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

Optimal result

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

[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] (verified)

Time = 1.02 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 30, 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} \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {8 b i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))}{9 d f^5}+\frac {i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}-\frac {3 b i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))}{d f^5}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))^2}{d f^5}-\frac {4 b i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))}{d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}+\frac {(h+i x)^3 (f h-e i) (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {4 a b i x (f h-e i)^3}{d f^4}-\frac {4 b^2 i (e+f x) (f h-e i)^3 \log (c (e+f x))}{d f^5}+\frac {8 b^2 i^3 (e+f x)^3 (f h-e i)}{27 d f^5}+\frac {3 b^2 i^2 (e+f x)^2 (f h-e i)^2}{2 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {8 b^2 i x (f h-e i)^3}{d f^4} \]

[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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i 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 {f h-e i}{f}+\frac {i x}{f}\right )^4 (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 )^3 (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 )^3 (a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^2} \\ & = \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^4 (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{2 d f}+\frac {(i (f h-e 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^3}+\frac {(f h-e i)^2 \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^3} \\ & = -\frac {2 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)))}{2 d f^5}-\frac {2 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{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 {b^2 \text {Subst}\left (\int \frac {48 i (f h-e i)^3+36 i^2 (f h-e i)^2 x+16 i^3 (f h-e i) x^2+3 i^4 x^3+\frac {12 (f h-e i)^4 \log (x)}{x}}{12 f^4} \, dx,x,e+f x\right )}{2 d f}-\frac {(2 b (f h-e i)) \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^2}+\frac {\left (i (f h-e i)^2\right ) \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^4}+\frac {(f h-e i)^3 \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^4} \\ & = -\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {5 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 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 {(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 {b^2 \text {Subst}\left (\int \left (48 i (f h-e i)^3+36 i^2 (f h-e i)^2 x+16 i^3 (f h-e i) x^2+3 i^4 x^3+\frac {12 (f h-e i)^4 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{24 d f^5}+\frac {\left (2 b^2 (f h-e i)\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^2}+\frac {\left (i (f h-e i)^2\right ) \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^4}+\frac {\left (i (f h-e i)^3\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}+\frac {(f h-e i)^4 \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d f^5} \\ & = \frac {2 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}{4 d f^5}+\frac {2 b^2 i^3 (f h-e i) (e+f x)^3}{9 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 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 {5 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 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 {i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^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 {\left (b^2 (f h-e i)\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)}{x} \, dx,x,e+f x\right )}{9 d f^5}+\frac {\left (i^2 (f h-e i)^2\right ) \text {Subst}\left (\int x (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}+\frac {\left (i (f h-e i)^3\right ) \text {Subst}\left (\int (a+b \log (c x))^2 \, dx,x,e+f x\right )}{d f^5}-\frac {\left (2 b i (f h-e i)^3\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}+\frac {(f h-e i)^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 (f h-e i)^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{2 d f^5} \\ & = -\frac {2 a b i (f h-e i)^3 x}{d f^4}+\frac {2 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}{4 d f^5}+\frac {2 b^2 i^3 (f h-e i) (e+f x)^3}{9 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {b^2 (f h-e i)^4 \log ^2(e+f x)}{4 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 {5 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 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}+\frac {\left (b^2 (f h-e i)\right ) \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^5}-\frac {\left (b i^2 (f h-e i)^2\right ) \text {Subst}(\int x (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}-\frac {\left (2 b i (f h-e i)^3\right ) \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^5}-\frac {\left (2 b^2 i (f h-e i)^3\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^5} \\ & = -\frac {4 a b i (f h-e i)^3 x}{d f^4}+\frac {4 b^2 i (f h-e i)^3 x}{d f^4}+\frac {b^2 i^2 (f h-e i)^2 (e+f x)^2}{d f^5}+\frac {2 b^2 i^3 (f h-e i) (e+f x)^3}{9 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {b^2 (f h-e i)^4 \log ^2(e+f x)}{4 d f^5}-\frac {2 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}+\frac {\left (b^2 i (f h-e 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^5}-\frac {\left (2 b^2 i (f h-e i)^3\right ) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^5}+\frac {\left (2 b^2 (f h-e i)^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{3 d f^5} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.65 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {3456 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2+2592 i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2+1152 i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))^2+216 i^4 (e+f x)^4 (a+b \log (c (e+f x)))^2+\frac {288 (f h-e i)^4 (a+b \log (c (e+f x)))^3}{b}-6912 b i (f h-e i)^3 ((a-b) f x+b (e+f x) \log (c (e+f x)))+1296 b i^2 (f h-e i)^2 \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )+256 b i^3 (f h-e i) \left (b f x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 (a+b \log (c (e+f x)))\right )+27 b i^4 \left (b f x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{864 d f^5} \]

