Integrand size = 21, antiderivative size = 107 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}-\frac {x \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{2 e^2 (d+e x)}+\frac {\left (2 a+3 b n+2 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^3}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3} \]
-1/2*x^2*(a+b*ln(c*x^n))/e/(e*x+d)^2-1/2*x*(2*a+b*n+2*b*ln(c*x^n))/e^2/(e* x+d)+1/2*(2*a+3*b*n+2*b*ln(c*x^n))*ln(1+e*x/d)/e^3+b*n*polylog(2,-e*x/d)/e ^3
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\frac {-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {4 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}-4 b n (\log (x)-\log (d+e x))+b n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 e^3} \]
(-((d^2*(a + b*Log[c*x^n]))/(d + e*x)^2) + (4*d*(a + b*Log[c*x^n]))/(d + e *x) - 4*b*n*(Log[x] - Log[d + e*x]) + b*n*(d/(d + e*x) + Log[x] - Log[d + e*x]) + 2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] + 2*b*n*PolyLog[2, -((e*x)/d )])/(2*e^3)
Time = 0.41 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2784, 2784, 2754, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\int \frac {x \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{2 e}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\int \frac {2 a+3 b n+2 b \log \left (c x^n\right )}{d+e x}dx}{e}-\frac {x \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{e (d+e x)}}{2 e}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\frac {\frac {\log \left (\frac {e x}{d}+1\right ) \left (2 a+2 b \log \left (c x^n\right )+3 b n\right )}{e}-\frac {2 b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}}{e}-\frac {x \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{e (d+e x)}}{2 e}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {\log \left (\frac {e x}{d}+1\right ) \left (2 a+2 b \log \left (c x^n\right )+3 b n\right )}{e}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}}{e}-\frac {x \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{e (d+e x)}}{2 e}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}\) |
-1/2*(x^2*(a + b*Log[c*x^n]))/(e*(d + e*x)^2) + (-((x*(2*a + b*n + 2*b*Log [c*x^n]))/(e*(d + e*x))) + (((2*a + 3*b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/ d])/e + (2*b*n*PolyLog[2, -((e*x)/d)])/e)/e)/(2*e)
3.1.47.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.43 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.41
| method | result | size |
| risch | \(\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{3}}+\frac {2 b \ln \left (x^{n}\right ) d}{e^{3} \left (e x +d \right )}-\frac {b \ln \left (x^{n}\right ) d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {b n d}{2 e^{3} \left (e x +d \right )}+\frac {3 b n \ln \left (e x +d \right )}{2 e^{3}}-\frac {3 b n \ln \left (e x \right )}{2 e^{3}}-\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}-\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\ln \left (e x +d \right )}{e^{3}}+\frac {2 d}{e^{3} \left (e x +d \right )}-\frac {d^{2}}{2 e^{3} \left (e x +d \right )^{2}}\right )\) | \(258\) |
b*ln(x^n)/e^3*ln(e*x+d)+2*b*ln(x^n)/e^3*d/(e*x+d)-1/2*b*ln(x^n)/e^3*d^2/(e *x+d)^2+1/2*b*n/e^3*d/(e*x+d)+3/2*b*n/e^3*ln(e*x+d)-3/2*b*n/e^3*ln(e*x)-b* n/e^3*ln(e*x+d)*ln(-e*x/d)-b*n/e^3*dilog(-e*x/d)+(-1/2*I*b*Pi*csgn(I*c)*cs gn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*cs gn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(1/e^3*ln( e*x+d)+2/e^3*d/(e*x+d)-1/2/e^3*d^2/(e*x+d)^2)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
Time = 19.77 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.24 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\frac {a d^{2} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 a d \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {a \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {b d^{2} n \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 d^{2} e + 2 d e^{2} x} - \frac {\log {\left (x \right )}}{2 d^{2} e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} + \frac {2 b d n \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 b d \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\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 (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} \]
a*d**2*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e**2 - 2*a*d*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e**2 + a*P iecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**2 - b*d**2*n*Piecewis e((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log(x)/(2*d**2*e) + lo g(d/e + x)/(2*d**2*e), True))/e**2 + b*d**2*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e**2 + 2*b*d*n*Piecewise((x/d** 2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/e**2 - 2*b*d*Pie cewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**2 - b *n*Piecewise((x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_polar(I*pi)/ d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x*exp_po lar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I* pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + m eijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I* pi)/d), True))/e, True))/e**2 + b*Piecewise((x/d, Eq(e, 0)), (log(d + e*x) /e, True))*log(c*x**n)/e**2
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
1/2*a*((4*d*e*x + 3*d^2)/(e^5*x^2 + 2*d*e^4*x + d^2*e^3) + 2*log(e*x + d)/ e^3) + b*integrate((x^2*log(c) + x^2*log(x^n))/(e^3*x^3 + 3*d*e^2*x^2 + 3* d^2*e*x + d^3), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \]