Integrand size = 30, antiderivative size = 244 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=-\frac {3 b i (f h-e i)^2 x}{d f^3}-\frac {3 b i^2 (f h-e i) (e+f x)^2}{4 d f^4}-\frac {b i^3 (e+f x)^3}{9 d f^4}-\frac {b (f h-e i)^3 \log ^2(e+f x)}{2 d f^4}+\frac {3 i (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}{d f^4}+\frac {3 i^2 (f h-e i) (e+f x)^2 (a+b \log (c (e+f x)))}{2 d f^4}+\frac {i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{3 d f^4}+\frac {(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))}{d f^4} \]
-3*b*i*(-e*i+f*h)^2*x/d/f^3-3/4*b*i^2*(-e*i+f*h)*(f*x+e)^2/d/f^4-1/9*b*i^3 *(f*x+e)^3/d/f^4-1/2*b*(-e*i+f*h)^3*ln(f*x+e)^2/d/f^4+3*i*(-e*i+f*h)^2*(f* x+e)*(a+b*ln(c*(f*x+e)))/d/f^4+3/2*i^2*(-e*i+f*h)*(f*x+e)^2*(a+b*ln(c*(f*x +e)))/d/f^4+1/3*i^3*(f*x+e)^3*(a+b*ln(c*(f*x+e)))/d/f^4+(-e*i+f*h)^3*ln(f* x+e)*(a+b*ln(c*(f*x+e)))/d/f^4
Time = 0.19 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.54 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {18 a^2 f^3 h^3-54 a^2 e f^2 h^2 i+54 a^2 e^2 f h i^2-18 a^2 e^3 i^3+108 a b f^3 h^2 i x-108 b^2 f^3 h^2 i x-108 a b e f^2 h i^2 x+162 b^2 e f^2 h i^2 x+36 a b e^2 f i^3 x-66 b^2 e^2 f i^3 x+54 a b f^3 h i^2 x^2-27 b^2 f^3 h i^2 x^2-18 a b e f^2 i^3 x^2+15 b^2 e f^2 i^3 x^2+12 a b f^3 i^3 x^3-4 b^2 f^3 i^3 x^3+6 b^2 e^2 i^2 (-9 f h+5 e i) \log (e+f x)+6 b \left (6 a (f h-e i)^3+b i \left (6 e^3 i^2+6 e^2 f i (-3 h+i x)+3 e f^2 \left (6 h^2-6 h i x-i^2 x^2\right )+f^3 x \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right ) \log (c (e+f x))+18 b^2 (f h-e i)^3 \log ^2(c (e+f x))}{36 b d f^4} \]
(18*a^2*f^3*h^3 - 54*a^2*e*f^2*h^2*i + 54*a^2*e^2*f*h*i^2 - 18*a^2*e^3*i^3 + 108*a*b*f^3*h^2*i*x - 108*b^2*f^3*h^2*i*x - 108*a*b*e*f^2*h*i^2*x + 162 *b^2*e*f^2*h*i^2*x + 36*a*b*e^2*f*i^3*x - 66*b^2*e^2*f*i^3*x + 54*a*b*f^3* h*i^2*x^2 - 27*b^2*f^3*h*i^2*x^2 - 18*a*b*e*f^2*i^3*x^2 + 15*b^2*e*f^2*i^3 *x^2 + 12*a*b*f^3*i^3*x^3 - 4*b^2*f^3*i^3*x^3 + 6*b^2*e^2*i^2*(-9*f*h + 5* e*i)*Log[e + f*x] + 6*b*(6*a*(f*h - e*i)^3 + b*i*(6*e^3*i^2 + 6*e^2*f*i*(- 3*h + i*x) + 3*e*f^2*(6*h^2 - 6*h*i*x - i^2*x^2) + f^3*x*(18*h^2 + 9*h*i*x + 2*i^2*x^2)))*Log[c*(e + f*x)] + 18*b^2*(f*h - e*i)^3*Log[c*(e + f*x)]^2 )/(36*b*d*f^4)
Time = 0.46 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2858, 27, 2772, 2009}
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.
| \(\displaystyle \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^3 (a+b \log (c (e+f x)))}{d f^3 (e+f x)}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))}{e+f x}d(e+f x)}{d f^4}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-b \int \left (\frac {1}{3} (e+f x)^2 i^3+\frac {3}{2} (f h-e i) (e+f x) i^2+3 (f h-e i)^2 i+\frac {(f h-e i)^3 \log (e+f x)}{e+f x}\right )d(e+f x)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))+(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))+3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))}{d f^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{2} i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))+(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))+3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))-b \left (\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+3 i (e+f x) (f h-e i)^2+\frac {1}{2} (f h-e i)^3 \log ^2(e+f x)+\frac {1}{9} i^3 (e+f x)^3\right )}{d f^4}\) |
(-(b*(3*i*(f*h - e*i)^2*(e + f*x) + (3*i^2*(f*h - e*i)*(e + f*x)^2)/4 + (i ^3*(e + f*x)^3)/9 + ((f*h - e*i)^3*Log[e + f*x]^2)/2)) + 3*i*(f*h - e*i)^2 *(e + f*x)*(a + b*Log[c*(e + f*x)]) + (3*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/2 + (i^3*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/3 + (f *h - e*i)^3*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(d*f^4)
3.2.76.3.1 Defintions of rubi rules used
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_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[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])
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 = 0.65 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.46
| method | result | size |
| norman | \(\frac {b i \left (e^{2} i^{2}-3 e f h i +3 f^{2} h^{2}\right ) x \ln \left (c \left (f x +e \right )\right )}{d \,f^{3}}-\frac {\left (6 a \,e^{3} i^{3}-18 a \,e^{2} f h \,i^{2}+18 a e \,f^{2} h^{2} i -6 a \,f^{3} h^{3}-11 b \,e^{3} i^{3}+27 b \,e^{2} f h \,i^{2}-18 b e \,f^{2} h^{2} i \right ) \ln \left (c \left (f x +e \right )\right )}{6 d \,f^{4}}-\frac {b \left (e^{3} i^{3}-3 e^{2} f h \,i^{2}+3 e \,f^{2} h^{2} i -f^{3} h^{3}\right ) \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{4}}+\frac {i \left (6 a \,e^{2} i^{2}-18 a e f h i +18 a \,f^{2} h^{2}-11 b \,e^{2} i^{2}+27 b e f h i -18 b \,f^{2} h^{2}\right ) x}{6 d \,f^{3}}-\frac {i^{2} \left (6 a e i -18 a f h -5 b e i +9 b f h \right ) x^{2}}{12 d \,f^{2}}+\frac {i^{3} \left (3 a -b \right ) x^{3}}{9 d f}+\frac {b \,i^{3} x^{3} \ln \left (c \left (f x +e \right )\right )}{3 d f}-\frac {b \,i^{2} \left (e i -3 f h \right ) x^{2} \ln \left (c \left (f x +e \right )\right )}{2 d \,f^{2}}\) | \(356\) |
| parts | \(\frac {a \left (\frac {i \left (\frac {1}{3} f^{2} i^{2} x^{3}-\frac {1}{2} e f \,i^{2} x^{2}+\frac {3}{2} f^{2} h i \,x^{2}+x \,e^{2} i^{2}-3 x e f h i +3 x \,f^{2} h^{2}\right )}{f^{3}}+\frac {\left (-e^{3} i^{3}+3 e^{2} f h \,i^{2}-3 e \,f^{2} h^{2} i +f^{3} h^{3}\right ) \ln \left (f x +e \right )}{f^{4}}\right )}{d}+\frac {b \left (-\frac {c \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3}}+\frac {3 c \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2}}-\frac {3 c e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f}+\frac {c \,h^{3} \ln \left (c f x +c e \right )^{2}}{2}+\frac {3 e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3}}-\frac {6 e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2}}+\frac {3 h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f}-\frac {3 e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3}}+\frac {3 h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2}}+\frac {i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3}}\right )}{d c f}\) | \(463\) |
| risch | \(-\frac {b \ln \left (c \left (f x +e \right )\right )^{2} e^{3} i^{3}}{2 d \,f^{4}}+\frac {3 b \ln \left (c \left (f x +e \right )\right )^{2} e^{2} h \,i^{2}}{2 d \,f^{3}}-\frac {3 b \ln \left (c \left (f x +e \right )\right )^{2} e \,h^{2} i}{2 d \,f^{2}}+\frac {b \ln \left (c \left (f x +e \right )\right )^{2} h^{3}}{2 d f}+\frac {b i x \left (2 f^{2} i^{2} x^{2}-3 e f \,i^{2} x +9 f^{2} h i x +6 e^{2} i^{2}-18 e f h i +18 f^{2} h^{2}\right ) \ln \left (c \left (f x +e \right )\right )}{6 d \,f^{3}}+\frac {a \,i^{3} x^{3}}{3 d f}-\frac {b \,i^{3} x^{3}}{9 d f}-\frac {a e \,i^{3} x^{2}}{2 d \,f^{2}}+\frac {3 a h \,i^{2} x^{2}}{2 d f}+\frac {5 b e \,i^{3} x^{2}}{12 d \,f^{2}}-\frac {3 b h \,i^{2} x^{2}}{4 d f}-\frac {\ln \left (f x +e \right ) a \,e^{3} i^{3}}{d \,f^{4}}+\frac {3 \ln \left (f x +e \right ) a \,e^{2} h \,i^{2}}{d \,f^{3}}-\frac {3 \ln \left (f x +e \right ) a e \,h^{2} i}{d \,f^{2}}+\frac {\ln \left (f x +e \right ) a \,h^{3}}{d f}+\frac {11 \ln \left (f x +e \right ) b \,e^{3} i^{3}}{6 d \,f^{4}}-\frac {9 \ln \left (f x +e \right ) b \,e^{2} h \,i^{2}}{2 d \,f^{3}}+\frac {3 \ln \left (f x +e \right ) b e \,h^{2} i}{d \,f^{2}}+\frac {a \,e^{2} i^{3} x}{d \,f^{3}}-\frac {3 a e h \,i^{2} x}{d \,f^{2}}+\frac {3 a \,h^{2} i x}{d f}-\frac {11 b \,e^{2} i^{3} x}{6 d \,f^{3}}+\frac {9 b e h \,i^{2} x}{2 d \,f^{2}}-\frac {3 b \,h^{2} i x}{d f}\) | \(494\) |
