Integrand size = 32, antiderivative size = 238 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {4 a b i (f h-e i) x}{d f^2}+\frac {4 b^2 i (f h-e i) x}{d f^2}+\frac {b^2 i^2 (e+f x)^2}{4 d f^3}-\frac {4 b^2 i (f h-e i) (e+f x) \log (c (e+f x))}{d f^3}-\frac {b i^2 (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^3}+\frac {2 i (f h-e i) (e+f x) (a+b \log (c (e+f x)))^2}{d f^3}+\frac {i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^3}+\frac {(f h-e i)^2 (a+b \log (c (e+f x)))^3}{3 b d f^3} \] Output:
-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*l n(c*(f*x+e)))^3/b/d/f^3
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {24 i (f h-e i) (e+f x) (a+b \log (c (e+f x)))^2+6 i^2 (e+f x)^2 (a+b \log (c (e+f x)))^2+\frac {4 (f h-e i)^2 (a+b \log (c (e+f x)))^3}{b}-48 b i (f h-e i) ((a-b) f x+b (e+f x) \log (c (e+f x)))+3 b i^2 \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )}{12 d f^3} \] Input:
Integrate[((h + i*x)^2*(a + b*Log[c*(e + f*x)])^2)/(d*e + d*f*x),x]
Output:
(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)
Time = 1.52 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2858, 27, 2788, 2767, 2009, 2788, 2733, 2009, 2739, 15}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^2 (a+b \log (c (e+f x)))^2}{d f^2 (e+f x)}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {i \int (f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2d(e+f x)+(f h-e i) \int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2767 |
| \(\displaystyle \frac {(f h-e i) \int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \int \left (f h \left (1-\frac {e i}{f h}\right ) (a+b \log (c (e+f x)))^2+i (e+f x) (a+b \log (c (e+f x)))^2\right )d(e+f x)}{d f^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f h-e i) \int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \left ((e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-\frac {1}{2} b i (e+f x)^2 (a+b \log (c (e+f x)))+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2-2 a b (e+f x) (f h-e i)-2 b^2 (e+f x) (f h-e i) \log (c (e+f x))+2 b^2 (e+f x) (f h-e i)+\frac {1}{4} b^2 i (e+f x)^2\right )}{d f^3}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {(f h-e i) \left ((f h-e i) \int \frac {(a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \int (a+b \log (c (e+f x)))^2d(e+f x)\right )+i \left ((e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-\frac {1}{2} b i (e+f x)^2 (a+b \log (c (e+f x)))+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2-2 a b (e+f x) (f h-e i)-2 b^2 (e+f x) (f h-e i) \log (c (e+f x))+2 b^2 (e+f x) (f h-e i)+\frac {1}{4} b^2 i (e+f x)^2\right )}{d f^3}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle \frac {(f h-e i) \left ((f h-e i) \int \frac {(a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b \int (a+b \log (c (e+f x)))d(e+f x)\right )\right )+i \left ((e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-\frac {1}{2} b i (e+f x)^2 (a+b \log (c (e+f x)))+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2-2 a b (e+f x) (f h-e i)-2 b^2 (e+f x) (f h-e i) \log (c (e+f x))+2 b^2 (e+f x) (f h-e i)+\frac {1}{4} b^2 i (e+f x)^2\right )}{d f^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f h-e i) \left ((f h-e i) \int \frac {(a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b (a (e+f x)+b (e+f x) \log (c (e+f x))-b (e+f x))\right )\right )+i \left ((e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-\frac {1}{2} b i (e+f x)^2 (a+b \log (c (e+f x)))+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2-2 a b (e+f x) (f h-e i)-2 b^2 (e+f x) (f h-e i) \log (c (e+f x))+2 b^2 (e+f x) (f h-e i)+\frac {1}{4} b^2 i (e+f x)^2\right )}{d f^3}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle \frac {(f h-e i) \left (\frac {(f h-e i) \int (a+b \log (c (e+f x)))^2d(a+b \log (c (e+f x)))}{b}+i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b (a (e+f x)+b (e+f x) \log (c (e+f x))-b (e+f x))\right )\right )+i \left ((e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-\frac {1}{2} b i (e+f x)^2 (a+b \log (c (e+f x)))+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2-2 a b (e+f x) (f h-e i)-2 b^2 (e+f x) (f h-e i) \log (c (e+f x))+2 b^2 (e+f x) (f h-e i)+\frac {1}{4} b^2 i (e+f x)^2\right )}{d f^3}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {i \left ((e+f x) (f h-e i) (a+b \log (c (e+f x)))^2-\frac {1}{2} b i (e+f x)^2 (a+b \log (c (e+f x)))+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2-2 a b (e+f x) (f h-e i)-2 b^2 (e+f x) (f h-e i) \log (c (e+f x))+2 b^2 (e+f x) (f h-e i)+\frac {1}{4} b^2 i (e+f x)^2\right )+(f h-e i) \left (\frac {(f h-e i) (a+b \log (c (e+f x)))^3}{3 b}+i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b (a (e+f x)+b (e+f x) \log (c (e+f x))-b (e+f x))\right )\right )}{d f^3}\) |
Input:
Int[((h + i*x)^2*(a + b*Log[c*(e + f*x)])^2)/(d*e + d*f*x),x]
Output:
(i*(-2*a*b*(f*h - e*i)*(e + f*x) + 2*b^2*(f*h - e*i)*(e + f*x) + (b^2*i*(e + f*x)^2)/4 - 2*b^2*(f*h - e*i)*(e + f*x)*Log[c*(e + f*x)] - (b*i*(e + f* x)^2*(a + b*Log[c*(e + f*x)]))/2 + (f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)])^2 + (i*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/2) + (f*h - e*i)*(( (f*h - e*i)*(a + b*Log[c*(e + f*x)])^3)/(3*b) + i*((e + f*x)*(a + b*Log[c* (e + f*x)])^2 - 2*b*(a*(e + f*x) - b*(e + f*x) + b*(e + f*x)*Log[c*(e + f* x)]))))/(d*f^3)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( q_.), x_Symbol] :> With[{u = ExpandIntegrand[(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]))
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(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]
Time = 1.58 (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 | \(\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e^{2} i^{2}}{3 d \,f^{3}}-\frac {2 b^{2} \ln \left (c \left (f x +e \right )\right )^{3} e h i}{3 d \,f^{2}}+\frac {b^{2} \ln \left (c \left (f x +e \right )\right )^{3} h^{2}}{3 d f}+\frac {b \left (b \,f^{2} i^{2} x^{2}-2 b e f \,i^{2} x +4 b \,f^{2} h i x +2 a \,e^{2} i^{2}-4 a e f h i +2 a \,f^{2} h^{2}-3 b \,e^{2} i^{2}+4 b e f h i \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{3}}-\frac {b i x \left (-2 a f i x +b f i x +4 a e i -8 a f h -6 b e i +8 b f h \right ) \ln \left (c \left (f x +e \right )\right )}{2 d \,f^{2}}+\frac {a^{2} i^{2} x^{2}}{2 d f}-\frac {a b \,i^{2} x^{2}}{2 d f}+\frac {b^{2} i^{2} x^{2}}{4 d f}+\frac {\ln \left (f x +e \right ) a^{2} e^{2} i^{2}}{d \,f^{3}}-\frac {2 \ln \left (f x +e \right ) a^{2} e h i}{d \,f^{2}}+\frac {\ln \left (f x +e \right ) a^{2} h^{2}}{d f}-\frac {3 \ln \left (f x +e \right ) a b \,e^{2} i^{2}}{d \,f^{3}}+\frac {4 \ln \left (f x +e \right ) a b e h i}{d \,f^{2}}+\frac {7 \ln \left (f x +e \right ) b^{2} e^{2} i^{2}}{2 d \,f^{3}}-\frac {4 \ln \left (f x +e \right ) b^{2} e h i}{d \,f^{2}}-\frac {a^{2} e \,i^{2} x}{d \,f^{2}}+\frac {2 a^{2} h i x}{d f}+\frac {3 a b e \,i^{2} x}{d \,f^{2}}-\frac {4 a b h i x}{d f}-\frac {7 b^{2} e \,i^{2} x}{2 d \,f^{2}}+\frac {4 b^{2} h i x}{d f}\) | \(501\) |
| parts | \(\frac {a^{2} \left (\frac {i \left (\frac {1}{2} f i \,x^{2}-e i x +2 f h x \right )}{f^{2}}+\frac {\left (e^{2} i^{2}-2 e f h i +f^{2} h^{2}\right ) \ln \left (f x +e \right )}{f^{3}}\right )}{d}+\frac {b^{2} \left (\frac {c \,e^{2} i^{2} \ln \left (c f x +c e \right )^{3}}{3 f^{2}}-\frac {2 c e h i \ln \left (c f x +c e \right )^{3}}{3 f}+\frac {c \,h^{2} \ln \left (c f x +c e \right )^{3}}{3}-\frac {2 e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f^{2}}+\frac {2 h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )^{2}-2 \left (c f x +c e \right ) \ln \left (c f x +c e \right )+2 c f x +2 c e \right )}{f}+\frac {i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}+\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}\right )}{d c f}+\frac {2 a b \left (\frac {c \,e^{2} i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2}}-\frac {c e h i \ln \left (c f x +c e \right )^{2}}{f}+\frac {c \,h^{2} \ln \left (c f x +c e \right )^{2}}{2}-\frac {2 e \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2}}+\frac {2 h i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f}+\frac {i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}\right )}{d c f}\) | \(509\) |
