*I*b^2*f*h*e^3*log(c*f*x + c*e)^2 - 864*b^2*f*x*e^3*log(c*f*x
+ c*e)^2 - 1728*a*b*f*x*e^3*log(c*f*x + c*e) + 3600*b^2*f*x*e^3*log(c*f*x + c*e) + 288*b^2*e^4*log(c*f*x + c*e
)^3 + 3456*I*a^2*f*h*e^3*log(f*x + e) - 12672*I*a*b*f*h*e^3*log(f*x + e) + 16320*I*b^2*f*h*e^3*log(f*x + e) -
864*a^2*f*x*e^3 + 3600*a*b*f*x*e^3 - 4980*b^2*f*x*e^3 + 864*a*b*e^4*log(c*f*x + c*e)^2 - 1800*b^2*e^4*log(c*f*
x + c*e)^2 + 864*a^2*e^4*log(f*x + e) - 3600*a*b*e^4*log(f*x + e) + 4980*b^2*e^4*log(f*x + e))/(d*f^5)

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Mupad [B]
time = 0.96, size = 1346, normalized size = 2.32 \begin {gather*} {\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^4\,x^4}{4\,d\,f^2}-\frac {b^2\,i^3\,x^3\,\left (e\,i-4\,f\,h\right )}{3\,d\,f^3}-\frac {b^2\,i\,x\,\left (e^3\,i^3-4\,e^2\,f\,h\,i^2+6\,e\,f^2\,h^2\,i-4\,f^3\,h^3\right )}{d\,f^5}+\frac {b^2\,i^2\,x^2\,\left (e^2\,i^2-4\,e\,f\,h\,i+6\,f^2\,h^2\right )}{2\,d\,f^4}\right )+\frac {-25\,b^2\,e^4\,i^4+88\,b^2\,e^3\,f\,h\,i^3-108\,b^2\,e^2\,f^2\,h^2\,i^2+48\,b^2\,e\,f^3\,h^3\,i+12\,a\,b\,e^4\,i^4-48\,a\,b\,e^3\,f\,h\,i^3+72\,a\,b\,e^2\,f^2\,h^2\,i^2-48\,a\,b\,e\,f^3\,h^3\,i+12\,a\,b\,f^4\,h^4}{12\,d\,f^5}\right )-x^2\,\left (\frac {e\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{18\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{8\,d\,f^2}\right )}{2\,f}-\frac {i^2\,\left (72\,a^2\,f^2\,h^2-12\,a\,b\,e^2\,i^2+48\,a\,b\,e\,f\,h\,i-72\,a\,b\,f^2\,h^2+13\,b^2\,e^2\,i^2-40\,b^2\,e\,f\,h\,i+36\,b^2\,f^2\,h^2\right )}{24\,d\,f^3}\right )+x^3\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{54\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{24\,d\,f^2}\right )+x\,\left (\frac {288\,a^2\,f^3\,h^3\,i+144\,a\,b\,e^3\,i^4-576\,a\,b\,e^2\,f\,h\,i^3+864\,a\,b\,e\,f^2\,h^2\,i^2-576\,a\,b\,f^3\,h^3\,i-300\,b^2\,e^3\,i^4+1056\,b^2\,e^2\,f\,h\,i^3-1296\,b^2\,e\,f^2\,h^2\,i^2+576\,b^2\,f^3\,h^3\,i}{72\,d\,f^4}+\frac {e\,\left (\frac {e\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{18\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{8\,d\,f^2}\right )}{f}-\frac {i^2\,\left (72\,a^2\,f^2\,h^2-12\,a\,b\,e^2\,i^2+48\,a\,b\,e\,f\,h\,i-72\,a\,b\,f^2\,h^2+13\,b^2\,e^2\,i^2-40\,b^2\,e\,f\,h\,i+36\,b^2\,f^2\,h^2\right )}{12\,d\,f^3}\right )}{f}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x^3\,\left (7\,e\,b^2\,i^4-16\,f\,h\,b^2\,i^3-12\,a\,e\,b\,i^4+48\,a\,f\,h\,b\,i^3\right )}{18\,d\,f^3}-\frac {x^2\,\left (13\,b^2\,e^2\,i^4-40\,b^2\,e\,f\,h\,i^3+36\,b^2\,f^2\,h^2\,i^2-12\,a\,b\,e^2\,i^4+48\,a\,b\,e\,f\,h\,i^3-72\,a\,b\,f^2\,h^2\,i^2\right )}{12\,d\,f^4}+\frac {x\,\left (25\,b^2\,e^3\,i^4-88\,b^2\,e^2\,f\,h\,i^3+108\,b^2\,e\,f^2\,h^2\,i^2-48\,b^2\,f^3\,h^3\,i-12\,a\,b\,e^3\,i^4+48\,a\,b\,e^2\,f\,h\,i^3-72\,a\,b\,e\,f^2\,h^2\,i^2+48\,a\,b\,f^3\,h^3\,i\right )}{6\,d\,f^5}+\frac {b\,i^4\,x^4\,\left (4\,a-b\right )}{8\,d\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (72\,a^2\,e^4\,i^4-288\,a^2\,e^3\,f\,h\,i^3+432\,a^2\,e^2\,f^2\,h^2\,i^2-288\,a^2\,e\,f^3\,h^3\,i+72\,a^2\,f^4\,h^4-300\,a\,b\,e^4\,i^4+1056\,a\,b\,e^3\,f\,h\,i^3-1296\,a\,b\,e^2\,f^2\,h^2\,i^2+576\,a\,b\,e\,f^3\,h^3\,i+415\,b^2\,e^4\,i^4-1360\,b^2\,e^3\,f\,h\,i^3+1512\,b^2\,e^2\,f^2\,h^2\,i^2-576\,b^2\,e\,f^3\,h^3\,i\right )}{72\,d\,f^5}+\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^4\,i^4-4\,e^3\,f\,h\,i^3+6\,e^2\,f^2\,h^2\,i^2-4\,e\,f^3\,h^3\,i+f^4\,h^4\right )}{3\,d\,f^5}+\frac {i^4\,x^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{32\,d\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((h + i*x)^4*(a + b*log(c*(e + f*x)))^2)/(d*e + d*f*x),x)

[Out]

log(c*(e + f*x))^2*(f*((b^2*i^4*x^4)/(4*d*f^2) - (b^2*i^3*x^3*(e*i - 4*f*h))/(3*d*f^3) - (b^2*i*x*(e^3*i^3 - 4
*f^3*h^3 + 6*e*f^2*h^2*i - 4*e^2*f*h*i^2))/(d*f^5) + (b^2*i^2*x^2*(e^2*i^2 + 6*f^2*h^2 - 4*e*f*h*i))/(2*d*f^4)
) + (12*a*b*e^4*i^4 - 25*b^2*e^4*i^4 + 12*a*b*f^4*h^4 - 108*b^2*e^2*f^2*h^2*i^2 + 48*b^2*e*f^3*h^3*i + 88*b^2*
e^3*f*h*i^3 + 72*a*b*e^2*f^2*h^2*i^2 - 48*a*b*e*f^3*h^3*i - 48*a*b*e^3*f*h*i^3)/(12*d*f^5)) - x^2*((e*((i^3*(7
2*a^2*f*h - 7*b^2*e*i + 16*b^2*f*h + 12*a*b*e*i - 48*a*b*f*h))/(18*d*f^2) - (e*i^4*(8*a^2 - 4*a*b + b^2))/(8*d
*f^2)))/(2*f) - (i^2*(72*a^2*f^2*h^2 + 13*b^2*e^2*i^2 + 36*b^2*f^2*h^2 - 12*a*b*e^2*i^2 - 72*a*b*f^2*h^2 - 40*
b^2*e*f*h*i + 48*a*b*e*f*h*i))/(24*d*f^3)) + x^3*((i^3*(72*a^2*f*h - 7*b^2*e*i + 16*b^2*f*h + 12*a*b*e*i - 48*
a*b*f*h))/(54*d*f^2) - (e*i^4*(8*a^2 - 4*a*b + b^2))/(24*d*f^2)) + x*((288*a^2*f^3*h^3*i - 300*b^2*e^3*i^4 + 5
76*b^2*f^3*h^3*i + 144*a*b*e^3*i^4 - 576*a*b*f^3*h^3*i + 1056*b^2*e^2*f*h*i^3 - 1296*b^2*e*f^2*h^2*i^2 - 576*a
*b*e^2*f*h*i^3 + 864*a*b*e*f^2*h^2*i^2)/(72*d*f^4) + (e*((e*((i^3*(72*a^2*f*h - 7*b^2*e*i + 16*b^2*f*h + 12*a*
b*e*i - 48*a*b*f*h))/(18*d*f^2) - (e*i^4*(8*a^2 - 4*a*b + b^2))/(8*d*f^2)))/f - (i^2*(72*a^2*f^2*h^2 + 13*b^2*
e^2*i^2 + 36*b^2*f^2*h^2 - 12*a*b*e^2*i^2 - 72*a*b*f^2*h^2 - 40*b^2*e*f*h*i + 48*a*b*e*f*h*i))/(12*d*f^3)))/f)
+ f*log(c*(e + f*x))*((x^3*(7*b^2*e*i^4 - 12*a*b*e*i^4 - 16*b^2*f*h*i^3 + 48*a*b*f*h*i^3))/(18*d*f^3) - (x^2*
(13*b^2*e^2*i^4 + 36*b^2*f^2*h^2*i^2 - 12*a*b*e^2*i^4 - 40*b^2*e*f*h*i^3 - 72*a*b*f^2*h^2*i^2 + 48*a*b*e*f*h*i
^3))/(12*d*f^4) + (x*(25*b^2*e^3*i^4 - 48*b^2*f^3*h^3*i - 12*a*b*e^3*i^4 + 48*a*b*f^3*h^3*i - 88*b^2*e^2*f*h*i
^3 + 108*b^2*e*f^2*h^2*i^2 + 48*a*b*e^2*f*h*i^3 - 72*a*b*e*f^2*h^2*i^2))/(6*d*f^5) + (b*i^4*x^4*(4*a - b))/(8*
d*f^2)) + (log(e + f*x)*(72*a^2*e^4*i^4 + 72*a^2*f^4*h^4 + 415*b^2*e^4*i^4 - 300*a*b*e^4*i^4 + 432*a^2*e^2*f^2
*h^2*i^2 + 1512*b^2*e^2*f^2*h^2*i^2 - 288*a^2*e*f^3*h^3*i - 288*a^2*e^3*f*h*i^3 - 576*b^2*e*f^3*h^3*i - 1360*b
^2*e^3*f*h*i^3 - 1296*a*b*e^2*f^2*h^2*i^2 + 576*a*b*e*f^3*h^3*i + 1056*a*b*e^3*f*h*i^3))/(72*d*f^5) + (b^2*log
(c*(e + f*x))^3*(e^4*i^4 + f^4*h^4 + 6*e^2*f^2*h^2*i^2 - 4*e*f^3*h^3*i - 4*e^3*f*h*i^3))/(3*d*f^5) + (i^4*x^4*
(8*a^2 - 4*a*b + b^2))/(32*d*f)

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