Integrand size = 18, antiderivative size = 63 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\frac {a x}{d}-\frac {b x}{d}+\frac {b x \log (c x)}{d}-\frac {e (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2} \]
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\frac {a x}{d}-\frac {b x}{d}+\frac {b x \log (c x)}{d}-\frac {e (a+b \log (c x)) \log \left (\frac {e+d x}{e}\right )}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2} \]
(a*x)/d - (b*x)/d + (b*x*Log[c*x])/d - (e*(a + b*Log[c*x])*Log[(e + d*x)/e ])/d^2 - (b*e*PolyLog[2, -((d*x)/e)])/d^2
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2767, 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.
\(\displaystyle \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx\) |
\(\Big \downarrow \) 2767 |
\(\displaystyle \int \left (\frac {a+b \log (c x)}{d}-\frac {e (a+b \log (c x))}{d (d x+e)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^2}+\frac {a x}{d}+\frac {b x \log (c x)}{d}-\frac {b e \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2}-\frac {b x}{d}\) |
(a*x)/d - (b*x)/d + (b*x*Log[c*x])/d - (e*(a + b*Log[c*x])*Log[1 + (d*x)/e ])/d^2 - (b*e*PolyLog[2, -((d*x)/e)])/d^2
3.4.40.3.1 Defintions of rubi rules used
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]))
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40
method | result | size |
risch | \(\frac {a x}{d}-\frac {a e \ln \left (d x +e \right )}{d^{2}}+\frac {b x \ln \left (x c \right )}{d}-\frac {b x}{d}-\frac {b e \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{2}}-\frac {b e \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{2}}\) | \(88\) |
parts | \(\frac {a x}{d}-\frac {a e \ln \left (d x +e \right )}{d^{2}}+\frac {b x \ln \left (x c \right )}{d}-\frac {b x}{d}-\frac {b e \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{2}}-\frac {b e \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{2}}\) | \(88\) |
derivativedivides | \(\frac {\frac {a x c}{d}-\frac {a e c \ln \left (c d x +c e \right )}{d^{2}}+b \left (\frac {x c \ln \left (x c \right )-x c}{d}-\frac {e c \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d}\right )}{c}\) | \(101\) |
default | \(\frac {\frac {a x c}{d}-\frac {a e c \ln \left (c d x +c e \right )}{d^{2}}+b \left (\frac {x c \ln \left (x c \right )-x c}{d}-\frac {e c \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d}\right )}{c}\) | \(101\) |
a*x/d-a*e/d^2*ln(d*x+e)+b*x*ln(x*c)/d-b*x/d-b*e/d^2*dilog((c*d*x+c*e)/e/c) -b*e/d^2*ln(x*c)*ln((c*d*x+c*e)/e/c)
\[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\int { \frac {b \log \left (c x\right ) + a}{d + \frac {e}{x}} \,d x } \]
Time = 43.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.48 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=- \frac {a e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d} + \frac {a x}{d} + \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d} - \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d} + \frac {b x \log {\left (c x \right )}}{d} - \frac {b x}{d} \]
-a*e*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, True))/d + a*x/d + b*e*Pi ecewise((x/e, Eq(d, 0)), (Piecewise((-polylog(2, d*x*exp_polar(I*pi)/e), ( Abs(x) < 1) & (1/Abs(x) < 1)), (log(e)*log(x) - polylog(2, d*x*exp_polar(I *pi)/e), Abs(x) < 1), (-log(e)*log(1/x) - polylog(2, d*x*exp_polar(I*pi)/e ), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijer g(((1, 1), ()), ((), (0, 0)), x)*log(e) - polylog(2, d*x*exp_polar(I*pi)/e ), True))/d, True))/d - b*e*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, Tr ue))*log(c*x)/d + b*x*log(c*x)/d - b*x/d
Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=-\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e}{d^{2}} + \frac {b x \log \left (x\right ) + {\left (b {\left (\log \left (c\right ) - 1\right )} + a\right )} x}{d} - \frac {{\left (b e \log \left (c\right ) + a e\right )} \log \left (d x + e\right )}{d^{2}} \]
-(log(d*x/e + 1)*log(x) + dilog(-d*x/e))*b*e/d^2 + (b*x*log(x) + (b*(log(c ) - 1) + a)*x)/d - (b*e*log(c) + a*e)*log(d*x + e)/d^2
\[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\int { \frac {b \log \left (c x\right ) + a}{d + \frac {e}{x}} \,d x } \]
Timed out. \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\int \frac {a+b\,\ln \left (c\,x\right )}{d+\frac {e}{x}} \,d x \]