| 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 | \(\frac {6 a^{2} f^{2} i^{2} x^{2}+3 b^{2} f^{2} i^{2} x^{2}+96 a b e f h i +36 a b e f \,i^{2} x -48 a b \,f^{2} h i x -66 a b \,e^{2} i^{2}-48 a^{2} e f h i -96 b^{2} e f h i -24 \ln \left (c \left (f x +e \right )\right )^{2} a b e f h i +48 \ln \left (c \left (f x +e \right )\right ) a b e f h i -24 x \ln \left (c \left (f x +e \right )\right ) a b e f \,i^{2}+48 x \ln \left (c \left (f x +e \right )\right ) a b \,f^{2} h i +4 \ln \left (c \left (f x +e \right )\right )^{3} b^{2} f^{2} h^{2}-18 \ln \left (c \left (f x +e \right )\right )^{2} b^{2} e^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right ) a^{2} e^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right ) a^{2} f^{2} h^{2}+42 \ln \left (c \left (f x +e \right )\right ) b^{2} e^{2} i^{2}-6 a b \,f^{2} i^{2} x^{2}-12 a^{2} e f \,i^{2} x +24 a^{2} f^{2} h i x -42 b^{2} e f \,i^{2} x +48 b^{2} f^{2} h i x +4 \ln \left (c \left (f x +e \right )\right )^{3} b^{2} e^{2} i^{2}-24 \ln \left (c \left (f x +e \right )\right ) a^{2} e f h i +12 x^{2} \ln \left (c \left (f x +e \right )\right ) a b \,f^{2} i^{2}-12 x \ln \left (c \left (f x +e \right )\right )^{2} b^{2} e f \,i^{2}+24 x \ln \left (c \left (f x +e \right )\right )^{2} b^{2} f^{2} h i -8 \ln \left (c \left (f x +e \right )\right )^{3} b^{2} e f h i +36 x \ln \left (c \left (f x +e \right )\right ) b^{2} e f \,i^{2}-48 x \ln \left (c \left (f x +e \right )\right ) b^{2} f^{2} h i +24 \ln \left (c \left (f x +e \right )\right )^{2} b^{2} e f h i -36 \ln \left (c \left (f x +e \right )\right ) a b \,e^{2} i^{2}+6 x^{2} \ln \left (c \left (f x +e \right )\right )^{2} b^{2} f^{2} i^{2}-6 x^{2} \ln \left (c \left (f x +e \right )\right ) b^{2} f^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right )^{2} a b \,e^{2} i^{2}+12 \ln \left (c \left (f x +e \right )\right )^{2} a b \,f^{2} h^{2}-48 \ln \left (c \left (f x +e \right )\right ) b^{2} e f h i +18 a^{2} e^{2} i^{2}+81 b^{2} e^{2} i^{2}}{12 d \,f^{3}}\) | \(640\) |
Input:
int((i*x+h)^2*(a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e),x,method=_RETURNVERBOSE)
Output:
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
Time = 0.08 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.41 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {3 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{2} i^{2} x^{2} + 4 \, {\left (b^{2} f^{2} h^{2} - 2 \, b^{2} e f h i + b^{2} e^{2} i^{2}\right )} \log \left (c f x + c e\right )^{3} + 6 \, {\left (b^{2} f^{2} i^{2} x^{2} + 2 \, a b f^{2} h^{2} - 4 \, {\left (a b - b^{2}\right )} e f h i + {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} i^{2} + 2 \, {\left (2 \, b^{2} f^{2} h i - b^{2} e f i^{2}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 6 \, {\left (4 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{2} h i - {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f i^{2}\right )} x + 6 \, {\left ({\left (2 \, a b - b^{2}\right )} f^{2} i^{2} x^{2} + 2 \, a^{2} f^{2} h^{2} - 4 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f h i + {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} i^{2} + 2 \, {\left (4 \, {\left (a b - b^{2}\right )} f^{2} h i - {\left (2 \, a b - 3 \, b^{2}\right )} e f i^{2}\right )} x\right )} \log \left (c f x + c e\right )}{12 \, d f^{3}} \] Input:
integrate((i*x+h)^2*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="frica s")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (218) = 436\).
