Integrand size = 13, antiderivative size = 107 \[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=-\frac {x}{3 d^2 e (d+e x)}+\frac {x \log (x)}{3 d e (d+e x)^2}+\frac {x^2 (3 d+e x) \log ^2(x)}{6 d^2 (d+e x)^3}-\frac {\log (x) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^2 e^2} \]
-1/3*x/d^2/e/(e*x+d)+1/3*x*ln(x)/d/e/(e*x+d)^2+1/6*x^2*(e*x+3*d)*ln(x)^2/d ^2/(e*x+d)^3-1/3*ln(x)*ln(1+e*x/d)/d^2/e^2-1/3*polylog(2,-e*x/d)/d^2/e^2
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=\frac {2 d (d+e x)^2+e^2 x^2 (3 d+e x) \log ^2(x)-2 (d+e x) \log (x) \left (-d e x+(d+e x)^2 \log \left (1+\frac {e x}{d}\right )\right )-2 (d+e x)^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 d^2 e^2 (d+e x)^3} \]
(2*d*(d + e*x)^2 + e^2*x^2*(3*d + e*x)*Log[x]^2 - 2*(d + e*x)*Log[x]*(-(d* e*x) + (d + e*x)^2*Log[1 + (e*x)/d]) - 2*(d + e*x)^3*PolyLog[2, -((e*x)/d) ])/(6*d^2*e^2*(d + e*x)^3)
Time = 0.59 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.53, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2783, 2773, 49, 2009, 2781, 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.
| \(\displaystyle \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle \frac {\int \frac {x \log ^2(x)}{(d+e x)^3}dx}{3 d}-\frac {2 \int \frac {x \log (x)}{(d+e x)^3}dx}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle -\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\int \frac {x}{(d+e x)^2}dx}{2 d}\right )}{3 d}+\frac {\int \frac {x \log ^2(x)}{(d+e x)^3}dx}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\int \left (\frac {1}{e (d+e x)}-\frac {d}{e (d+e x)^2}\right )dx}{2 d}\right )}{3 d}+\frac {\int \frac {x \log ^2(x)}{(d+e x)^3}dx}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {x \log ^2(x)}{(d+e x)^3}dx}{3 d}-\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}}{2 d}\right )}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle \frac {\frac {x^2 \log ^2(x)}{2 d (d+e x)^2}-\frac {\int \frac {x \log (x)}{(d+e x)^2}dx}{d}}{3 d}-\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}}{2 d}\right )}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {x^2 \log ^2(x)}{2 d (d+e x)^2}-\frac {\frac {\int \frac {\log (x)+1}{d+e x}dx}{e}-\frac {x \log (x)}{e (d+e x)}}{d}}{3 d}-\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}}{2 d}\right )}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\frac {x^2 \log ^2(x)}{2 d (d+e x)^2}-\frac {\frac {\frac {(\log (x)+1) \log \left (\frac {e x}{d}+1\right )}{e}-\frac {\int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}}{e}-\frac {x \log (x)}{e (d+e x)}}{d}}{3 d}-\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}}{2 d}\right )}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 \left (\frac {x^2 \log (x)}{2 d (d+e x)^2}-\frac {\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}}{2 d}\right )}{3 d}+\frac {\frac {x^2 \log ^2(x)}{2 d (d+e x)^2}-\frac {\frac {\frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}+\frac {(\log (x)+1) \log \left (\frac {e x}{d}+1\right )}{e}}{e}-\frac {x \log (x)}{e (d+e x)}}{d}}{3 d}+\frac {x^2 \log ^2(x)}{3 d (d+e x)^3}\) |
(x^2*Log[x]^2)/(3*d*(d + e*x)^3) - (2*((x^2*Log[x])/(2*d*(d + e*x)^2) - (d /(e^2*(d + e*x)) + Log[d + e*x]/e^2)/(2*d)))/(3*d) + ((x^2*Log[x]^2)/(2*d* (d + e*x)^2) - (-((x*Log[x])/(e*(d + e*x))) + (((1 + Log[x])*Log[1 + (e*x) /d])/e + PolyLog[2, -((e*x)/d)]/e)/e)/d)/(3*d)
3.2.20.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x) ^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Simp[(m + q + 2)/(d*(q + 1)) Int[ (f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && L tQ[q, -1] && GtQ[m, 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]
Time = 0.32 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.68
| method | result | size |
| parts | \(-\frac {\ln \left (x \right )^{2}}{2 e^{2} \left (e x +d \right )^{2}}+\frac {\ln \left (x \right )^{2} d}{3 e^{2} \left (e x +d \right )^{3}}+\frac {\ln \left (x \right )^{2}}{6 d^{2} e^{2}}-\frac {\frac {\operatorname {dilog}\left (\frac {e x +d}{d}\right )}{e}+\frac {\ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{e}}{3 e \,d^{2}}+\frac {-\frac {\ln \left (e x +d \right )}{3 d^{2} e}+\frac {1}{3 d e \left (e x +d \right )}+\frac {\ln \left (x \right ) x \left (e x +2 d \right )}{3 d^{2} \left (e x +d \right )^{2}}}{e}-\frac {-\frac {\ln \left (e x +d \right )}{d e}+\frac {\ln \left (x \right ) x}{d \left (e x +d \right )}}{3 e d}\) | \(180\) |