| parallelrisch | \(\frac {-108 x \ln \left (c \left (f x +e \right )\right ) b e \,f^{2} h \,i^{2}+12 a \,f^{3} i^{3} x^{3}-4 b \,f^{3} i^{3} x^{3}+108 \ln \left (c \left (f x +e \right )\right ) a \,e^{2} f h \,i^{2}-108 \ln \left (c \left (f x +e \right )\right ) a e \,f^{2} h^{2} i -162 \ln \left (c \left (f x +e \right )\right ) b \,e^{2} f h \,i^{2}+108 \ln \left (c \left (f x +e \right )\right ) b e \,f^{2} h^{2} i -18 x^{2} \ln \left (c \left (f x +e \right )\right ) b e \,f^{2} i^{3}+54 x^{2} \ln \left (c \left (f x +e \right )\right ) b \,f^{3} h \,i^{2}+36 x \ln \left (c \left (f x +e \right )\right ) b \,e^{2} f \,i^{3}+108 x \ln \left (c \left (f x +e \right )\right ) b \,f^{3} h^{2} i +54 \ln \left (c \left (f x +e \right )\right )^{2} b \,e^{2} f h \,i^{2}-54 \ln \left (c \left (f x +e \right )\right )^{2} b e \,f^{2} h^{2} i -54 a \,e^{3} i^{3}+117 b \,e^{3} i^{3}-18 \ln \left (c \left (f x +e \right )\right )^{2} b \,e^{3} i^{3}+18 \ln \left (c \left (f x +e \right )\right )^{2} b \,f^{3} h^{3}-36 \ln \left (c \left (f x +e \right )\right ) a \,e^{3} i^{3}+36 \ln \left (c \left (f x +e \right )\right ) a \,f^{3} h^{3}+66 \ln \left (c \left (f x +e \right )\right ) b \,e^{3} i^{3}-108 a e \,f^{2} h \,i^{2} x +162 a \,e^{2} f h \,i^{2}-216 a e \,f^{2} h^{2} i -297 b \,e^{2} f h \,i^{2}+216 b e \,f^{2} h^{2} i +12 x^{3} \ln \left (c \left (f x +e \right )\right ) b \,f^{3} i^{3}-18 a e \,f^{2} i^{3} x^{2}+54 a \,f^{3} h \,i^{2} x^{2}+15 b e \,f^{2} i^{3} x^{2}-27 b \,f^{3} h \,i^{2} x^{2}+36 a \,e^{2} f \,i^{3} x +108 a \,f^{3} h^{2} i x -66 b \,e^{2} f \,i^{3} x -108 b \,f^{3} h^{2} i x +162 b e \,f^{2} h \,i^{2} x}{36 d \,f^{4}}\) | \(543\) |
| derivativedivides | \(\frac {-\frac {c a \,e^{3} i^{3} \ln \left (c f x +c e \right )}{f^{3} d}+\frac {3 c a \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )}{f^{2} d}-\frac {3 c a e \,h^{2} i \ln \left (c f x +c e \right )}{f d}+\frac {c a \,h^{3} \ln \left (c f x +c e \right )}{d}+\frac {3 a \,e^{2} i^{3} \left (c f x +c e \right )}{f^{3} d}-\frac {6 a e h \,i^{2} \left (c f x +c e \right )}{f^{2} d}+\frac {3 a \,h^{2} i \left (c f x +c e \right )}{f d}-\frac {3 a e \,i^{3} \left (c f x +c e \right )^{2}}{2 c \,f^{3} d}+\frac {3 a h \,i^{2} \left (c f x +c e \right )^{2}}{2 c \,f^{2} d}+\frac {a \,i^{3} \left (c f x +c e \right )^{3}}{3 c^{2} f^{3} d}-\frac {c b \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3} d}+\frac {3 c b \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2} d}-\frac {3 c b e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f d}+\frac {c b \,h^{3} \ln \left (c f x +c e \right )^{2}}{2 d}+\frac {3 b \,e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3} d}-\frac {6 b e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2} d}+\frac {3 b \,h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}-\frac {3 b e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3} d}+\frac {3 b h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}+\frac {b \,i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3} d}}{c f}\) | \(621\) |