Time = 0.59 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.99 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=x^{2} \left (\frac {a^{2} i^{2}}{2 d f} - \frac {a b i^{2}}{2 d f} + \frac {b^{2} i^{2}}{4 d f}\right ) + x \left (- \frac {a^{2} e i^{2}}{d f^{2}} + \frac {2 a^{2} h i}{d f} + \frac {3 a b e i^{2}}{d f^{2}} - \frac {4 a b h i}{d f} - \frac {7 b^{2} e i^{2}}{2 d f^{2}} + \frac {4 b^{2} h i}{d f}\right ) + \frac {\left (- 4 a b e i^{2} x + 8 a b f h i x + 2 a b f i^{2} x^{2} + 6 b^{2} e i^{2} x - 8 b^{2} f h i x - b^{2} f i^{2} x^{2}\right ) \log {\left (c \left (e + f x\right ) \right )}}{2 d f^{2}} + \frac {\left (b^{2} e^{2} i^{2} - 2 b^{2} e f h i + b^{2} f^{2} h^{2}\right ) \log {\left (c \left (e + f x\right ) \right )}^{3}}{3 d f^{3}} + \frac {\left (2 a^{2} e^{2} i^{2} - 4 a^{2} e f h i + 2 a^{2} f^{2} h^{2} - 6 a b e^{2} i^{2} + 8 a b e f h i + 7 b^{2} e^{2} i^{2} - 8 b^{2} e f h i\right ) \log {\left (e + f x \right )}}{2 d f^{3}} + \frac {\left (2 a b e^{2} i^{2} - 4 a b e f h i + 2 a b f^{2} h^{2} - 3 b^{2} e^{2} i^{2} + 4 b^{2} e f h i - 2 b^{2} e f i^{2} x + 4 b^{2} f^{2} h i x + b^{2} f^{2} i^{2} x^{2}\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{3}} \] Input:
integrate((i*x+h)**2*(a+b*ln(c*(f*x+e)))**2/(d*f*x+d*e),x)
Output:
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)*log(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)
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (230) = 460\).
Time = 0.06 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.46 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=4 \, a b h i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) + a b i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} \log \left (c f x + c e\right ) - a b h^{2} {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + 2 \, a^{2} h i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} + \frac {1}{2} \, a^{2} i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} + \frac {b^{2} h^{2} \log \left (c f x + c e\right )^{3}}{3 \, d f} + \frac {2 \, a b h^{2} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a^{2} h^{2} \log \left (d f x + d e\right )}{d f} + \frac {2 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} a b h i}{d f^{2}} - \frac {{\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} a b i^{2}}{2 \, d f^{3}} - \frac {2 \, {\left (c^{2} e \log \left (c f x + c e\right )^{3} - 3 \, {\left (c f x + c e\right )} {\left (c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + 2 \, c\right )}\right )} b^{2} h i}{3 \, c^{2} d f^{2}} + \frac {{\left (4 \, c^{3} e^{2} \log \left (c f x + c e\right )^{3} + 3 \, {\left (c f x + c e\right )}^{2} {\left (2 \, c \log \left (c f x + c e\right )^{2} - 2 \, c \log \left (c f x + c e\right ) + c\right )} - 24 \, {\left (c^{2} e \log \left (c f x + c e\right )^{2} - 2 \, c^{2} e \log \left (c f x + c e\right ) + 2 \, c^{2} e\right )} {\left (c f x + c e\right )}\right )} b^{2} i^{2}}{12 \, c^{3} d f^{3}} \] Input:
integrate((i*x+h)^2*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="maxim a")
Output:
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*log (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)
Time = 0.13 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.76 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {1}{2} \, {\left (\frac {b^{2} i^{2} x^{2}}{d f} + \frac {2 \, {\left (2 \, b^{2} f h i - b^{2} e i^{2}\right )} x}{d f^{2}} + \frac {2 \, a b f^{2} h^{2} - 4 \, a b e f h i + 4 \, b^{2} e f h i + 2 \, a b e^{2} i^{2} - 3 \, b^{2} e^{2} i^{2}}{d f^{3}}\right )} \log \left (c f x + c e\right )^{2} + \frac {1}{2} \, {\left (\frac {{\left (2 \, a b i^{2} - b^{2} i^{2}\right )} x^{2}}{d f} + \frac {2 \, {\left (4 \, a b f h i - 4 \, b^{2} f h i - 2 \, a b e i^{2} + 3 \, b^{2} e i^{2}\right )} x}{d f^{2}}\right )} \log \left (c f x + c e\right ) + \frac {{\left (2 \, a^{2} i^{2} - 2 \, a b i^{2} + b^{2} i^{2}\right )} x^{2}}{4 \, d f} + \frac {{\left (b^{2} f^{2} h^{2} - 2 \, b^{2} e f h i + b^{2} e^{2} i^{2}\right )} \log \left (c f x + c e\right )^{3}}{3 \, d f^{3}} + \frac {{\left (4 \, a^{2} f h i - 8 \, a b f h i + 8 \, b^{2} f h i - 2 \, a^{2} e i^{2} + 6 \, a b e i^{2} - 7 \, b^{2} e i^{2}\right )} x}{2 \, d f^{2}} + \frac {{\left (2 \, a^{2} f^{2} h^{2} - 4 \, a^{2} e f h i + 8 \, a b e f h i - 8 \, b^{2} e f h i + 2 \, a^{2} e^{2} i^{2} - 6 \, a b e^{2} i^{2} + 7 \, b^{2} e^{2} i^{2}\right )} \log \left (f x + e\right )}{2 \, d f^{3}} \] Input:
integrate((i*x+h)^2*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x, algorithm="giac" )
Output:
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)
Time = 26.92 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.71 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=x\,\left (\frac {i\,\left (2\,a^2\,f\,h-3\,b^2\,e\,i+4\,b^2\,f\,h+2\,a\,b\,e\,i-4\,a\,b\,f\,h\right )}{d\,f^2}-\frac {e\,i^2\,\left (2\,a^2-2\,a\,b+b^2\right )}{2\,d\,f^2}\right )+{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^2\,x^2}{2\,d\,f^2}-\frac {b^2\,i\,x\,\left (e\,i-2\,f\,h\right )}{d\,f^3}\right )+\frac {-3\,b^2\,e^2\,i^2+4\,b^2\,e\,f\,h\,i+2\,a\,b\,e^2\,i^2-4\,a\,b\,e\,f\,h\,i+2\,a\,b\,f^2\,h^2}{2\,d\,f^3}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x\,\left (3\,e\,b^2\,i^2-4\,f\,h\,b^2\,i-2\,a\,e\,b\,i^2+4\,a\,f\,h\,b\,i\right )}{d\,f^3}+\frac {b\,i^2\,x^2\,\left (2\,a-b\right )}{2\,d\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (2\,a^2\,e^2\,i^2-4\,a^2\,e\,f\,h\,i+2\,a^2\,f^2\,h^2-6\,a\,b\,e^2\,i^2+8\,a\,b\,e\,f\,h\,i+7\,b^2\,e^2\,i^2-8\,b^2\,e\,f\,h\,i\right )}{2\,d\,f^3}+\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^2\,i^2-2\,e\,f\,h\,i+f^2\,h^2\right )}{3\,d\,f^3}+\frac {i^2\,x^2\,\left (2\,a^2-2\,a\,b+b^2\right )}{4\,d\,f} \] Input:
int(((h + i*x)^2*(a + b*log(c*(e + f*x)))^2)/(d*e + d*f*x),x)
Output:
