Integrand size = 21, antiderivative size = 152 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {3 b d n x}{e^3}-\frac {d (3 a+b n) x}{e^3}-\frac {3 b n x^2}{4 e^2}-\frac {3 b d x \log \left (c x^n\right )}{e^3}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {3 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4} \]
3*b*d*n*x/e^3-d*(b*n+3*a)*x/e^3-3/4*b*n*x^2/e^2-3*b*d*x*ln(c*x^n)/e^3-x^3* (a+b*ln(c*x^n))/e/(e*x+d)+1/2*x^2*(3*a+b*n+3*b*ln(c*x^n))/e^2+d^2*(3*a+b*n +3*b*ln(c*x^n))*ln(1+e*x/d)/e^4+3*b*d^2*n*polylog(2,-e*x/d)/e^4
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {-8 a d e x+8 b d e n x-b e^2 n x^2-8 b d e x \log \left (c x^n\right )+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-4 b d^2 n (\log (x)-\log (d+e x))+12 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+12 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^4} \]
(-8*a*d*e*x + 8*b*d*e*n*x - b*e^2*n*x^2 - 8*b*d*e*x*Log[c*x^n] + 2*e^2*x^2 *(a + b*Log[c*x^n]) + (4*d^3*(a + b*Log[c*x^n]))/(d + e*x) - 4*b*d^2*n*(Lo g[x] - Log[d + e*x]) + 12*d^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] + 12*b*d ^2*n*PolyLog[2, -((e*x)/d)])/(4*e^4)
Time = 0.43 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2784, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\int \frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{d+e x}dx}{e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \frac {\int \left (\frac {\left (3 a+b n+3 b \log \left (c x^n\right )\right ) d^2}{e^2 (d+e x)}-\frac {\left (3 a+b n+3 b \log \left (c x^n\right )\right ) d}{e^2}+\frac {x \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{e}\right )dx}{e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e^3}+\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e}-\frac {d x (3 a+b n)}{e^2}-\frac {3 b d x \log \left (c x^n\right )}{e^2}+\frac {3 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {3 b d n x}{e^2}-\frac {3 b n x^2}{4 e}}{e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\) |
-((x^3*(a + b*Log[c*x^n]))/(e*(d + e*x))) + ((3*b*d*n*x)/e^2 - (d*(3*a + b *n)*x)/e^2 - (3*b*n*x^2)/(4*e) - (3*b*d*x*Log[c*x^n])/e^2 + (x^2*(3*a + b* n + 3*b*Log[c*x^n]))/(2*e) + (d^2*(3*a + b*n + 3*b*Log[c*x^n])*Log[1 + (e* x)/d])/e^3 + (3*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^3)/e
3.1.39.3.1 Defintions of rubi rules used
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[((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]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.41 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e^{2}}-\frac {2 b \ln \left (x^{n}\right ) d x}{e^{3}}+\frac {3 b \ln \left (x^{n}\right ) d^{2} \ln \left (e x +d \right )}{e^{4}}+\frac {b \ln \left (x^{n}\right ) d^{3}}{e^{4} \left (e x +d \right )}-\frac {3 b n \,d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {3 b n \,d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4}}-\frac {b n \,x^{2}}{4 e^{2}}+\frac {2 b d n x}{e^{3}}+\frac {9 b n \,d^{2}}{4 e^{4}}+\frac {b n \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {b n \,d^{2} \ln \left (e x \right )}{e^{4}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} e \,x^{2}-2 d x}{e^{3}}+\frac {3 d^{2} \ln \left (e x +d \right )}{e^{4}}+\frac {d^{3}}{e^{4} \left (e x +d \right )}\right )\) | \(298\) |
1/2*b*ln(x^n)/e^2*x^2-2*b*ln(x^n)/e^3*d*x+3*b*ln(x^n)/e^4*d^2*ln(e*x+d)+b* ln(x^n)*d^3/e^4/(e*x+d)-3*b*n/e^4*d^2*ln(e*x+d)*ln(-e*x/d)-3*b*n/e^4*d^2*d ilog(-e*x/d)-1/4*b*n*x^2/e^2+2*b*d*n*x/e^3+9/4*b*n/e^4*d^2+b*n/e^4*d^2*ln( e*x+d)-b*n/e^4*d^2*ln(e*x)+(-1/2*I*b*Pi*csgn(I*c)*csgn(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*csgn(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*(1/2*e*x^2-2*d*x)+3/e^4*d ^2*ln(e*x+d)+d^3/e^4/(e*x+d))
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
Time = 24.56 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.12 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=- \frac {a d^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 a d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {2 a d x}{e^{3}} + \frac {a x^{2}}{2 e^{2}} + \frac {b d^{3} 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^{3}} - \frac {b d^{3} \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^{3}} - \frac {3 b d^{2} 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^{3}} + \frac {3 b d^{2} \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^{3}} + \frac {2 b d n x}{e^{3}} - \frac {2 b d x \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b n x^{2}}{4 e^{2}} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e^{2}} \]
-a*d**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e**3 + 3* a*d**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**3 - 2*a*d*x/e **3 + a*x**2/(2*e**2) + b*d**3*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d *e) + log(d/e + x)/(d*e), True))/e**3 - b*d**3*Piecewise((x/d**2, Eq(e, 0) ), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**3 - 3*b*d**2*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_polar(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) + meijerg(((1, 1), ( )), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/e**3 + 3*b*d**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True)) *log(c*x**n)/e**3 + 2*b*d*n*x/e**3 - 2*b*d*x*log(c*x**n)/e**3 - b*n*x**2/( 4*e**2) + b*x**2*log(c*x**n)/(2*e**2)
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
1/2*(2*d^3/(e^5*x + d*e^4) + 6*d^2*log(e*x + d)/e^4 + (e*x^2 - 4*d*x)/e^3) *a + b*integrate((x^3*log(c) + x^3*log(x^n))/(e^2*x^2 + 2*d*e*x + d^2), x)
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]