[In]

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

[Out]

(3456*i*(f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)])^2 + 2592*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(
e + f*x)])^2 + 1152*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)])^2 + 216*i^4*(e + f*x)^4*(a + b*Log[c*
(e + f*x)])^2 + (288*(f*h - e*i)^4*(a + b*Log[c*(e + f*x)])^3)/b - 6912*b*i*(f*h - e*i)^3*((a - b)*f*x + b*(e
+ f*x)*Log[c*(e + f*x)]) + 1296*b*i^2*(f*h - e*i)^2*(b*f*x*(2*e + f*x) - 2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]
)) + 256*b*i^3*(f*h - e*i)*(b*f*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 3*(e + f*x)^3*(a + b*Log[c*(e + f*x)])) + 27*b
*i^4*(b*f*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) - 4*(e + f*x)^4*(a + b*Log[c*(e + f*x)])))/(864*d*f^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1134\) vs. \(2(557)=1114\).

Time = 0.92 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.96

method	result	size
norman	\(\text {Expression too large to display}\)	\(1135\)
risch	\(\text {Expression too large to display}\)	\(1475\)
parts	\(\text {Expression too large to display}\)	\(1541\)
derivativedivides	\(\text {Expression too large to display}\)	\(1891\)
default	\(\text {Expression too large to display}\)	\(1891\)
parallelrisch	\(\text {Expression too large to display}\)	\(1912\)

[In]

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

[Out]

1/72*(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+1
512*b^2*e^2*f^2*h^2*i^2-576*b^2*e*f^3*h^3*i)/d/f^5*ln(c*(f*x+e))+1/12*b*(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*f^4*h^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)/d/f^5*ln(c*(f*x+e))^2+1/3*b^2*(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)/d/f^5*ln(c*(
f*x+e))^3-1/72*i*(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)/d/f^4*x+1/144*i^2*(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)/d/f^3*x^2-1/216*i^3*(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)/f^2/d*x^3+1/32*i^4*(8*a^2-4*a*b+b^2)/d/f*x^4+1/4*b^2*i^4/d/
f*x^4*ln(c*(f*x+e))^2-1/6*b*i*(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)/d/f^4*x*ln(c*(f*x+e))+1/12*b*i^2*(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)/d/f^3*x^2*ln(c*(f*x+e))-1/18*b*i^3*(12*a*e*i-48*a*f*h-7*b*e*i+16
*b*f*h)/d/f^2*x^3*ln(c*(f*x+e))+1/8*b*i^4*(4*a-b)/d/f*x^4*ln(c*(f*x+e))-b^2*i*(e^3*i^3-4*e^2*f*h*i^2+6*e*f^2*h
^2*i-4*f^3*h^3)/d/f^4*x*ln(c*(f*x+e))^2+1/2*b^2*i^2*(e^2*i^2-4*e*f*h*i+6*f^2*h^2)/d/f^3*x^2*ln(c*(f*x+e))^2-1/
3*b^2*i^3*(e*i-4*f*h)/d/f^2*x^3*ln(c*(f*x+e))^2