| default | \(\frac {-\frac {c a \,e^{3} i^{3} \ln \left (c f x +c e \right )}{f^{3} d}+\frac {3 c a \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )}{f^{2} d}-\frac {3 c a e \,h^{2} i \ln \left (c f x +c e \right )}{f d}+\frac {c a \,h^{3} \ln \left (c f x +c e \right )}{d}+\frac {3 a \,e^{2} i^{3} \left (c f x +c e \right )}{f^{3} d}-\frac {6 a e h \,i^{2} \left (c f x +c e \right )}{f^{2} d}+\frac {3 a \,h^{2} i \left (c f x +c e \right )}{f d}-\frac {3 a e \,i^{3} \left (c f x +c e \right )^{2}}{2 c \,f^{3} d}+\frac {3 a h \,i^{2} \left (c f x +c e \right )^{2}}{2 c \,f^{2} d}+\frac {a \,i^{3} \left (c f x +c e \right )^{3}}{3 c^{2} f^{3} d}-\frac {c b \,e^{3} i^{3} \ln \left (c f x +c e \right )^{2}}{2 f^{3} d}+\frac {3 c b \,e^{2} h \,i^{2} \ln \left (c f x +c e \right )^{2}}{2 f^{2} d}-\frac {3 c b e \,h^{2} i \ln \left (c f x +c e \right )^{2}}{2 f d}+\frac {c b \,h^{3} \ln \left (c f x +c e \right )^{2}}{2 d}+\frac {3 b \,e^{2} i^{3} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{3} d}-\frac {6 b e h \,i^{2} \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f^{2} d}+\frac {3 b \,h^{2} i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}-\frac {3 b e \,i^{3} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{3} d}+\frac {3 b h \,i^{2} \left (\frac {\left (c f x +c e \right )^{2} \ln \left (c f x +c e \right )}{2}-\frac {\left (c f x +c e \right )^{2}}{4}\right )}{c \,f^{2} d}+\frac {b \,i^{3} \left (\frac {\left (c f x +c e \right )^{3} \ln \left (c f x +c e \right )}{3}-\frac {\left (c f x +c e \right )^{3}}{9}\right )}{c^{2} f^{3} d}}{c f}\) | \(621\) |
b*i*(e^2*i^2-3*e*f*h*i+3*f^2*h^2)/d/f^3*x*ln(c*(f*x+e))-1/6*(6*a*e^3*i^3-1 8*a*e^2*f*h*i^2+18*a*e*f^2*h^2*i-6*a*f^3*h^3-11*b*e^3*i^3+27*b*e^2*f*h*i^2 -18*b*e*f^2*h^2*i)/d/f^4*ln(c*(f*x+e))-1/2*b*(e^3*i^3-3*e^2*f*h*i^2+3*e*f^ 2*h^2*i-f^3*h^3)/d/f^4*ln(c*(f*x+e))^2+1/6*i*(6*a*e^2*i^2-18*a*e*f*h*i+18* a*f^2*h^2-11*b*e^2*i^2+27*b*e*f*h*i-18*b*f^2*h^2)/d/f^3*x-1/12*i^2*(6*a*e* i-18*a*f*h-5*b*e*i+9*b*f*h)/d/f^2*x^2+1/9*i^3*(3*a-b)/d/f*x^3+1/3*b*i^3/d/ f*x^3*ln(c*(f*x+e))-1/2*b*i^2*(e*i-3*f*h)/d/f^2*x^2*ln(c*(f*x+e))
Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.26 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {4 \, {\left (3 \, a - b\right )} f^{3} i^{3} x^{3} + 3 \, {\left (9 \, {\left (2 \, a - b\right )} f^{3} h i^{2} - {\left (6 \, a - 5 \, b\right )} e f^{2} i^{3}\right )} x^{2} + 18 \, {\left (b f^{3} h^{3} - 3 \, b e f^{2} h^{2} i + 3 \, b e^{2} f h i^{2} - b e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{2} + 6 \, {\left (18 \, {\left (a - b\right )} f^{3} h^{2} i - 9 \, {\left (2 \, a - 3 \, b\right )} e f^{2} h i^{2} + {\left (6 \, a - 11 \, b\right )} e^{2} f i^{3}\right )} x + 6 \, {\left (2 \, b f^{3} i^{3} x^{3} + 6 \, a f^{3} h^{3} - 18 \, {\left (a - b\right )} e f^{2} h^{2} i + 9 \, {\left (2 \, a - 3 \, b\right )} e^{2} f h i^{2} - {\left (6 \, a - 11 \, b\right )} e^{3} i^{3} + 3 \, {\left (3 \, b f^{3} h i^{2} - b e f^{2} i^{3}\right )} x^{2} + 6 \, {\left (3 \, b f^{3} h^{2} i - 3 \, b e f^{2} h i^{2} + b e^{2} f i^{3}\right )} x\right )} \log \left (c f x + c e\right )}{36 \, d f^{4}} \]
