Integrand size = 23, antiderivative size = 285 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=-\frac {b^2 n^2}{4 d^2 x^2}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {3 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {6 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4} \] Output:
-1/4*b^2*n^2/d^2/x^2+4*b^2*e*n^2/d^3/x-1/2*b*n*(a+b*ln(c*x^n))/d^2/x^2+4*b *e*n*(a+b*ln(c*x^n))/d^3/x-1/2*(a+b*ln(c*x^n))^2/d^2/x^2+2*e*(a+b*ln(c*x^n ))^2/d^3/x-e^3*x*(a+b*ln(c*x^n))^2/d^4/(e*x+d)-3*e^2*ln(1+d/e/x)*(a+b*ln(c *x^n))^2/d^4+2*b*e^2*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^4+6*b*e^2*n*(a+b*ln(c *x^n))*polylog(2,-d/e/x)/d^4+2*b^2*e^2*n^2*polylog(2,-e*x/d)/d^4+6*b^2*e^2 *n^2*polylog(3,-d/e/x)/d^4
Time = 0.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\frac {-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac {8 d e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {4 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+\frac {16 b d e n \left (a+b n+b \log \left (c x^n\right )\right )}{x}-\frac {b d^2 n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{x^2}-12 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+4 e^2 \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (1+\frac {e x}{d}\right )\right )\right )+2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-24 b e^2 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{4 d^4} \] Input:
Integrate[(a + b*Log[c*x^n])^2/(x^3*(d + e*x)^2),x]
Output:
((-2*d^2*(a + b*Log[c*x^n])^2)/x^2 + (8*d*e*(a + b*Log[c*x^n])^2)/x + (4*d *e^2*(a + b*Log[c*x^n])^2)/(d + e*x) + (4*e^2*(a + b*Log[c*x^n])^3)/(b*n) + (16*b*d*e*n*(a + b*n + b*Log[c*x^n]))/x - (b*d^2*n*(2*a + b*n + 2*b*Log[ c*x^n]))/x^2 - 12*e^2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 4*e^2*(-((a + b*Log[c*x^n])*(a + b*Log[c*x^n] - 2*b*n*Log[1 + (e*x)/d])) + 2*b^2*n^2*P olyLog[2, -((e*x)/d)]) - 24*b*e^2*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x) /d)] - b*n*PolyLog[3, -((e*x)/d)]))/(4*d^4)
Time = 0.70 (sec) , antiderivative size = 285, 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.
\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx\) |
\(\Big \downarrow \) 2795 |
\(\displaystyle \int \left (-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x (d+e x)}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {6 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {3 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {2 b e^2 n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b^2 n^2}{4 d^2 x^2}\) |
Input:
Int[(a + b*Log[c*x^n])^2/(x^3*(d + e*x)^2),x]
Output:
-1/4*(b^2*n^2)/(d^2*x^2) + (4*b^2*e*n^2)/(d^3*x) - (b*n*(a + b*Log[c*x^n]) )/(2*d^2*x^2) + (4*b*e*n*(a + b*Log[c*x^n]))/(d^3*x) - (a + b*Log[c*x^n])^ 2/(2*d^2*x^2) + (2*e*(a + b*Log[c*x^n])^2)/(d^3*x) - (e^3*x*(a + b*Log[c*x ^n])^2)/(d^4*(d + e*x)) - (3*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^ 4 + (2*b*e^2*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^4 + (6*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^4 + (2*b^2*e^2*n^2*PolyLog[2, -((e *x)/d)])/d^4 + (6*b^2*e^2*n^2*PolyLog[3, -(d/(e*x))])/d^4
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 = 0.94 (sec) , antiderivative size = 924, normalized size of antiderivative = 3.24
Input:
int((a+b*ln(c*x^n))^2/x^3/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
b^2/d^4*e^2*ln(x)^3*n^2+6*b^2/d^4*e^2*n^2*polylog(3,-e*x/d)+2*b^2*n*ln(x^n )/d^4*e^2*ln(e*x+d)-2*b^2*n*ln(x^n)/d^4*e^2*ln(x)-2*b^2/d^4*n^2*e^2*ln(e*x +d)*ln(-e*x/d)-3*b^2*n/d^4*e^2*ln(x^n)*ln(x)^2-6*b^2/d^4*e^2*ln(x)*dilog(- e*x/d)*n^2+6*b^2*n/d^4*e^2*ln(x^n)*dilog(-e*x/d)+3*b^2/d^4*e^2*n^2*ln(e*x+ d)*ln(x)^2-3*b^2/d^4*e^2*n^2*ln(x)^2*ln(1+e*x/d)-6*b^2/d^4*e^2*n^2*ln(x)*p olylog(2,-e*x/d)-2*b^2/d^4*n^2*e^2*dilog(-e*x/d)-3*b^2*ln(x^n)^2/d^4*e^2*l n(e*x+d)+3*b^2*ln(x^n)^2/d^4*e^2*ln(x)-1/2*b^2*n*ln(x^n)/d^2/x^2+b^2/d^4*n ^2*e^2*ln(x)^2+1/4*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)* csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn (I*c)+2*b*ln(c)+2*a)^2*(-3/d^4*e^2*ln(e*x+d)+1/d^3*e^2/(e*x+d)-1/2/d^2/x^2 +3/d^4*e^2*ln(x)+2/d^3*e/x)-6*b^2/d^4*e^2*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2+6 *b^2*n/d^4*e^2*ln(x^n)*ln(e*x+d)*ln(-e*x/d)-1/2*b^2*ln(x^n)^2/d^2/x^2+2*b^ 2*ln(x^n)^2/d^3*e/x+4*b^2*n*ln(x^n)/d^3*e/x+b^2*ln(x^n)^2/d^3*e^2/(e*x+d)+ (I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn( I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a )*b*(-3*ln(x^n)/d^4*e^2*ln(e*x+d)+ln(x^n)/d^3*e^2/(e*x+d)-1/2*ln(x^n)/d^2/ x^2+3*ln(x^n)/d^4*e^2*ln(x)+2*ln(x^n)/d^3*e/x-1/2*n*(-2/d^4*e^2*ln(e*x+d)+ 1/2/d^2/x^2-4/d^3*e/x+2/d^4*e^2*ln(x)+3/d^4*e^2*ln(x)^2-6/d^4*e^2*(dilog(- e*x/d)+ln(e*x+d)*ln(-e*x/d))))+4*b^2*e*n^2/d^3/x-1/4*b^2*n^2/d^2/x^2
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*log(c*x^n))^2/x^3/(e*x+d)^2,x, algorithm="fricas")
Output:
integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^2*x^5 + 2*d*e*x^4 + d^2*x^3), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{3} \left (d + e x\right )^{2}}\, dx \] Input:
integrate((a+b*ln(c*x**n))**2/x**3/(e*x+d)**2,x)
Output:
Integral((a + b*log(c*x**n))**2/(x**3*(d + e*x)**2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*log(c*x^n))^2/x^3/(e*x+d)^2,x, algorithm="maxima")
Output:
1/2*a^2*((6*e^2*x^2 + 3*d*e*x - d^2)/(d^3*e*x^3 + d^4*x^2) - 6*e^2*log(e*x + d)/d^4 + 6*e^2*log(x)/d^4) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e^2*x^5 + 2*d*e*x^4 + d^2* x^3), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*log(c*x^n))^2/x^3/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)^2/((e*x + d)^2*x^3), x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3\,{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + b*log(c*x^n))^2/(x^3*(d + e*x)^2),x)
Output:
int((a + b*log(c*x^n))^2/(x^3*(d + e*x)^2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{2} x^{5}+2 d e \,x^{4}+d^{2} x^{3}}d x \right ) b^{2} d^{5} x^{2}+2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{2} x^{5}+2 d e \,x^{4}+d^{2} x^{3}}d x \right ) b^{2} d^{4} e \,x^{3}+4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{5}+2 d e \,x^{4}+d^{2} x^{3}}d x \right ) a b \,d^{5} x^{2}+4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{5}+2 d e \,x^{4}+d^{2} x^{3}}d x \right ) a b \,d^{4} e \,x^{3}-6 \,\mathrm {log}\left (e x +d \right ) a^{2} d \,e^{2} x^{2}-6 \,\mathrm {log}\left (e x +d \right ) a^{2} e^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) a^{2} d \,e^{2} x^{2}+6 \,\mathrm {log}\left (x \right ) a^{2} e^{3} x^{3}-a^{2} d^{3}+3 a^{2} d^{2} e x -6 a^{2} e^{3} x^{3}}{2 d^{4} x^{2} \left (e x +d \right )} \] Input:
int((a+b*log(c*x^n))^2/x^3/(e*x+d)^2,x)
Output:
(2*int(log(x**n*c)**2/(d**2*x**3 + 2*d*e*x**4 + e**2*x**5),x)*b**2*d**5*x* *2 + 2*int(log(x**n*c)**2/(d**2*x**3 + 2*d*e*x**4 + e**2*x**5),x)*b**2*d** 4*e*x**3 + 4*int(log(x**n*c)/(d**2*x**3 + 2*d*e*x**4 + e**2*x**5),x)*a*b*d **5*x**2 + 4*int(log(x**n*c)/(d**2*x**3 + 2*d*e*x**4 + e**2*x**5),x)*a*b*d **4*e*x**3 - 6*log(d + e*x)*a**2*d*e**2*x**2 - 6*log(d + e*x)*a**2*e**3*x* *3 + 6*log(x)*a**2*d*e**2*x**2 + 6*log(x)*a**2*e**3*x**3 - a**2*d**3 + 3*a **2*d**2*e*x - 6*a**2*e**3*x**3)/(2*d**4*x**2*(d + e*x))