-1/2*ln(x)^2/e^2/(e*x+d)^2+1/3*ln(x)^2/e^2*d/(e*x+d)^3+1/6/d^2/e^2*ln(x)^2 -1/3/e/d^2*(dilog((e*x+d)/d)/e+ln(x)*ln((e*x+d)/d)/e)+2/3/e*(-1/2/d^2*ln(e *x+d)/e+1/2/d/e/(e*x+d)+1/2*ln(x)*x*(e*x+2*d)/d^2/(e*x+d)^2)-1/3/e/d*(-1/d /e*ln(e*x+d)+ln(x)*x/d/(e*x+d))
\[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=\int { \frac {x \log \left (x\right )^{2}}{{\left (e x + d\right )}^{4}} \,d x } \]
Time = 20.97 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.50 \[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=\frac {\left (- d - 3 e x\right ) \log {\left (x \right )}^{2}}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} x^{3}} + \frac {\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 (x \right )}}{e} - \frac {\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}}{e} + \frac {\begin {cases} - \frac {1}{e^{3} x} & \text {for}\: d = 0 \\- \frac {1}{2 d e^{2} + 2 e^{3} x} - \frac {\log {\left (d + e x \right )}}{2 d e^{2}} & \text {otherwise} \end {cases}}{3 d} - \frac {\left (\begin {cases} \frac {1}{e^{3} x} & \text {for}\: d = 0 \\- \frac {1}{2 d \left (\frac {d}{x} + e\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (x \right )}}{3 d} - \frac {2 \left (\begin {cases} - \frac {1}{e^{2} x} & \text {for}\: d = 0 \\- \frac {\log {\left (d^{2} + d e x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{3 d e} + \frac {2 \left (\begin {cases} \frac {1}{e^{2} x} & \text {for}\: d = 0 \\- \frac {1}{\frac {d^{2}}{x} + d e} & \text {otherwise} \end {cases}\right ) \log {\left (x \right )}}{3 d e} + \frac {\begin {cases} - \frac {1}{e x} & \text {for}\: d = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x}\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 e^{i \pi }}{e x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x}\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 e^{i \pi }}{e x}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}}{3 d e^{2}} - \frac {\left (\begin {cases} \frac {1}{e x} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {d}{x} + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (x \right )}}{3 d e^{2}} \]
(-d - 3*e*x)*log(x)**2/(6*d**3*e**2 + 18*d**2*e**3*x + 18*d*e**4*x**2 + 6* e**5*x**3) + Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))* log(x)/e - Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log (x)/(2*d**2*e) + log(d/e + x)/(2*d**2*e), True))/e + Piecewise((-1/(e**3*x ), Eq(d, 0)), (-1/(2*d*e**2 + 2*e**3*x) - log(d + e*x)/(2*d*e**2), True))/ (3*d) - Piecewise((1/(e**3*x), Eq(d, 0)), (-1/(2*d*(d/x + e)**2), True))*l og(x)/(3*d) - 2*Piecewise((-1/(e**2*x), Eq(d, 0)), (-log(d**2 + d*e*x)/(d* e), True))/(3*d*e) + 2*Piecewise((1/(e**2*x), Eq(d, 0)), (-1/(d**2/x + d*e ), True))*log(x)/(3*d*e) + Piecewise((-1/(e*x), Eq(d, 0)), (Piecewise((pol ylog(2, d*exp_polar(I*pi)/(e*x)), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(e)* log(x) + polylog(2, d*exp_polar(I*pi)/(e*x)), Abs(x) < 1), (-log(e)*log(1/ x) + polylog(2, d*exp_polar(I*pi)/(e*x)), 1/Abs(x) < 1), (-meijerg(((), (1 , 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1), ()), ((), (0, 0)), x)*lo g(e) + polylog(2, d*exp_polar(I*pi)/(e*x)), True))/d, True))/(3*d*e**2) - Piecewise((1/(e*x), Eq(d, 0)), (log(d/x + e)/d, True))*log(x)/(3*d*e**2)
Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.23 \[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=-\frac {d^{2} \log \left (x\right )^{2} - 2 \, {\left (e^{2} \log \left (x\right ) + e^{2}\right )} x^{2} - 2 \, d^{2} + {\left (3 \, d e \log \left (x\right )^{2} - 2 \, d e \log \left (x\right ) - 4 \, d e\right )} x}{6 \, {\left (d e^{5} x^{3} + 3 \, d^{2} e^{4} x^{2} + 3 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} + \frac {\log \left (x\right )^{2}}{6 \, d^{2} e^{2}} - \frac {\log \left (\frac {e x}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x}{d}\right )}{3 \, d^{2} e^{2}} \]
-1/6*(d^2*log(x)^2 - 2*(e^2*log(x) + e^2)*x^2 - 2*d^2 + (3*d*e*log(x)^2 - 2*d*e*log(x) - 4*d*e)*x)/(d*e^5*x^3 + 3*d^2*e^4*x^2 + 3*d^3*e^3*x + d^4*e^ 2) + 1/6*log(x)^2/(d^2*e^2) - 1/3*(log(e*x/d + 1)*log(x) + dilog(-e*x/d))/ (d^2*e^2)
\[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=\int { \frac {x \log \left (x\right )^{2}}{{\left (e x + d\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {x \log ^2(x)}{(d+e x)^4} \, dx=\int \frac {x\,{\ln \left (x\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]