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 939, normalized size of antiderivative = 1.62 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {27 \, {\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} f^{4} i^{4} x^{4} + 4 \, {\left (32 \, {\left (9 \, a^{2} - 6 \, a b + 2 \, b^{2}\right )} f^{4} h i^{3} - {\left (72 \, a^{2} - 84 \, a b + 37 \, b^{2}\right )} e f^{3} i^{4}\right )} x^{3} + 288 \, {\left (b^{2} f^{4} h^{4} - 4 \, b^{2} e f^{3} h^{3} i + 6 \, b^{2} e^{2} f^{2} h^{2} i^{2} - 4 \, b^{2} e^{3} f h i^{3} + b^{2} e^{4} i^{4}\right )} \log \left (c f x + c e\right )^{3} + 6 \, {\left (216 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{4} h^{2} i^{2} - 16 \, {\left (18 \, a^{2} - 30 \, a b + 19 \, b^{2}\right )} e f^{3} h i^{3} + {\left (72 \, a^{2} - 156 \, a b + 115 \, b^{2}\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 72 \, {\left (3 \, b^{2} f^{4} i^{4} x^{4} + 12 \, a b f^{4} h^{4} - 48 \, {\left (a b - b^{2}\right )} e f^{3} h^{3} i + 36 \, {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} f^{2} h^{2} i^{2} - 8 \, {\left (6 \, a b - 11 \, b^{2}\right )} e^{3} f h i^{3} + {\left (12 \, a b - 25 \, b^{2}\right )} e^{4} i^{4} + 4 \, {\left (4 \, b^{2} f^{4} h i^{3} - b^{2} e f^{3} i^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{2} f^{4} h^{2} i^{2} - 4 \, b^{2} e f^{3} h i^{3} + b^{2} e^{2} f^{2} i^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{2} f^{4} h^{3} i - 6 \, b^{2} e f^{3} h^{2} i^{2} + 4 \, b^{2} e^{2} f^{2} h i^{3} - b^{2} e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 12 \, {\left (288 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{4} h^{3} i - 216 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f^{3} h^{2} i^{2} + 16 \, {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{2} f^{2} h i^{3} - {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{3} f i^{4}\right )} x + 12 \, {\left (9 \, {\left (4 \, a b - b^{2}\right )} f^{4} i^{4} x^{4} + 72 \, a^{2} f^{4} h^{4} - 288 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f^{3} h^{3} i + 216 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} f^{2} h^{2} i^{2} - 16 \, {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{3} f h i^{3} + {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{4} i^{4} + 4 \, {\left (16 \, {\left (3 \, a b - b^{2}\right )} f^{4} h i^{3} - {\left (12 \, a b - 7 \, b^{2}\right )} e f^{3} i^{4}\right )} x^{3} + 6 \, {\left (36 \, {\left (2 \, a b - b^{2}\right )} f^{4} h^{2} i^{2} - 8 \, {\left (6 \, a b - 5 \, b^{2}\right )} e f^{3} h i^{3} + {\left (12 \, a b - 13 \, b^{2}\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 12 \, {\left (48 \, {\left (a b - b^{2}\right )} f^{4} h^{3} i - 36 \, {\left (2 \, a b - 3 \, b^{2}\right )} e f^{3} h^{2} i^{2} + 8 \, {\left (6 \, a b - 11 \, b^{2}\right )} e^{2} f^{2} h i^{3} - {\left (12 \, a b - 25 \, b^{2}\right )} e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )}{864 \, d f^{5}} \]

[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*(27*(8*a^2 - 4*a*b + b^2)*f^4*i^4*x^4 + 4*(32*(9*a^2 - 6*a*b + 2*b^2)*f^4*h*i^3 - (72*a^2 - 84*a*b + 37*
b^2)*e*f^3*i^4)*x^3 + 288*(b^2*f^4*h^4 - 4*b^2*e*f^3*h^3*i + 6*b^2*e^2*f^2*h^2*i^2 - 4*b^2*e^3*f*h*i^3 + b^2*e
^4*i^4)*log(c*f*x + c*e)^3 + 6*(216*(2*a^2 - 2*a*b + b^2)*f^4*h^2*i^2 - 16*(18*a^2 - 30*a*b + 19*b^2)*e*f^3*h*
i^3 + (72*a^2 - 156*a*b + 115*b^2)*e^2*f^2*i^4)*x^2 + 72*(3*b^2*f^4*i^4*x^4 + 12*a*b*f^4*h^4 - 48*(a*b - b^2)*
e*f^3*h^3*i + 36*(2*a*b - 3*b^2)*e^2*f^2*h^2*i^2 - 8*(6*a*b - 11*b^2)*e^3*f*h*i^3 + (12*a*b - 25*b^2)*e^4*i^4
+ 4*(4*b^2*f^4*h*i^3 - b^2*e*f^3*i^4)*x^3 + 6*(6*b^2*f^4*h^2*i^2 - 4*b^2*e*f^3*h*i^3 + b^2*e^2*f^2*i^4)*x^2 +
12*(4*b^2*f^4*h^3*i - 6*b^2*e*f^3*h^2*i^2 + 4*b^2*e^2*f^2*h*i^3 - b^2*e^3*f*i^4)*x)*log(c*f*x + c*e)^2 + 12*(2
88*(a^2 - 2*a*b + 2*b^2)*f^4*h^3*i - 216*(2*a^2 - 6*a*b + 7*b^2)*e*f^3*h^2*i^2 + 16*(18*a^2 - 66*a*b + 85*b^2)
*e^2*f^2*h*i^3 - (72*a^2 - 300*a*b + 415*b^2)*e^3*f*i^4)*x + 12*(9*(4*a*b - b^2)*f^4*i^4*x^4 + 72*a^2*f^4*h^4
- 288*(a^2 - 2*a*b + 2*b^2)*e*f^3*h^3*i + 216*(2*a^2 - 6*a*b + 7*b^2)*e^2*f^2*h^2*i^2 - 16*(18*a^2 - 66*a*b +
85*b^2)*e^3*f*h*i^3 + (72*a^2 - 300*a*b + 415*b^2)*e^4*i^4 + 4*(16*(3*a*b - b^2)*f^4*h*i^3 - (12*a*b - 7*b^2)*
e*f^3*i^4)*x^3 + 6*(36*(2*a*b - b^2)*f^4*h^2*i^2 - 8*(6*a*b - 5*b^2)*e*f^3*h*i^3 + (12*a*b - 13*b^2)*e^2*f^2*i
^4)*x^2 + 12*(48*(a*b - b^2)*f^4*h^3*i - 36*(2*a*b - 3*b^2)*e*f^3*h^2*i^2 + 8*(6*a*b - 11*b^2)*e^2*f^2*h*i^3 -
(12*a*b - 25*b^2)*e^3*f*i^4)*x)*log(c*f*x + c*e))/(d*f^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (534) = 1068\).

Time = 1.88 (sec) , antiderivative size = 1479, normalized size of antiderivative = 2.55 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (557) = 1114\).

Time = 0.27 (sec) , antiderivative size = 1427, normalized size of antiderivative = 2.46 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

8*a*b*h^3*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + 1/6*a*b*i^4*(12*e^4*log(f*x + e)/(d*f^5) + (
3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/(d*f^4))*log(c*f*x + c*e) - 4/3*a*b*h*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) + 6*a*b*h^2*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^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)) + 4*a^2*h^3*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) + 1/12*a^2*i^4*(12
*e^4*log(f*x + e)/(d*f^5) + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/(d*f^4)) - 2/3*a^2*h*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*a^2*h^2*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^4*log(c*f*x + c*e)^3/(d*f) + 2*a*b*h^4*log(c*f*x + c*e)*log(d*f*x + d*
e)/(d*f) + a^2*h^4*log(d*f*x + d*e)/(d*f) + 4*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*a*b*h^3*i/(d*f^2)
- 3*(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^2*i^2/(d*f^3) - 4/3*(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^3*i/(c^2*d*f^2) - 2/9*(
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*h*i^3/(d*f^4) - 1/72*
(9*f^4*x^4 - 28*e*f^3*x^3 + 78*e^2*f^2*x^2 + 72*e^4*log(f*x + e)^2 - 300*e^3*f*x + 300*e^4*log(f*x + e))*a*b*i
^4/(d*f^5) + 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*i^2/(c^3*
d*f^3) - 1/27*(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*l
og(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*h*i^3/(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*log(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*lo
g(c*f*x + c*e) + 2*c^4*e^3)*(c*f*x + c*e))*b^2*i^4/(c^5*d*f^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1213 vs. \(2 (557) = 1114\).

Time = 0.33 (sec) , antiderivative size = 1213, normalized size of antiderivative = 2.09 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*(8*a^2*i^4 - 4*a*b*i^4 + b^2*i^4)*x^4/(d*f) + 1/12*(3*b^2*i^4*x^4/(d*f) + 4*(4*b^2*f*h*i^3 - b^2*e*i^4)*x
^3/(d*f^2) + 6*(6*b^2*f^2*h^2*i^2 - 4*b^2*e*f*h*i^3 + b^2*e^2*i^4)*x^2/(d*f^3) + 12*(4*b^2*f^3*h^3*i - 6*b^2*e
*f^2*h^2*i^2 + 4*b^2*e^2*f*h*i^3 - b^2*e^3*i^4)*x/(d*f^4) + (12*a*b*f^4*h^4 - 48*a*b*e*f^3*h^3*i + 48*b^2*e*f^
3*h^3*i + 72*a*b*e^2*f^2*h^2*i^2 - 108*b^2*e^2*f^2*h^2*i^2 - 48*a*b*e^3*f*h*i^3 + 88*b^2*e^3*f*h*i^3 + 12*a*b*
e^4*i^4 - 25*b^2*e^4*i^4)/(d*f^5))*log(c*f*x + c*e)^2 + 1/72*(9*(4*a*b*i^4 - b^2*i^4)*x^4/(d*f) + 4*(48*a*b*f*
h*i^3 - 16*b^2*f*h*i^3 - 12*a*b*e*i^4 + 7*b^2*e*i^4)*x^3/(d*f^2) + 6*(72*a*b*f^2*h^2*i^2 - 36*b^2*f^2*h^2*i^2
- 48*a*b*e*f*h*i^3 + 40*b^2*e*f*h*i^3 + 12*a*b*e^2*i^4 - 13*b^2*e^2*i^4)*x^2/(d*f^3) + 12*(48*a*b*f^3*h^3*i -
48*b^2*f^3*h^3*i - 72*a*b*e*f^2*h^2*i^2 + 108*b^2*e*f^2*h^2*i^2 + 48*a*b*e^2*f*h*i^3 - 88*b^2*e^2*f*h*i^3 - 12
*a*b*e^3*i^4 + 25*b^2*e^3*i^4)*x/(d*f^4))*log(c*f*x + c*e) + 1/216*(288*a^2*f*h*i^3 - 192*a*b*f*h*i^3 + 64*b^2
*f*h*i^3 - 72*a^2*e*i^4 + 84*a*b*e*i^4 - 37*b^2*e*i^4)*x^3/(d*f^2) + 1/144*(432*a^2*f^2*h^2*i^2 - 432*a*b*f^2*
h^2*i^2 + 216*b^2*f^2*h^2*i^2 - 288*a^2*e*f*h*i^3 + 480*a*b*e*f*h*i^3 - 304*b^2*e*f*h*i^3 + 72*a^2*e^2*i^4 - 1
56*a*b*e^2*i^4 + 115*b^2*e^2*i^4)*x^2/(d*f^3) + 1/3*(b^2*f^4*h^4 - 4*b^2*e*f^3*h^3*i + 6*b^2*e^2*f^2*h^2*i^2 -
4*b^2*e^3*f*h*i^3 + b^2*e^4*i^4)*log(c*f*x + c*e)^3/(d*f^5) + 1/72*(288*a^2*f^3*h^3*i - 576*a*b*f^3*h^3*i + 5
76*b^2*f^3*h^3*i - 432*a^2*e*f^2*h^2*i^2 + 1296*a*b*e*f^2*h^2*i^2 - 1512*b^2*e*f^2*h^2*i^2 + 288*a^2*e^2*f*h*i
^3 - 1056*a*b*e^2*f*h*i^3 + 1360*b^2*e^2*f*h*i^3 - 72*a^2*e^3*i^4 + 300*a*b*e^3*i^4 - 415*b^2*e^3*i^4)*x/(d*f^
4) + 1/72*(72*a^2*f^4*h^4 - 288*a^2*e*f^3*h^3*i + 576*a*b*e*f^3*h^3*i - 576*b^2*e*f^3*h^3*i + 432*a^2*e^2*f^2*
h^2*i^2 - 1296*a*b*e^2*f^2*h^2*i^2 + 1512*b^2*e^2*f^2*h^2*i^2 - 288*a^2*e^3*f*h*i^3 + 1056*a*b*e^3*f*h*i^3 - 1
360*b^2*e^3*f*h*i^3 + 72*a^2*e^4*i^4 - 300*a*b*e^4*i^4 + 415*b^2*e^4*i^4)*log(f*x + e)/(d*f^5)

Mupad [B] (verification not implemented)

Time = 2.07 (sec) , antiderivative size = 1346, normalized size of antiderivative = 2.32 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx={\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} \]

[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)