Integrand size = 30, antiderivative size = 250 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=-\frac {b f}{2 d (f h-e i)^2 (h+i x)}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {b f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \]
-1/2*b*f/d/(-e*i+f*h)^2/(i*x+h)-1/2*b*f^2*ln(f*x+e)/d/(-e*i+f*h)^3+1/2*(a+ b*ln(c*(f*x+e)))/d/(-e*i+f*h)/(i*x+h)^2-f*i*(f*x+e)*(a+b*ln(c*(f*x+e)))/d/ (-e*i+f*h)^3/(i*x+h)+3/2*b*f^2*ln(i*x+h)/d/(-e*i+f*h)^3-f^2*(a+b*ln(c*(f*x +e)))*ln(1+(-e*i+f*h)/i/(f*x+e))/d/(-e*i+f*h)^3+b*f^2*polylog(2,(e*i-f*h)/ i/(f*x+e))/d/(-e*i+f*h)^3
Time = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\frac {\frac {(f h-e i)^2 (a+b \log (c (e+f x)))}{(h+i x)^2}+\frac {2 f (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac {f^2 (a+b \log (c (e+f x)))^2}{b}+2 b f^2 (-\log (e+f x)+\log (h+i x))-\frac {b f (f h-e i+f (h+i x) \log (e+f x)-f (h+i x) \log (h+i x))}{h+i x}-2 f^2 (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )-2 b f^2 \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )}{2 d (f h-e i)^3} \]
(((f*h - e*i)^2*(a + b*Log[c*(e + f*x)]))/(h + i*x)^2 + (2*f*(f*h - e*i)*( a + b*Log[c*(e + f*x)]))/(h + i*x) + (f^2*(a + b*Log[c*(e + f*x)])^2)/b + 2*b*f^2*(-Log[e + f*x] + Log[h + i*x]) - (b*f*(f*h - e*i + f*(h + i*x)*Log [e + f*x] - f*(h + i*x)*Log[h + i*x]))/(h + i*x) - 2*f^2*(a + b*Log[c*(e + f*x)])*Log[(f*(h + i*x))/(f*h - e*i)] - 2*b*f^2*PolyLog[2, (i*(e + f*x))/ (-(f*h) + e*i)])/(2*d*(f*h - e*i)^3)
Time = 1.08 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.34, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2858, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int \frac {a+b \log (c (e+f x))}{(h+i x)^3 (d e+d f x)} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {f^3 (a+b \log (c (e+f x)))}{d (e+f x) \left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^3}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^2 \int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))^3}d(e+f x)}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {f^2 \left (\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))^2}d(e+f x)}{f h-e i}-\frac {i \int \frac {a+b \log (c (e+f x))}{(f h-e i+i (e+f x))^3}d(e+f x)}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {f^2 \left (\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))^2}d(e+f x)}{f h-e i}-\frac {i \left (\frac {b \int \frac {1}{(e+f x) (f h-e i+i (e+f x))^2}d(e+f x)}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {f^2 \left (\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))^2}d(e+f x)}{f h-e i}-\frac {i \left (\frac {b \int \left (-\frac {i}{(f h-e i)^2 (f h-e i+i (e+f x))}-\frac {i}{(f h-e i) (f h-e i+i (e+f x))^2}+\frac {1}{(f h-e i)^2 (e+f x)}\right )d(e+f x)}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \left (\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))^2}d(e+f x)}{f h-e i}-\frac {i \left (\frac {b \left (\frac {1}{(f h-e i) (i (e+f x)-e i+f h)}+\frac {\log (e+f x)}{(f h-e i)^2}-\frac {\log (i (e+f x)-e i+f h)}{(f h-e i)^2}\right )}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))}d(e+f x)}{f h-e i}-\frac {i \int \frac {a+b \log (c (e+f x))}{(f h-e i+i (e+f x))^2}d(e+f x)}{f h-e i}}{f h-e i}-\frac {i \left (\frac {b \left (\frac {1}{(f h-e i) (i (e+f x)-e i+f h)}+\frac {\log (e+f x)}{(f h-e i)^2}-\frac {\log (i (e+f x)-e i+f h)}{(f h-e i)^2}\right )}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))}d(e+f x)}{f h-e i}-\frac {i \left (\frac {(e+f x) (a+b \log (c (e+f x)))}{(f h-e i) (i (e+f x)-e i+f h)}-\frac {b \int \frac {1}{f h-e i+i (e+f x)}d(e+f x)}{f h-e i}\right )}{f h-e i}}{f h-e i}-\frac {i \left (\frac {b \left (\frac {1}{(f h-e i) (i (e+f x)-e i+f h)}+\frac {\log (e+f x)}{(f h-e i)^2}-\frac {\log (i (e+f x)-e i+f h)}{(f h-e i)^2}\right )}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\int \frac {a+b \log (c (e+f x))}{(e+f x) (f h-e i+i (e+f x))}d(e+f x)}{f h-e i}-\frac {i \left (\frac {(e+f x) (a+b \log (c (e+f x)))}{(f h-e i) (i (e+f x)-e i+f h)}-\frac {b \log (i (e+f x)-e i+f h)}{i (f h-e i)}\right )}{f h-e i}}{f h-e i}-\frac {i \left (\frac {b \left (\frac {1}{(f h-e i) (i (e+f x)-e i+f h)}+\frac {\log (e+f x)}{(f h-e i)^2}-\frac {\log (i (e+f x)-e i+f h)}{(f h-e i)^2}\right )}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\frac {b \int \frac {\log \left (\frac {f h-e i}{i (e+f x)}+1\right )}{e+f x}d(e+f x)}{f h-e i}-\frac {\log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{f h-e i}}{f h-e i}-\frac {i \left (\frac {(e+f x) (a+b \log (c (e+f x)))}{(f h-e i) (i (e+f x)-e i+f h)}-\frac {b \log (i (e+f x)-e i+f h)}{i (f h-e i)}\right )}{f h-e i}}{f h-e i}-\frac {i \left (\frac {b \left (\frac {1}{(f h-e i) (i (e+f x)-e i+f h)}+\frac {\log (e+f x)}{(f h-e i)^2}-\frac {\log (i (e+f x)-e i+f h)}{(f h-e i)^2}\right )}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\frac {b \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{f h-e i}-\frac {\log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{f h-e i}}{f h-e i}-\frac {i \left (\frac {(e+f x) (a+b \log (c (e+f x)))}{(f h-e i) (i (e+f x)-e i+f h)}-\frac {b \log (i (e+f x)-e i+f h)}{i (f h-e i)}\right )}{f h-e i}}{f h-e i}-\frac {i \left (\frac {b \left (\frac {1}{(f h-e i) (i (e+f x)-e i+f h)}+\frac {\log (e+f x)}{(f h-e i)^2}-\frac {\log (i (e+f x)-e i+f h)}{(f h-e i)^2}\right )}{2 i}-\frac {a+b \log (c (e+f x))}{2 i (i (e+f x)-e i+f h)^2}\right )}{f h-e i}\right )}{d}\) |
(f^2*(-((i*(-1/2*(a + b*Log[c*(e + f*x)])/(i*(f*h - e*i + i*(e + f*x))^2) + (b*(1/((f*h - e*i)*(f*h - e*i + i*(e + f*x))) + Log[e + f*x]/(f*h - e*i) ^2 - Log[f*h - e*i + i*(e + f*x)]/(f*h - e*i)^2))/(2*i)))/(f*h - e*i)) + ( -((i*(((e + f*x)*(a + b*Log[c*(e + f*x)]))/((f*h - e*i)*(f*h - e*i + i*(e + f*x))) - (b*Log[f*h - e*i + i*(e + f*x)])/(i*(f*h - e*i))))/(f*h - e*i)) + (-(((a + b*Log[c*(e + f*x)])*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(f*h - e*i)) + (b*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(f*h - e*i))/(f*h - e*i))/(f*h - e*i)))/d
3.2.82.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
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] - Simp[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] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(241)=482\).
Time = 1.20 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.04
| method | result | size |
| parts | \(\frac {a \left (-\frac {1}{2 \left (e i -f h \right ) \left (i x +h \right )^{2}}+\frac {f^{2} \ln \left (i x +h \right )}{\left (e i -f h \right )^{3}}+\frac {f}{\left (e i -f h \right )^{2} \left (i x +h \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (e i -f h \right )^{3}}\right )}{d}+\frac {b \left (-\frac {c \,f^{3} \ln \left (c f x +c e \right )^{2}}{2 \left (e i -f h \right )^{3}}+\frac {c^{3} f^{3} i \left (-\frac {\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{i}+\frac {c \left (e i -f h \right )}{i \left (-c e i +h c f +i \left (c f x +c e \right )\right )}}{2 c^{2} \left (e i -f h \right )^{2}}+\frac {\ln \left (c f x +c e \right ) \left (-2 c e i +2 h c f +i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 \left (-c e i +h c f +i \left (c f x +c e \right )\right )^{2} c^{2} \left (e i -f h \right )^{2}}\right )}{e i -f h}+\frac {c \,f^{3} i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{\left (e i -f h \right )^{3}}-\frac {c^{2} f^{3} i \left (\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}-\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (-c e i +h c f +i \left (c f x +c e \right )\right )}\right )}{\left (e i -f h \right )^{2}}\right )}{d c f}\) | \(510\) |
| derivativedivides | \(\frac {-\frac {c^{4} f^{3} a \left (\frac {\ln \left (c f x +c e \right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{2 c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{d}-\frac {c^{4} f^{3} b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{3} \left (e i -f h \right )^{3}}+\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{3} \left (e i -f h \right )^{3}}-\frac {i \left (-\frac {-\frac {c \left (e i -f h \right )}{i \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{i}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {\ln \left (c f x +c e \right ) \left (2 c e i -2 h c f -i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\) | \(580\) |
| default | \(\frac {-\frac {c^{4} f^{3} a \left (\frac {\ln \left (c f x +c e \right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{2 c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{d}-\frac {c^{4} f^{3} b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{3} \left (e i -f h \right )^{3}}+\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{3} \left (e i -f h \right )^{3}}-\frac {i \left (-\frac {-\frac {c \left (e i -f h \right )}{i \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{i}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {\ln \left (c f x +c e \right ) \left (2 c e i -2 h c f -i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\) | \(580\) |
| risch | \(-\frac {a}{2 d \left (e i -f h \right ) \left (i x +h \right )^{2}}+\frac {a \,f^{2} \ln \left (i x +h \right )}{d \left (e i -f h \right )^{3}}+\frac {a f}{d \left (e i -f h \right )^{2} \left (i x +h \right )}-\frac {a \,f^{2} \ln \left (f x +e \right )}{d \left (e i -f h \right )^{3}}-\frac {b \,f^{2} \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{3}}-\frac {3 b \,f^{2} \ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{2 d \left (e i -f h \right )^{3}}-\frac {b c \,f^{2} i e}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b c \,f^{3} h}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b \,c^{2} f^{4} i^{2} \ln \left (c f x +c e \right ) x^{2}}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,c^{2} f^{4} i \ln \left (c f x +c e \right ) h x}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}-\frac {b \,c^{2} f^{2} i^{2} \ln \left (c f x +c e \right ) e^{2}}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,c^{2} f^{3} i \ln \left (c f x +c e \right ) e h}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,f^{2} \operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{3}}+\frac {b \,f^{2} \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{3}}+\frac {b c \,f^{3} i \ln \left (c f x +c e \right ) x}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b c \,f^{2} i \ln \left (c f x +c e \right ) e}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}\) | \(619\) |
a/d*(-1/2/(e*i-f*h)/(i*x+h)^2+f^2/(e*i-f*h)^3*ln(i*x+h)+f/(e*i-f*h)^2/(i*x +h)-f^2/(e*i-f*h)^3*ln(f*x+e))+b/d/c/f*(-1/2*c*f^3/(e*i-f*h)^3*ln(c*f*x+c* e)^2+c^3*f^3/(e*i-f*h)*i*(-1/2/c^2/(e*i-f*h)^2*(ln(-c*e*i+h*c*f+i*(c*f*x+c *e))/i+c*(e*i-f*h)/i/(-c*e*i+h*c*f+i*(c*f*x+c*e)))+1/2*ln(c*f*x+c*e)*(-2*c *e*i+2*h*c*f+i*(c*f*x+c*e))*(c*f*x+c*e)/(-c*e*i+h*c*f+i*(c*f*x+c*e))^2/c^2 /(e*i-f*h)^2)+c*f^3/(e*i-f*h)^3*i*(dilog((-c*e*i+h*c*f+i*(c*f*x+c*e))/(-c* e*i+c*f*h))/i+ln(c*f*x+c*e)*ln((-c*e*i+h*c*f+i*(c*f*x+c*e))/(-c*e*i+c*f*h) )/i)-c^2*f^3/(e*i-f*h)^2*i*(1/c/(e*i-f*h)*ln(-c*e*i+h*c*f+i*(c*f*x+c*e))/i -ln(c*f*x+c*e)*(c*f*x+c*e)/c/(e*i-f*h)/(-c*e*i+h*c*f+i*(c*f*x+c*e))))
\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]
integral((b*log(c*f*x + c*e) + a)/(d*f*i^3*x^4 + d*e*h^3 + (3*d*f*h*i^2 + d*e*i^3)*x^3 + 3*(d*f*h^2*i + d*e*h*i^2)*x^2 + (d*f*h^3 + 3*d*e*h^2*i)*x), x)
\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\frac {\int \frac {a}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx + \int \frac {b \log {\left (c e + c f x \right )}}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx}{d} \]
(Integral(a/(e*h**3 + 3*e*h**2*i*x + 3*e*h*i**2*x**2 + e*i**3*x**3 + f*h** 3*x + 3*f*h**2*i*x**2 + 3*f*h*i**2*x**3 + f*i**3*x**4), x) + Integral(b*lo g(c*e + c*f*x)/(e*h**3 + 3*e*h**2*i*x + 3*e*h*i**2*x**2 + e*i**3*x**3 + f* h**3*x + 3*f*h**2*i*x**2 + 3*f*h*i**2*x**3 + f*i**3*x**4), x))/d
\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]
1/2*(2*f^2*log(f*x + e)/(d*f^3*h^3 - 3*d*e*f^2*h^2*i + 3*d*e^2*f*h*i^2 - d *e^3*i^3) - 2*f^2*log(i*x + h)/(d*f^3*h^3 - 3*d*e*f^2*h^2*i + 3*d*e^2*f*h* i^2 - d*e^3*i^3) + (2*f*i*x + 3*f*h - e*i)/(d*f^2*h^4 - 2*d*e*f*h^3*i + d* e^2*h^2*i^2 + (d*f^2*h^2*i^2 - 2*d*e*f*h*i^3 + d*e^2*i^4)*x^2 + 2*(d*f^2*h ^3*i - 2*d*e*f*h^2*i^2 + d*e^2*h*i^3)*x))*a + b*integrate((log(f*x + e) + log(c))/(d*f*i^3*x^4 + d*e*h^3 + (3*f*h*i^2 + e*i^3)*d*x^3 + 3*(f*h^2*i + e*h*i^2)*d*x^2 + (f*h^3 + 3*e*h^2*i)*d*x), x)
\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int \frac {a+b\,\ln \left (c\,\left (e+f\,x\right )\right )}{{\left (h+i\,x\right )}^3\,\left (d\,e+d\,f\,x\right )} \,d x \]