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)
Time = 0.19 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.28 \[ \int \frac {(h+i x)^2 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {-4 \mathrm {log}\left (c f x +c e \right )^{3} b^{2} e^{2}+18 \mathrm {log}\left (c f x +c e \right )^{2} b^{2} e^{2}-42 \,\mathrm {log}\left (c f x +c e \right ) b^{2} e^{2}-12 \,\mathrm {log}\left (f x +e \right ) a^{2} e^{2}-6 a^{2} f^{2} x^{2}-3 b^{2} f^{2} x^{2}-48 a b \,f^{2} h i x +4 \mathrm {log}\left (c f x +c e \right )^{3} b^{2} f^{2} h^{2}-12 \mathrm {log}\left (c f x +c e \right )^{2} a b \,e^{2}-6 \mathrm {log}\left (c f x +c e \right )^{2} b^{2} f^{2} x^{2}+36 \,\mathrm {log}\left (c f x +c e \right ) a b \,e^{2}+6 \,\mathrm {log}\left (c f x +c e \right ) b^{2} f^{2} x^{2}+12 \,\mathrm {log}\left (f x +e \right ) a^{2} f^{2} h^{2}+6 a b \,f^{2} x^{2}+42 b^{2} e f x +12 a^{2} e f x -24 \mathrm {log}\left (c f x +c e \right )^{2} a b e f h i +48 \,\mathrm {log}\left (c f x +c e \right ) a b e f h i +48 \,\mathrm {log}\left (c f x +c e \right ) a b \,f^{2} h i x -8 \mathrm {log}\left (c f x +c e \right )^{3} b^{2} e f h i +24 \mathrm {log}\left (c f x +c e \right )^{2} b^{2} e f h i +24 \mathrm {log}\left (c f x +c e \right )^{2} b^{2} f^{2} h i x +24 \,\mathrm {log}\left (c f x +c e \right ) a b e f x -48 \,\mathrm {log}\left (c f x +c e \right ) b^{2} e f h i -48 \,\mathrm {log}\left (c f x +c e \right ) b^{2} f^{2} h i x -24 \,\mathrm {log}\left (f x +e \right ) a^{2} e f h i +12 \mathrm {log}\left (c f x +c e \right )^{2} a b \,f^{2} h^{2}+12 \mathrm {log}\left (c f x +c e \right )^{2} b^{2} e f x -12 \,\mathrm {log}\left (c f x +c e \right ) a b \,f^{2} x^{2}-36 \,\mathrm {log}\left (c f x +c e \right ) b^{2} e f x +24 a^{2} f^{2} h i x -36 a b e f x +48 b^{2} f^{2} h i x}{12 d \,f^{3}} \] Input:
int((i*x+h)^2*(a+b*log(c*(f*x+e)))^2/(d*f*x+d*e),x)
Output:
( - 4*log(c*e + c*f*x)**3*b**2*e**2 - 8*log(c*e + c*f*x)**3*b**2*e*f*h*i + 4*log(c*e + c*f*x)**3*b**2*f**2*h**2 - 12*log(c*e + c*f*x)**2*a*b*e**2 - 24*log(c*e + c*f*x)**2*a*b*e*f*h*i + 12*log(c*e + c*f*x)**2*a*b*f**2*h**2 + 18*log(c*e + c*f*x)**2*b**2*e**2 + 24*log(c*e + c*f*x)**2*b**2*e*f*h*i + 12*log(c*e + c*f*x)**2*b**2*e*f*x + 24*log(c*e + c*f*x)**2*b**2*f**2*h*i* x - 6*log(c*e + c*f*x)**2*b**2*f**2*x**2 + 36*log(c*e + c*f*x)*a*b*e**2 + 48*log(c*e + c*f*x)*a*b*e*f*h*i + 24*log(c*e + c*f*x)*a*b*e*f*x + 48*log(c *e + c*f*x)*a*b*f**2*h*i*x - 12*log(c*e + c*f*x)*a*b*f**2*x**2 - 42*log(c* e + c*f*x)*b**2*e**2 - 48*log(c*e + c*f*x)*b**2*e*f*h*i - 36*log(c*e + c*f *x)*b**2*e*f*x - 48*log(c*e + c*f*x)*b**2*f**2*h*i*x + 6*log(c*e + c*f*x)* b**2*f**2*x**2 - 12*log(e + f*x)*a**2*e**2 - 24*log(e + f*x)*a**2*e*f*h*i + 12*log(e + f*x)*a**2*f**2*h**2 + 12*a**2*e*f*x + 24*a**2*f**2*h*i*x - 6* a**2*f**2*x**2 - 36*a*b*e*f*x - 48*a*b*f**2*h*i*x + 6*a*b*f**2*x**2 + 42*b **2*e*f*x + 48*b**2*f**2*h*i*x - 3*b**2*f**2*x**2)/(12*d*f**3)