Integrand size = 23, antiderivative size = 271 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {2 a b d^2 n x}{e^3}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {2 b d^3 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^3 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4} \]
-2*a*b*d^2*n*x/e^3+2*b^2*d^2*n^2*x/e^3-1/4*b^2*d*n^2*x^2/e^2+2/27*b^2*n^2* x^3/e-2*b^2*d^2*n*x*ln(c*x^n)/e^3+1/2*b*d*n*x^2*(a+b*ln(c*x^n))/e^2-2/9*b* n*x^3*(a+b*ln(c*x^n))/e+d^2*x*(a+b*ln(c*x^n))^2/e^3-1/2*d*x^2*(a+b*ln(c*x^ n))^2/e^2+1/3*x^3*(a+b*ln(c*x^n))^2/e-d^3*(a+b*ln(c*x^n))^2*ln(1+e*x/d)/e^ 4-2*b*d^3*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/e^4+2*b^2*d^3*n^2*polylog(3, -e*x/d)/e^4
Time = 0.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {-108 d^2 e x \left (a+b \log \left (c x^n\right )\right )^2+54 d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-36 e^3 x^3 \left (a+b \log \left (c x^n\right )\right )^2+216 b d^2 e n x \left (a-b n+b \log \left (c x^n\right )\right )-8 b e^3 n x^3 \left (b n-3 \left (a+b \log \left (c x^n\right )\right )\right )+27 b d e^2 n x^2 \left (b n-2 \left (a+b \log \left (c x^n\right )\right )\right )+108 d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+216 b d^3 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{108 e^4} \]
-1/108*(-108*d^2*e*x*(a + b*Log[c*x^n])^2 + 54*d*e^2*x^2*(a + b*Log[c*x^n] )^2 - 36*e^3*x^3*(a + b*Log[c*x^n])^2 + 216*b*d^2*e*n*x*(a - b*n + b*Log[c *x^n]) - 8*b*e^3*n*x^3*(b*n - 3*(a + b*Log[c*x^n])) + 27*b*d*e^2*n*x^2*(b* n - 2*(a + b*Log[c*x^n])) + 108*d^3*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 216*b*d^3*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/e^4
Time = 0.50 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2795, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \int \left (-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b d^3 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}-\frac {2 a b d^2 n x}{e^3}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {2 b^2 d^3 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}\) |
(-2*a*b*d^2*n*x)/e^3 + (2*b^2*d^2*n^2*x)/e^3 - (b^2*d*n^2*x^2)/(4*e^2) + ( 2*b^2*n^2*x^3)/(27*e) - (2*b^2*d^2*n*x*Log[c*x^n])/e^3 + (b*d*n*x^2*(a + b *Log[c*x^n]))/(2*e^2) - (2*b*n*x^3*(a + b*Log[c*x^n]))/(9*e) + (d^2*x*(a + b*Log[c*x^n])^2)/e^3 - (d*x^2*(a + b*Log[c*x^n])^2)/(2*e^2) + (x^3*(a + b *Log[c*x^n])^2)/(3*e) - (d^3*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 - (2*b*d^3*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 + (2*b^2*d^3*n^2 *PolyLog[3, -((e*x)/d)])/e^4
3.1.92.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.57 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.70
| method | result | size |
| risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} x^{3}}{3 e}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d \,x^{2}}{2 e^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} x \,d^{2}}{e^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x^{3}}{9 e}+\frac {b^{2} n \ln \left (x^{n}\right ) d \,x^{2}}{2 e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x \,d^{2}}{e^{3}}+\frac {2 b^{2} n^{2} x^{3}}{27 e}-\frac {b^{2} d \,n^{2} x^{2}}{4 e^{2}}+\frac {2 b^{2} d^{2} n^{2} x}{e^{3}}-\frac {2 b^{2} d^{3} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{e^{4}}-\frac {2 b^{2} d^{3} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{e^{4}}+\frac {2 b^{2} n \,d^{3} \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}+\frac {2 b^{2} n \,d^{3} \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4}}+\frac {b^{2} d^{3} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{4}}-\frac {b^{2} d^{3} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{4}}-\frac {2 b^{2} d^{3} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e^{4}}+\frac {2 b^{2} d^{3} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e^{4}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) x^{3}}{3 e}-\frac {\ln \left (x^{n}\right ) d \,x^{2}}{2 e^{2}}+\frac {\ln \left (x^{n}\right ) x \,d^{2}}{e^{3}}-\frac {\ln \left (x^{n}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-n \left (\frac {\frac {2 \left (e x +d \right )^{3}}{3}-\frac {7 d \left (e x +d \right )^{2}}{2}+11 d^{2} \left (e x +d \right )}{6 e^{4}}-\frac {d^{3} \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e^{4}}\right )\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (\frac {\frac {1}{3} e^{2} x^{3}-\frac {1}{2} d e \,x^{2}+d^{2} x}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right )}{e^{4}}\right )}{4}\) | \(732\) |
1/3*b^2*ln(x^n)^2/e*x^3-1/2*b^2*ln(x^n)^2/e^2*d*x^2+b^2*ln(x^n)^2/e^3*x*d^ 2-b^2*ln(x^n)^2*d^3/e^4*ln(e*x+d)-2/9*b^2*n*ln(x^n)/e*x^3+1/2*b^2*n*ln(x^n )/e^2*d*x^2-2*b^2*n*ln(x^n)/e^3*x*d^2+2/27*b^2*n^2*x^3/e-1/4*b^2*d*n^2*x^2 /e^2+2*b^2*d^2*n^2*x/e^3-2*b^2*d^3/e^4*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2-2*b^ 2*d^3/e^4*ln(x)*dilog(-e*x/d)*n^2+2*b^2*n*d^3/e^4*ln(x^n)*ln(e*x+d)*ln(-e* x/d)+2*b^2*n*d^3/e^4*ln(x^n)*dilog(-e*x/d)+b^2*d^3/e^4*n^2*ln(e*x+d)*ln(x) ^2-b^2*d^3/e^4*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2*d^3/e^4*n^2*ln(x)*polylog(2,- e*x/d)+2*b^2*d^3*n^2*polylog(3,-e*x/d)/e^4+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)* csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c *x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(1/3*ln(x^n)/e*x^3-1/2*ln( x^n)/e^2*d*x^2+ln(x^n)/e^3*x*d^2-ln(x^n)*d^3/e^4*ln(e*x+d)-n*(1/6/e^4*(2/3 *(e*x+d)^3-7/2*d*(e*x+d)^2+11*d^2*(e*x+d))-d^3/e^4*(dilog(-e*x/d)+ln(e*x+d )*ln(-e*x/d))))+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*cs gn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c *x^n)^3+2*b*ln(c)+2*a)^2*(1/e^3*(1/3*e^2*x^3-1/2*d*e*x^2+d^2*x)-d^3/e^4*ln (e*x+d))
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d} \,d x } \]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d} \,d x } \]
-1/6*a^2*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + integrate((b^2*x^3*log(x^n)^2 + 2*(b^2*log(c) + a*b)*x^3*log(x^n) + (b^2 *log(c)^2 + 2*a*b*log(c))*x^3)/(e*x + d), x)
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int \frac {x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \]