\(\int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) [185]

Optimal result

Integrand size = 32, antiderivative size = 238 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {4 a b i (f h-e i) x}{d f^2}+\frac {4 b^2 i (f h-e i) x}{d f^2}+\frac {b^2 i^2 (e+f x)^2}{4 d f^3}-\frac {4 b^2 i (f h-e i) (e+f x) \log (c (e+f x))}{d f^3}-\frac {b i^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}+\frac {2 i (f h-e i) (e+f x) (a+b \log (c (e+f x)))^2}{d f^3}+\frac {i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}+\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2458, 12, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341} \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3}+\frac {2 i (e+f x) (f h-e i) (a+b \log (c (e+f x)))^2}{d f^3}+\frac {i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}-\frac {b i^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}-\frac {4 a b i x (f h-e i)}{d f^2}-\frac {4 b^2 i (e+f x) (f h-e i) \log (c (e+f x))}{d f^3}+\frac {b^2 i^2 (e+f x)^2}{4 d f^3}+\frac {4 b^2 i x (f h-e i)}{d f^2} \]

[In]

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

[Out]

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

Rule 12

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

Rule 30

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {24 i (f h-e i) (e+f x) (a+b \log (c (e+f x)))^2+6 i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2+\frac {4 (f h-e i)^2 (a+b \log (c (e+f x)))^3}{b}-48 b i (f h-e i) ((a-b) f x+b (e+f x) \log (c (e+f x)))+3 b i^2 \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )}{12 d f^3} \]

[In]

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

[Out]

(24*i*(f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)])^2 + 6*i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2 + (4*(f*
h - e*i)^2*(a + b*Log[c*(e + f*x)])^3)/b - 48*b*i*(f*h - e*i)*((a - b)*f*x + b*(e + f*x)*Log[c*(e + f*x)]) + 3
*b*i^2*(b*f*x*(2*e + f*x) - 2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])))/(12*d*f^3)

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.67

method	result	size
norman	\(\frac {\left (2 a^{2} e^{2} i^{2}-4 a^{2} e f h i +2 a^{2} f^{2} h^{2}-6 a b \,e^{2} i^{2}+8 a b e f h i +7 b^{2} e^{2} i^{2}-8 b^{2} e f h i \right ) \ln \left (c \left (f x +e \right )\right )}{2 d \,f^{3}}+\frac {b \left (2 a \,e^{2} i^{2}-4 a e f h i +2 a \,f^{2} h^{2}-3 b \,e^{2} i^{2}+4 b e f h i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{3}}+\frac {b^{2} \left (e^{2} i^{2}-2 e f h i +f^{2} h^{2}\right ) \ln \left (c \left (f x +e \right )\right )^{3}}{3 d \,f^{3}}-\frac {i \left (2 a^{2} e i -4 a^{2} f h -6 a b e i +8 a b f h +7 b^{2} e i -8 b^{2} f h \right ) x}{2 d \,f^{2}}+\frac {i^{2} \left (2 a^{2}-2 a b +b^{2}\right ) x^{2}}{4 d f}+\frac {b^{2} i^{2} x^{2} \ln \left (c \left (f x +e \right )\right )^{2}}{2 d f}-\frac {b i \left (2 a e i -4 a f h -3 b e i +4 b f h \right ) x \ln \left (c \left (f x +e \right )\right )}{d \,f^{2}}+\frac {b \,i^{2} \left (2 a -b \right ) x^{2} \ln \left (c \left (f x +e \right )\right )}{2 d f}-\frac {b^{2} i \left (e i -2 f h \right ) x \ln \left (c \left (f x +e \right )\right )^{2}}{d \,f^{2}}\)	\(397\)
risch	\(\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e^{2} i^{2}}{3 d \,f^{3}}-\frac {2 b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e h i}{3 d \,f^{2}}+\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} h^{2}}{3 d f}+\frac {b \left (b \,f^{2} i^{2} x^{2}-2 b e f \,i^{2} x +4 b \,f^{2} h i x +2 a \,e^{2} i^{2}-4 a e f h i +2 a \,f^{2} h^{2}-3 b \,e^{2} i^{2}+4 b e f h i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{3}}-\frac {b i x \left (-2 f i x a +b f i x +4 a e i -8 a f h -6 b e i +8 b f h \right ) \ln \left (c \left (f x +e \right )\right )}{2 d \,f^{2}}+\frac {a^{2} i^{2} x^{2}}{2 d f}-\frac {a b \,i^{2} x^{2}}{2 d f}+\frac {b^{2} i^{2} x^{2}}{4 d f}+\frac {\ln \left (f x +e \right ) a^{2} e^{2} i^{2}}{d \,f^{3}}-\frac {2 \ln \left (f x +e \right ) a^{2} e h i}{d \,f^{2}}+\frac {\ln \left (f x +e \right ) a^{2} h^{2}}{d f}-\frac {3 \ln \left (f x +e \right ) a b \,e^{2} i^{2}}{d \,f^{3}}+\frac {4 \ln \left (f x +e \right ) a b e h i}{d \,f^{2}}+\frac {7 \ln \left (f x +e \right ) b^{2} e^{2} i^{2}}{2 d \,f^{3}}-\frac {4 \ln \left (f x +e \right ) b^{2} e h i}{d \,f^{2}}-\frac {a^{2} e \,i^{2} x}{d \,f^{2}}+\frac {2 a^{2} h i x}{d f}+\frac {3 a b e \,i^{2} x}{d \,f^{2}}-\frac {4 a b h i x}{d f}-\frac {7 b^{2} e \,i^{2} x}{2 d \,f^{2}}+\frac {4 b^{2} h i x}{d f}\)	\(501\)
parts	\(\frac {a^{2} \left (\frac {i \left (\frac {1}{2} f i \,x^{2}-x e i +2 x f h \right )}{f^{2}}+\frac {\left (e^{2} i^{2}-2 e f h i +f^{2} h^{2}\right ) \ln \left (f x +e \right )}{f^{3}}\right )}{d}+\frac {b^{2} \left (\frac {c \,e^{2} i^{2} \ln \left (c f x +c e \right )^{3}}{3 f^{2}}-\frac {2 c e h i \ln \left (c f x +c e \right )^{3}}{3 f}+\frac {c \,h^{2} \ln \left (c f x +c e \right )^{3}}{3}-\frac {2 e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f^{2}}+\frac {2 h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f}+\frac {i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}+\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}\right )}{d c f}+\frac {2 a b \left (\frac {c \,e^{2} i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2}}-\frac {c e h i \ln \left (c f x +c e \right )^{2}}{f}+\frac {c \,h^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {2 e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2}}+\frac {2 h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f}+\frac {i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}\right )}{d c f}\)	\(509\)
derivativedivides	\(\frac {\frac {c \,a^{2} e^{2} i^{2} \ln \left (c f x +c e \right )}{f^{2} d}-\frac {2 c \,a^{2} e h i \ln \left (c f x +c e \right )}{f d}+\frac {c \,a^{2} h^{2} \ln \left (c f x +c e \right )}{d}-\frac {2 a^{2} e \,i^{2} \left (c f x +c e \right )}{f^{2} d}+\frac {2 a^{2} h i \left (c f x +c e \right )}{f d}+\frac {a^{2} i^{2} \left (c f x +c e \right )^{2}}{2 c \,f^{2} d}+\frac {c a b \,e^{2} i^{2} \ln \left (c f x +c e \right )^{2}}{f^{2} d}-\frac {2 c a b e h i \ln \left (c f x +c e \right )^{2}}{f d}+\frac {c a b \,h^{2} \ln \left (c f x +c e \right )^{2}}{d}-\frac {4 a b e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2} d}+\frac {4 a b h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}+\frac {2 a b \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}+\frac {c \,b^{2} e^{2} i^{2} \ln \left (c f x +c e \right )^{3}}{3 f^{2} d}-\frac {2 c \,b^{2} e h i \ln \left (c f x +c e \right )^{3}}{3 f d}+\frac {c \,b^{2} h^{2} \ln \left (c f x +c e \right )^{3}}{3 d}-\frac {2 b^{2} e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f^{2} d}+\frac {2 b^{2} h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f d}+\frac {b^{2} i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}+\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}}{c f}\)	\(632\)
default	\(\frac {\frac {c \,a^{2} e^{2} i^{2} \ln \left (c f x +c e \right )}{f^{2} d}-\frac {2 c \,a^{2} e h i \ln \left (c f x +c e \right )}{f d}+\frac {c \,a^{2} h^{2} \ln \left (c f x +c e \right )}{d}-\frac {2 a^{2} e \,i^{2} \left (c f x +c e \right )}{f^{2} d}+\frac {2 a^{2} h i \left (c f x +c e \right )}{f d}+\frac {a^{2} i^{2} \left (c f x +c e \right )^{2}}{2 c \,f^{2} d}+\frac {c a b \,e^{2} i^{2} \ln \left (c f x +c e \right )^{2}}{f^{2} d}-\frac {2 c a b e h i \ln \left (c f x +c e \right )^{2}}{f d}+\frac {c a b \,h^{2} \ln \left (c f x +c e \right )^{2}}{d}-\frac {4 a b e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2} d}+\frac {4 a b h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}+\frac {2 a b \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}+\frac {c \,b^{2} e^{2} i^{2} \ln \left (c f x +c e \right )^{3}}{3 f^{2} d}-\frac {2 c \,b^{2} e h i \ln \left (c f x +c e \right )^{3}}{3 f d}+\frac {c \,b^{2} h^{2} \ln \left (c f x +c e \right )^{3}}{3 d}-\frac {2 b^{2} e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f^{2} d}+\frac {2 b^{2} h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f d}+\frac {b^{2} i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}+\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}}{c f}\)	\(632\)
parallelrisch	\(\frac {-66 a b \,e^{2} i^{2}-12 a^{2} e f \,i^{2} x +24 a^{2} f^{2} h i x -42 b^{2} e f \,i^{2} x +48 b^{2} f^{2} h i x -48 a^{2} e f h i +36 a b e f \,i^{2} x +4 \ln \left (c \left (f x +e \right )\right )^{3} b^{2} e^{2} i^{2}+4 \ln \left (c \left (f x +e \right )\right )^{3} b^{2} f^{2} h^{2}-24 x \ln \left (c \left (f x +e \right )\right ) a b e f \,i^{2}+48 x \ln \left (c \left (f x +e \right )\right ) a b \,f^{2} h i -24 \ln \left (c \left (f x +e \right )\right )^{2} a b e f h i +48 \ln \left (c \left (f x +e \right )\right ) a b e f h i -48 a b \,f^{2} h i x -96 b^{2} e f h i +96 a b e f h i +6 a^{2} f^{2} i^{2} x^{2}+3 b^{2} f^{2} i^{2} x^{2}-18 \ln \left (c \left (f x +e \right )\right )^{2} b^{2} e^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right ) a^{2} e^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right ) a^{2} f^{2} h^{2}+42 \ln \left (c \left (f x +e \right )\right ) b^{2} e^{2} i^{2}+18 a^{2} e^{2} i^{2}+81 b^{2} e^{2} i^{2}+12 x^{2} \ln \left (c \left (f x +e \right )\right ) a b \,f^{2} i^{2}-12 x \ln \left (c \left (f x +e \right )\right )^{2} b^{2} e f \,i^{2}+24 x \ln \left (c \left (f x +e \right )\right )^{2} b^{2} f^{2} h i -8 \ln \left (c \left (f x +e \right )\right )^{3} b^{2} e f h i +36 x \ln \left (c \left (f x +e \right )\right ) b^{2} e f \,i^{2}-48 x \ln \left (c \left (f x +e \right )\right ) b^{2} f^{2} h i +24 \ln \left (c \left (f x +e \right )\right )^{2} b^{2} e f h i -24 \ln \left (c \left (f x +e \right )\right ) a^{2} e f h i -48 \ln \left (c \left (f x +e \right )\right ) b^{2} e f h i -6 a b \,f^{2} i^{2} x^{2}+6 x^{2} \ln \left (c \left (f x +e \right )\right )^{2} b^{2} f^{2} i^{2}-6 x^{2} \ln \left (c \left (f x +e \right )\right ) b^{2} f^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right )^{2} a b \,e^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right )^{2} a b \,f^{2} h^{2}-36 \ln \left (c \left (f x +e \right )\right ) a b \,e^{2} i^{2}}{12 d \,f^{3}}\)	\(640\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.41 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {3 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{2} i^{2} x^{2} + 4 \, {\left (b^{2} f^{2} h^{2} - 2 \, b^{2} e f h i + b^{2} e^{2} i^{2}\right )} \log \left (c f x + c e\right )^{3} + 6 \, {\left (b^{2} f^{2} i^{2} x^{2} + 2 \, a b f^{2} h^{2} - 4 \, {\left (a b - b^{2}\right )} e f h i + {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} i^{2} + 2 \, {\left (2 \, b^{2} f^{2} h i - b^{2} e f i^{2}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 6 \, {\left (4 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{2} h i - {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f i^{2}\right )} x + 6 \, {\left ({\left (2 \, a b - b^{2}\right )} f^{2} i^{2} x^{2} + 2 \, a^{2} f^{2} h^{2} - 4 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f h i + {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} i^{2} + 2 \, {\left (4 \, {\left (a b - b^{2}\right )} f^{2} h i - {\left (2 \, a b - 3 \, b^{2}\right )} e f i^{2}\right )} x\right )} \log \left (c f x + c e\right )}{12 \, d f^{3}} \]

[In]

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

[Out]

1/12*(3*(2*a^2 - 2*a*b + b^2)*f^2*i^2*x^2 + 4*(b^2*f^2*h^2 - 2*b^2*e*f*h*i + b^2*e^2*i^2)*log(c*f*x + c*e)^3 +
6*(b^2*f^2*i^2*x^2 + 2*a*b*f^2*h^2 - 4*(a*b - b^2)*e*f*h*i + (2*a*b - 3*b^2)*e^2*i^2 + 2*(2*b^2*f^2*h*i - b^2
*e*f*i^2)*x)*log(c*f*x + c*e)^2 + 6*(4*(a^2 - 2*a*b + 2*b^2)*f^2*h*i - (2*a^2 - 6*a*b + 7*b^2)*e*f*i^2)*x + 6*
((2*a*b - b^2)*f^2*i^2*x^2 + 2*a^2*f^2*h^2 - 4*(a^2 - 2*a*b + 2*b^2)*e*f*h*i + (2*a^2 - 6*a*b + 7*b^2)*e^2*i^2
+ 2*(4*(a*b - b^2)*f^2*h*i - (2*a*b - 3*b^2)*e*f*i^2)*x)*log(c*f*x + c*e))/(d*f^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (218) = 436\).

Time = 0.60 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.99 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=x^{2} \left (\frac {a^{2} i^{2}}{2 d f} - \frac {a b i^{2}}{2 d f} + \frac {b^{2} i^{2}}{4 d f}\right ) + x \left (- \frac {a^{2} e i^{2}}{d f^{2}} + \frac {2 a^{2} h i}{d f} + \frac {3 a b e i^{2}}{d f^{2}} - \frac {4 a b h i}{d f} - \frac {7 b^{2} e i^{2}}{2 d f^{2}} + \frac {4 b^{2} h i}{d f}\right ) + \frac {\left (- 4 a b e i^{2} x + 8 a b f h i x + 2 a b f i^{2} x^{2} + 6 b^{2} e i^{2} x - 8 b^{2} f h i x - b^{2} f i^{2} x^{2}\right ) \log {\left (c \left (e + f x\right ) \right )}}{2 d f^{2}} + \frac {\left (b^{2} e^{2} i^{2} - 2 b^{2} e f h i + b^{2} f^{2} h^{2}\right ) \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{3}} + \frac {\left (2 a^{2} e^{2} i^{2} - 4 a^{2} e f h i + 2 a^{2} f^{2} h^{2} - 6 a b e^{2} i^{2} + 8 a b e f h i + 7 b^{2} e^{2} i^{2} - 8 b^{2} e f h i\right ) \log {\left (e + f x \right )}}{2 d f^{3}} + \frac {\left (2 a b e^{2} i^{2} - 4 a b e f h i + 2 a b f^{2} h^{2} - 3 b^{2} e^{2} i^{2} + 4 b^{2} e f h i - 2 b^{2} e f i^{2} x + 4 b^{2} f^{2} h i x + b^{2} f^{2} i^{2} x^{2}\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{3}} \]

[In]

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

[Out]

x**2*(a**2*i**2/(2*d*f) - a*b*i**2/(2*d*f) + b**2*i**2/(4*d*f)) + x*(-a**2*e*i**2/(d*f**2) + 2*a**2*h*i/(d*f)
+ 3*a*b*e*i**2/(d*f**2) - 4*a*b*h*i/(d*f) - 7*b**2*e*i**2/(2*d*f**2) + 4*b**2*h*i/(d*f)) + (-4*a*b*e*i**2*x +
8*a*b*f*h*i*x + 2*a*b*f*i**2*x**2 + 6*b**2*e*i**2*x - 8*b**2*f*h*i*x - b**2*f*i**2*x**2)*log(c*(e + f*x))/(2*d
*f**2) + (b**2*e**2*i**2 - 2*b**2*e*f*h*i + b**2*f**2*h**2)*log(c*(e + f*x))**3/(3*d*f**3) + (2*a**2*e**2*i**2
- 4*a**2*e*f*h*i + 2*a**2*f**2*h**2 - 6*a*b*e**2*i**2 + 8*a*b*e*f*h*i + 7*b**2*e**2*i**2 - 8*b**2*e*f*h*i)*lo
g(e + f*x)/(2*d*f**3) + (2*a*b*e**2*i**2 - 4*a*b*e*f*h*i + 2*a*b*f**2*h**2 - 3*b**2*e**2*i**2 + 4*b**2*e*f*h*i
- 2*b**2*e*f*i**2*x + 4*b**2*f**2*h*i*x + b**2*f**2*i**2*x**2)*log(c*(e + f*x))**2/(2*d*f**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (230) = 460\).

Time = 0.24 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.46 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=4 \, a b h i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) + a b i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} \log \left (c f x + c e\right ) - a b h^{2} {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + 2 \, a^{2} h i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} + \frac {1}{2} \, a^{2} i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} + \frac {b^{2} h^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac {2 \, a b h^{2} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a^{2} h^{2} \log \left (d f x + d e\right )}{d f} + \frac {2 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b h i}{d f^{2}} - \frac {{\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} a b i^{2}}{2 \, d f^{3}} - \frac {2 \, {\left (c^{2} e \log \left (c f x + c e\right )^{3} - 3 \, {\left (c f x + c e\right )} {\left (c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + 2 \, c\right )}\right )} b^{2} h i}{3 \, c^{2} d f^{2}} + \frac {{\left (4 \, c^{3} e^{2} \log \left (c f x + c e\right )^{3} + 3 \, {\left (c f x + c e\right )}^{2} {\left (2 \, c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + c\right )} - 24 \, {\left (c^{2} e \log \left (c f x + c e\right )^{2} - 2 \, c^{2} e \log \left (c f x + c e\right ) + 2 \, c^{2} e\right )} {\left (c f x + c e\right )}\right )} b^{2} i^{2}}{12 \, c^{3} d f^{3}} \]

[In]

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

[Out]

4*a*b*h*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + a*b*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^2*(2*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - (log(f*x + e)^2 + 2*l
og(f*x + e)*log(c))/(d*f)) + 2*a^2*h*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) + 1/2*a^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^2*log(c*f*x + c*e)^3/(d*f) + 2*a*b*h^2*log(c*f*x + c*e)*log(d*f*x
+ d*e)/(d*f) + a^2*h^2*log(d*f*x + d*e)/(d*f) + 2*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*a*b*h*i/(d*f^
2) - 1/2*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*a*b*i^2/(d*f^3) - 2/3*(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*i/(c^2*d*f^2) + 1/12*(
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*i^2/(c^3*d*f^3)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.76 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {1}{2} \, {\left (\frac {b^{2} i^{2} x^{2}}{d f} + \frac {2 \, {\left (2 \, b^{2} f h i - b^{2} e i^{2}\right )} x}{d f^{2}} + \frac {2 \, a b f^{2} h^{2} - 4 \, a b e f h i + 4 \, b^{2} e f h i + 2 \, a b e^{2} i^{2} - 3 \, b^{2} e^{2} i^{2}}{d f^{3}}\right )} \log \left (c f x + c e\right )^{2} + \frac {1}{2} \, {\left (\frac {{\left (2 \, a b i^{2} - b^{2} i^{2}\right )} x^{2}}{d f} + \frac {2 \, {\left (4 \, a b f h i - 4 \, b^{2} f h i - 2 \, a b e i^{2} + 3 \, b^{2} e i^{2}\right )} x}{d f^{2}}\right )} \log \left (c f x + c e\right ) + \frac {{\left (2 \, a^{2} i^{2} - 2 \, a b i^{2} + b^{2} i^{2}\right )} x^{2}}{4 \, d f} + \frac {{\left (b^{2} f^{2} h^{2} - 2 \, b^{2} e f h i + b^{2} e^{2} i^{2}\right )} \log \left (c f x + c e\right )^{3}}{3 \, d f^{3}} + \frac {{\left (4 \, a^{2} f h i - 8 \, a b f h i + 8 \, b^{2} f h i - 2 \, a^{2} e i^{2} + 6 \, a b e i^{2} - 7 \, b^{2} e i^{2}\right )} x}{2 \, d f^{2}} + \frac {{\left (2 \, a^{2} f^{2} h^{2} - 4 \, a^{2} e f h i + 8 \, a b e f h i - 8 \, b^{2} e f h i + 2 \, a^{2} e^{2} i^{2} - 6 \, a b e^{2} i^{2} + 7 \, b^{2} e^{2} i^{2}\right )} \log \left (f x + e\right )}{2 \, d f^{3}} \]

[In]

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

[Out]

1/2*(b^2*i^2*x^2/(d*f) + 2*(2*b^2*f*h*i - b^2*e*i^2)*x/(d*f^2) + (2*a*b*f^2*h^2 - 4*a*b*e*f*h*i + 4*b^2*e*f*h*
i + 2*a*b*e^2*i^2 - 3*b^2*e^2*i^2)/(d*f^3))*log(c*f*x + c*e)^2 + 1/2*((2*a*b*i^2 - b^2*i^2)*x^2/(d*f) + 2*(4*a
*b*f*h*i - 4*b^2*f*h*i - 2*a*b*e*i^2 + 3*b^2*e*i^2)*x/(d*f^2))*log(c*f*x + c*e) + 1/4*(2*a^2*i^2 - 2*a*b*i^2 +
b^2*i^2)*x^2/(d*f) + 1/3*(b^2*f^2*h^2 - 2*b^2*e*f*h*i + b^2*e^2*i^2)*log(c*f*x + c*e)^3/(d*f^3) + 1/2*(4*a^2*
f*h*i - 8*a*b*f*h*i + 8*b^2*f*h*i - 2*a^2*e*i^2 + 6*a*b*e*i^2 - 7*b^2*e*i^2)*x/(d*f^2) + 1/2*(2*a^2*f^2*h^2 -
4*a^2*e*f*h*i + 8*a*b*e*f*h*i - 8*b^2*e*f*h*i + 2*a^2*e^2*i^2 - 6*a*b*e^2*i^2 + 7*b^2*e^2*i^2)*log(f*x + e)/(d
*f^3)

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.71 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=x\,\left (\frac {i\,\left (2\,a^2\,f\,h-3\,b^2\,e\,i+4\,b^2\,f\,h+2\,a\,b\,e\,i-4\,a\,b\,f\,h\right )}{d\,f^2}-\frac {e\,i^2\,\left (2\,a^2-2\,a\,b+b^2\right )}{2\,d\,f^2}\right )+{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^2\,x^2}{2\,d\,f^2}-\frac {b^2\,i\,x\,\left (e\,i-2\,f\,h\right )}{d\,f^3}\right )+\frac {-3\,b^2\,e^2\,i^2+4\,b^2\,e\,f\,h\,i+2\,a\,b\,e^2\,i^2-4\,a\,b\,e\,f\,h\,i+2\,a\,b\,f^2\,h^2}{2\,d\,f^3}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x\,\left (3\,e\,b^2\,i^2-4\,f\,h\,b^2\,i-2\,a\,e\,b\,i^2+4\,a\,f\,h\,b\,i\right )}{d\,f^3}+\frac {b\,i^2\,x^2\,\left (2\,a-b\right )}{2\,d\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (2\,a^2\,e^2\,i^2-4\,a^2\,e\,f\,h\,i+2\,a^2\,f^2\,h^2-6\,a\,b\,e^2\,i^2+8\,a\,b\,e\,f\,h\,i+7\,b^2\,e^2\,i^2-8\,b^2\,e\,f\,h\,i\right )}{2\,d\,f^3}+\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^2\,i^2-2\,e\,f\,h\,i+f^2\,h^2\right )}{3\,d\,f^3}+\frac {i^2\,x^2\,\left (2\,a^2-2\,a\,b+b^2\right )}{4\,d\,f} \]

[In]

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

[Out]

x*((i*(2*a^2*f*h - 3*b^2*e*i + 4*b^2*f*h + 2*a*b*e*i - 4*a*b*f*h))/(d*f^2) - (e*i^2*(2*a^2 - 2*a*b + b^2))/(2*
d*f^2)) + log(c*(e + f*x))^2*(f*((b^2*i^2*x^2)/(2*d*f^2) - (b^2*i*x*(e*i - 2*f*h))/(d*f^3)) + (2*a*b*e^2*i^2 -
3*b^2*e^2*i^2 + 2*a*b*f^2*h^2 + 4*b^2*e*f*h*i - 4*a*b*e*f*h*i)/(2*d*f^3)) + f*log(c*(e + f*x))*((x*(3*b^2*e*i
^2 - 2*a*b*e*i^2 - 4*b^2*f*h*i + 4*a*b*f*h*i))/(d*f^3) + (b*i^2*x^2*(2*a - b))/(2*d*f^2)) + (log(e + f*x)*(2*a
^2*e^2*i^2 + 2*a^2*f^2*h^2 + 7*b^2*e^2*i^2 - 6*a*b*e^2*i^2 - 4*a^2*e*f*h*i - 8*b^2*e*f*h*i + 8*a*b*e*f*h*i))/(
2*d*f^3) + (b^2*log(c*(e + f*x))^3*(e^2*i^2 + f^2*h^2 - 2*e*f*h*i))/(3*d*f^3) + (i^2*x^2*(2*a^2 - 2*a*b + b^2)
)/(4*d*f)