1/36*(4*(3*a - b)*f^3*i^3*x^3 + 3*(9*(2*a - b)*f^3*h*i^2 - (6*a - 5*b)*e*f ^2*i^3)*x^2 + 18*(b*f^3*h^3 - 3*b*e*f^2*h^2*i + 3*b*e^2*f*h*i^2 - b*e^3*i^ 3)*log(c*f*x + c*e)^2 + 6*(18*(a - b)*f^3*h^2*i - 9*(2*a - 3*b)*e*f^2*h*i^ 2 + (6*a - 11*b)*e^2*f*i^3)*x + 6*(2*b*f^3*i^3*x^3 + 6*a*f^3*h^3 - 18*(a - b)*e*f^2*h^2*i + 9*(2*a - 3*b)*e^2*f*h*i^2 - (6*a - 11*b)*e^3*i^3 + 3*(3* b*f^3*h*i^2 - b*e*f^2*i^3)*x^2 + 6*(3*b*f^3*h^2*i - 3*b*e*f^2*h*i^2 + b*e^ 2*f*i^3)*x)*log(c*f*x + c*e))/(d*f^4)
Time = 0.59 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.75 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=x^{3} \left (\frac {a i^{3}}{3 d f} - \frac {b i^{3}}{9 d f}\right ) + x^{2} \left (- \frac {a e i^{3}}{2 d f^{2}} + \frac {3 a h i^{2}}{2 d f} + \frac {5 b e i^{3}}{12 d f^{2}} - \frac {3 b h i^{2}}{4 d f}\right ) + x \left (\frac {a e^{2} i^{3}}{d f^{3}} - \frac {3 a e h i^{2}}{d f^{2}} + \frac {3 a h^{2} i}{d f} - \frac {11 b e^{2} i^{3}}{6 d f^{3}} + \frac {9 b e h i^{2}}{2 d f^{2}} - \frac {3 b h^{2} i}{d f}\right ) + \frac {\left (6 b e^{2} i^{3} x - 18 b e f h i^{2} x - 3 b e f i^{3} x^{2} + 18 b f^{2} h^{2} i x + 9 b f^{2} h i^{2} x^{2} + 2 b f^{2} i^{3} x^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}}{6 d f^{3}} + \frac {\left (- b e^{3} i^{3} + 3 b e^{2} f h i^{2} - 3 b e f^{2} h^{2} i + b f^{3} h^{3}\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{4}} - \frac {\left (6 a e^{3} i^{3} - 18 a e^{2} f h i^{2} + 18 a e f^{2} h^{2} i - 6 a f^{3} h^{3} - 11 b e^{3} i^{3} + 27 b e^{2} f h i^{2} - 18 b e f^{2} h^{2} i\right ) \log {\left (e + f x \right )}}{6 d f^{4}} \]
x**3*(a*i**3/(3*d*f) - b*i**3/(9*d*f)) + x**2*(-a*e*i**3/(2*d*f**2) + 3*a* h*i**2/(2*d*f) + 5*b*e*i**3/(12*d*f**2) - 3*b*h*i**2/(4*d*f)) + x*(a*e**2* i**3/(d*f**3) - 3*a*e*h*i**2/(d*f**2) + 3*a*h**2*i/(d*f) - 11*b*e**2*i**3/ (6*d*f**3) + 9*b*e*h*i**2/(2*d*f**2) - 3*b*h**2*i/(d*f)) + (6*b*e**2*i**3* x - 18*b*e*f*h*i**2*x - 3*b*e*f*i**3*x**2 + 18*b*f**2*h**2*i*x + 9*b*f**2* h*i**2*x**2 + 2*b*f**2*i**3*x**3)*log(c*(e + f*x))/(6*d*f**3) + (-b*e**3*i **3 + 3*b*e**2*f*h*i**2 - 3*b*e*f**2*h**2*i + b*f**3*h**3)*log(c*(e + f*x) )**2/(2*d*f**4) - (6*a*e**3*i**3 - 18*a*e**2*f*h*i**2 + 18*a*e*f**2*h**2*i - 6*a*f**3*h**3 - 11*b*e**3*i**3 + 27*b*e**2*f*h*i**2 - 18*b*e*f**2*h**2* i)*log(e + f*x)/(6*d*f**4)
Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (234) = 468\).
Time = 0.23 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.21 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=3 \, b h^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) - \frac {1}{6} \, b i^{3} {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac {3}{2} \, b h i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} \log \left (c f x + c e\right ) - \frac {1}{2} \, b h^{3} {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + 3 \, a h^{2} i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} - \frac {1}{6} \, a i^{3} {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{d f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{d f^{3}}\right )} + \frac {3}{2} \, a h i^{2} {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{d f^{3}} + \frac {f x^{2} - 2 \, e x}{d f^{2}}\right )} + \frac {b h^{3} \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a h^{3} \log \left (d f x + d e\right )}{d f} + \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} b h^{2} i}{2 \, d f^{2}} - \frac {3 \, {\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} b h i^{2}}{4 \, d f^{3}} - \frac {{\left (4 \, f^{3} x^{3} - 15 \, e f^{2} x^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} + 66 \, e^{2} f x - 66 \, e^{3} \log \left (f x + e\right )\right )} b i^{3}}{36 \, d f^{4}} \]
3*b*h^2*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) - 1/6*b*i^3* (6*e^3*log(f*x + e)/(d*f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3))*l og(c*f*x + c*e) + 3/2*b*h*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2*e*x )/(d*f^2))*log(c*f*x + c*e) - 1/2*b*h^3*(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)) + 3*a*h^2*i*(x/ (d*f) - e*log(f*x + e)/(d*f^2)) - 1/6*a*i^3*(6*e^3*log(f*x + e)/(d*f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3)) + 3/2*a*h*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2*e*x)/(d*f^2)) + b*h^3*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) + a*h^3*log(d*f*x + d*e)/(d*f) + 3/2*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*b*h^2*i/(d*f^2) - 3/4*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*b*h*i^2/(d*f^3) - 1/36*(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))*b*i^3/ (d*f^4)
Time = 0.31 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.51 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {{\left (3 \, a i^{3} - b i^{3}\right )} x^{3}}{9 \, d f} + \frac {1}{6} \, {\left (\frac {2 \, b i^{3} x^{3}}{d f} + \frac {3 \, {\left (3 \, b f h i^{2} - b e i^{3}\right )} x^{2}}{d f^{2}} + \frac {6 \, {\left (3 \, b f^{2} h^{2} i - 3 \, b e f h i^{2} + b e^{2} i^{3}\right )} x}{d f^{3}}\right )} \log \left (c f x + c e\right ) + \frac {{\left (18 \, a f h i^{2} - 9 \, b f h i^{2} - 6 \, a e i^{3} + 5 \, b e i^{3}\right )} x^{2}}{12 \, d f^{2}} + \frac {{\left (18 \, a f^{2} h^{2} i - 18 \, b f^{2} h^{2} i - 18 \, a e f h