Integrand size = 19, antiderivative size = 98 \[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=-\frac {a e x}{d^2}+\frac {b e x}{d^2}-\frac {b x^2}{4 d}-\frac {b e x \log (c x)}{d^2}+\frac {x^2 (a+b \log (c x))}{2 d}+\frac {e^2 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^3}+\frac {b e^2 \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^3} \]
-a*e*x/d^2+b*e*x/d^2-1/4*b*x^2/d-b*e*x*ln(c*x)/d^2+1/2*x^2*(a+b*ln(c*x))/d +e^2*(a+b*ln(c*x))*ln(1+d*x/e)/d^3+b*e^2*polylog(2,-d*x/e)/d^3
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=-\frac {a e x}{d^2}+\frac {b e x}{d^2}-\frac {b x^2}{4 d}-\frac {b e x \log (c x)}{d^2}+\frac {x^2 (a+b \log (c x))}{2 d}+\frac {e^2 (a+b \log (c x)) \log \left (\frac {e+d x}{e}\right )}{d^3}+\frac {b e^2 \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^3} \]
-((a*e*x)/d^2) + (b*e*x)/d^2 - (b*x^2)/(4*d) - (b*e*x*Log[c*x])/d^2 + (x^2 *(a + b*Log[c*x]))/(2*d) + (e^2*(a + b*Log[c*x])*Log[(e + d*x)/e])/d^3 + ( b*e^2*PolyLog[2, -((d*x)/e)])/d^3
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2005, 2793, 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.
| \(\displaystyle \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \int \frac {x^2 (a+b \log (c x))}{d x+e}dx\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \int \left (\frac {e^2 (a+b \log (c x))}{d^2 (d x+e)}-\frac {e (a+b \log (c x))}{d^2}+\frac {x (a+b \log (c x))}{d}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^3}+\frac {x^2 (a+b \log (c x))}{2 d}-\frac {a e x}{d^2}-\frac {b e x \log (c x)}{d^2}+\frac {b e^2 \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^3}+\frac {b e x}{d^2}-\frac {b x^2}{4 d}\) |
-((a*e*x)/d^2) + (b*e*x)/d^2 - (b*x^2)/(4*d) - (b*e*x*Log[c*x])/d^2 + (x^2 *(a + b*Log[c*x]))/(2*d) + (e^2*(a + b*Log[c*x])*Log[1 + (d*x)/e])/d^3 + ( b*e^2*PolyLog[2, -((d*x)/e)])/d^3
3.4.39.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer Q[r]))
Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(\frac {a \,x^{2}}{2 d}-\frac {a e x}{d^{2}}+\frac {a \,e^{2} \ln \left (d x +e \right )}{d^{3}}+\frac {b \,x^{2} \ln \left (x c \right )}{2 d}-\frac {b \,x^{2}}{4 d}-\frac {b e x \ln \left (x c \right )}{d^{2}}+\frac {b e x}{d^{2}}+\frac {b \,e^{2} \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{3}}+\frac {b \,e^{2} \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{3}}\) | \(126\) |
| parts | \(\frac {a \,x^{2}}{2 d}-\frac {a e x}{d^{2}}+\frac {a \,e^{2} \ln \left (d x +e \right )}{d^{3}}+\frac {b \,x^{2} \ln \left (x c \right )}{2 d}-\frac {b \,x^{2}}{4 d}-\frac {b e x \ln \left (x c \right )}{d^{2}}+\frac {b e x}{d^{2}}+\frac {b \,e^{2} \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{3}}+\frac {b \,e^{2} \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{3}}\) | \(126\) |
| derivativedivides | \(\frac {-\frac {a \,c^{2} e x}{d^{2}}+\frac {a \,x^{2} c^{2}}{2 d}+\frac {a \,c^{2} e^{2} \ln \left (c d x +c e \right )}{d^{3}}+b \left (\frac {\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}}{d}-\frac {e c \left (x c \ln \left (x c \right )-x c \right )}{d^{2}}+\frac {c^{2} e^{2} \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d^{2}}\right )}{c^{2}}\) | \(151\) |
| default | \(\frac {-\frac {a \,c^{2} e x}{d^{2}}+\frac {a \,x^{2} c^{2}}{2 d}+\frac {a \,c^{2} e^{2} \ln \left (c d x +c e \right )}{d^{3}}+b \left (\frac {\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}}{d}-\frac {e c \left (x c \ln \left (x c \right )-x c \right )}{d^{2}}+\frac {c^{2} e^{2} \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d^{2}}\right )}{c^{2}}\) | \(151\) |
1/2*a*x^2/d-a*e*x/d^2+a*e^2/d^3*ln(d*x+e)+1/2*b/d*x^2*ln(x*c)-1/4*b*x^2/d- b*e*x*ln(x*c)/d^2+b*e*x/d^2+b*e^2/d^3*dilog((c*d*x+c*e)/e/c)+b*e^2/d^3*ln( x*c)*ln((c*d*x+c*e)/e/c)
\[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )} x}{d + \frac {e}{x}} \,d x } \]
Time = 52.73 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.11 \[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\frac {a x^{2}}{2 d} + \frac {a e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e x}{d^{2}} + \frac {b x^{2} \log {\left (c x \right )}}{2 d} - \frac {b x^{2}}{4 d} - \frac {b e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d^{2}} - \frac {b e x \log {\left (c x \right )}}{d^{2}} + \frac {b e x}{d^{2}} \]
a*x**2/(2*d) + a*e**2*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, True))/d **2 - a*e*x/d**2 + b*x**2*log(c*x)/(2*d) - b*x**2/(4*d) - b*e**2*Piecewise ((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/Ab s(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(e) - polylog(2, d*x*exp_polar(I*pi)/e), True ))/d, True))/d**2 + b*e**2*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, Tru e))*log(c*x)/d**2 - b*e*x*log(c*x)/d**2 + b*e*x/d**2
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e^{2}}{d^{3}} + \frac {{\left ({\left (2 \, d \log \left (c\right ) - d\right )} b + 2 \, a d\right )} x^{2} - 4 \, {\left ({\left (e \log \left (c\right ) - e\right )} b + a e\right )} x + 2 \, {\left (b d x^{2} - 2 \, b e x\right )} \log \left (x\right )}{4 \, d^{2}} + \frac {{\left (b e^{2} \log \left (c\right ) + a e^{2}\right )} \log \left (d x + e\right )}{d^{3}} \]
(log(d*x/e + 1)*log(x) + dilog(-d*x/e))*b*e^2/d^3 + 1/4*(((2*d*log(c) - d) *b + 2*a*d)*x^2 - 4*((e*log(c) - e)*b + a*e)*x + 2*(b*d*x^2 - 2*b*e*x)*log (x))/d^2 + (b*e^2*log(c) + a*e^2)*log(d*x + e)/d^3
\[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )} x}{d + \frac {e}{x}} \,d x } \]
Timed out. \[ \int \frac {x (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x\right )\right )}{d+\frac {e}{x}} \,d x \]