i^{2} + 27 \, b e f h i^{2} + 6 \, a e^{2} i^{3} - 11 \, b e^{2} i^{3}\right )} x}{6 \, d f^{3}} + \frac {{\left (b f^{3} h^{3} - 3 \, b e f^{2} h^{2} i + 3 \, b e^{2} f h i^{2} - b e^{3} i^{3}\right )} \log \left (c f x + c e\right )^{2}}{2 \, d f^{4}} + \frac {{\left (6 \, a f^{3} h^{3} - 18 \, a e f^{2} h^{2} i + 18 \, b e f^{2} h^{2} i + 18 \, a e^{2} f h i^{2} - 27 \, b e^{2} f h i^{2} - 6 \, a e^{3} i^{3} + 11 \, b e^{3} i^{3}\right )} \log \left (f x + e\right )}{6 \, d f^{4}} \]
1/9*(3*a*i^3 - b*i^3)*x^3/(d*f) + 1/6*(2*b*i^3*x^3/(d*f) + 3*(3*b*f*h*i^2 - b*e*i^3)*x^2/(d*f^2) + 6*(3*b*f^2*h^2*i - 3*b*e*f*h*i^2 + b*e^2*i^3)*x/( d*f^3))*log(c*f*x + c*e) + 1/12*(18*a*f*h*i^2 - 9*b*f*h*i^2 - 6*a*e*i^3 + 5*b*e*i^3)*x^2/(d*f^2) + 1/6*(18*a*f^2*h^2*i - 18*b*f^2*h^2*i - 18*a*e*f*h *i^2 + 27*b*e*f*h*i^2 + 6*a*e^2*i^3 - 11*b*e^2*i^3)*x/(d*f^3) + 1/2*(b*f^3 *h^3 - 3*b*e*f^2*h^2*i + 3*b*e^2*f*h*i^2 - b*e^3*i^3)*log(c*f*x + c*e)^2/( d*f^4) + 1/6*(6*a*f^3*h^3 - 18*a*e*f^2*h^2*i + 18*b*e*f^2*h^2*i + 18*a*e^2 *f*h*i^2 - 27*b*e^2*f*h*i^2 - 6*a*e^3*i^3 + 11*b*e^3*i^3)*log(f*x + e)/(d* f^4)
Time = 1.54 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.61 \[ \int \frac {(h+i x)^3 (a+b \log (c (e+f x)))}{d e+d f x} \, dx=x^2\,\left (\frac {i^2\,\left (6\,a\,f\,h+b\,e\,i-3\,b\,f\,h\right )}{4\,d\,f^2}-\frac {e\,i^3\,\left (3\,a-b\right )}{6\,d\,f^2}\right )-x\,\left (\frac {e\,\left (\frac {i^2\,\left (6\,a\,f\,h+b\,e\,i-3\,b\,f\,h\right )}{2\,d\,f^2}-\frac {e\,i^3\,\left (3\,a-b\right )}{3\,d\,f^2}\right )}{f}-\frac {i\,\left (3\,a\,f^2\,h^2-b\,e^2\,i^2-3\,b\,f^2\,h^2+3\,b\,e\,f\,h\,i\right )}{d\,f^3}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {b\,i^3\,x^3}{3\,d\,f^2}+\frac {b\,i\,x\,\left (e^2\,i^2-3\,e\,f\,h\,i+3\,f^2\,h^2\right )}{d\,f^4}-\frac {b\,i^2\,x^2\,\left (e\,i-3\,f\,h\right )}{2\,d\,f^3}\right )+\frac {\ln \left (e+f\,x\right )\,\left (6\,a\,f^3\,h^3-6\,a\,e^3\,i^3+11\,b\,e^3\,i^3-18\,a\,e\,f^2\,h^2\,i+18\,a\,e^2\,f\,h\,i^2+18\,b\,e\,f^2\,h^2\,i-27\,b\,e^2\,f\,h\,i^2\right )}{6\,d\,f^4}+\frac {i^3\,x^3\,\left (3\,a-b\right )}{9\,d\,f}-\frac {b\,{\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (e^3\,i^3-3\,e^2\,f\,h\,i^2+3\,e\,f^2\,h^2\,i-f^3\,h^3\right )}{2\,d\,f^4} \]
x^2*((i^2*(6*a*f*h + b*e*i - 3*b*f*h))/(4*d*f^2) - (e*i^3*(3*a - b))/(6*d* f^2)) - x*((e*((i^2*(6*a*f*h + b*e*i - 3*b*f*h))/(2*d*f^2) - (e*i^3*(3*a - b))/(3*d*f^2)))/f - (i*(3*a*f^2*h^2 - b*e^2*i^2 - 3*b*f^2*h^2 + 3*b*e*f*h *i))/(d*f^3)) + f*log(c*(e + f*x))*((b*i^3*x^3)/(3*d*f^2) + (b*i*x*(e^2*i^ 2 + 3*f^2*h^2 - 3*e*f*h*i))/(d*f^4) - (b*i^2*x^2*(e*i - 3*f*h))/(2*d*f^3)) + (log(e + f*x)*(6*a*f^3*h^3 - 6*a*e^3*i^3 + 11*b*e^3*i^3 - 18*a*e*f^2*h^ 2*i + 18*a*e^2*f*h*i^2 + 18*b*e*f^2*h^2*i - 27*b*e^2*f*h*i^2))/(6*d*f^4) + (i^3*x^3*(3*a - b))/(9*d*f) - (b*log(c*(e + f*x))^2*(e^3*i^3 - f^3*h^3 + 3*e*f^2*h^2*i - 3*e^2*f*h*i^2))/(2*d*f^4)