Integrand size = 30, antiderivative size = 306 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=-\frac {2 b^2 d n^2}{x}+\frac {2 b^2 e h n^2 q \log (x)}{g}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 b e h n q \log \left (1+\frac {g}{h x}\right ) \left (a+b \log \left (c x^n\right )\right )}{g}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {e h q \log \left (1+\frac {g}{h x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{g}-\frac {2 b^2 e h n^2 q \log (g+h x)}{g}-\frac {2 b^2 e n^2 \log \left (f (g+h x)^q\right )}{x}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (f (g+h x)^q\right )}{x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (f (g+h x)^q\right )}{x}+\frac {2 b^2 e h n^2 q \operatorname {PolyLog}\left (2,-\frac {g}{h x}\right )}{g}+\frac {2 b e h n q \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g}{h x}\right )}{g}+\frac {2 b^2 e h n^2 q \operatorname {PolyLog}\left (3,-\frac {g}{h x}\right )}{g} \] Output:
-2*b^2*d*n^2/x+2*b^2*e*h*n^2*q*ln(x)/g-2*b*d*n*(a+b*ln(c*x^n))/x-2*b*e*h*n *q*ln(1+g/h/x)*(a+b*ln(c*x^n))/g-d*(a+b*ln(c*x^n))^2/x-e*h*q*ln(1+g/h/x)*( a+b*ln(c*x^n))^2/g-2*b^2*e*h*n^2*q*ln(h*x+g)/g-2*b^2*e*n^2*ln(f*(h*x+g)^q) /x-2*b*e*n*(a+b*ln(c*x^n))*ln(f*(h*x+g)^q)/x-e*(a+b*ln(c*x^n))^2*ln(f*(h*x +g)^q)/x+2*b^2*e*h*n^2*q*polylog(2,-g/h/x)/g+2*b*e*h*n*q*(a+b*ln(c*x^n))*p olylog(2,-g/h/x)/g+2*b^2*e*h*n^2*q*polylog(3,-g/h/x)/g
Leaf count is larger than twice the leaf count of optimal. \(697\) vs. \(2(306)=612\).
Time = 0.44 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=-\frac {3 a^2 d g+6 a b d g n+6 b^2 d g n^2-3 a^2 e h q x \log (x)-6 a b e h n q x \log (x)-6 b^2 e h n^2 q x \log (x)+3 a b e h n q x \log ^2(x)+3 b^2 e h n^2 q x \log ^2(x)-b^2 e h n^2 q x \log ^3(x)+6 a b d g \log \left (c x^n\right )+6 b^2 d g n \log \left (c x^n\right )-6 a b e h q x \log (x) \log \left (c x^n\right )-6 b^2 e h n q x \log (x) \log \left (c x^n\right )+3 b^2 e h n q x \log ^2(x) \log \left (c x^n\right )+3 b^2 d g \log ^2\left (c x^n\right )-3 b^2 e h q x \log (x) \log ^2\left (c x^n\right )+3 a^2 e h q x \log (g+h x)+6 a b e h n q x \log (g+h x)+6 b^2 e h n^2 q x \log (g+h x)-6 a b e h n q x \log (x) \log (g+h x)-6 b^2 e h n^2 q x \log (x) \log (g+h x)+3 b^2 e h n^2 q x \log ^2(x) \log (g+h x)+6 a b e h q x \log \left (c x^n\right ) \log (g+h x)+6 b^2 e h n q x \log \left (c x^n\right ) \log (g+h x)-6 b^2 e h n q x \log (x) \log \left (c x^n\right ) \log (g+h x)+3 b^2 e h q x \log ^2\left (c x^n\right ) \log (g+h x)+3 a^2 e g \log \left (f (g+h x)^q\right )+6 a b e g n \log \left (f (g+h x)^q\right )+6 b^2 e g n^2 \log \left (f (g+h x)^q\right )+6 a b e g \log \left (c x^n\right ) \log \left (f (g+h x)^q\right )+6 b^2 e g n \log \left (c x^n\right ) \log \left (f (g+h x)^q\right )+3 b^2 e g \log ^2\left (c x^n\right ) \log \left (f (g+h x)^q\right )+6 a b e h n q x \log (x) \log \left (1+\frac {h x}{g}\right )+6 b^2 e h n^2 q x \log (x) \log \left (1+\frac {h x}{g}\right )-3 b^2 e h n^2 q x \log ^2(x) \log \left (1+\frac {h x}{g}\right )+6 b^2 e h n q x \log (x) \log \left (c x^n\right ) \log \left (1+\frac {h x}{g}\right )+6 b e h n q x \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )-6 b^2 e h n^2 q x \operatorname {PolyLog}\left (3,-\frac {h x}{g}\right )}{3 g x} \] Input:
Integrate[((a + b*Log[c*x^n])^2*(d + e*Log[f*(g + h*x)^q]))/x^2,x]
Output:
-1/3*(3*a^2*d*g + 6*a*b*d*g*n + 6*b^2*d*g*n^2 - 3*a^2*e*h*q*x*Log[x] - 6*a *b*e*h*n*q*x*Log[x] - 6*b^2*e*h*n^2*q*x*Log[x] + 3*a*b*e*h*n*q*x*Log[x]^2 + 3*b^2*e*h*n^2*q*x*Log[x]^2 - b^2*e*h*n^2*q*x*Log[x]^3 + 6*a*b*d*g*Log[c* x^n] + 6*b^2*d*g*n*Log[c*x^n] - 6*a*b*e*h*q*x*Log[x]*Log[c*x^n] - 6*b^2*e* h*n*q*x*Log[x]*Log[c*x^n] + 3*b^2*e*h*n*q*x*Log[x]^2*Log[c*x^n] + 3*b^2*d* g*Log[c*x^n]^2 - 3*b^2*e*h*q*x*Log[x]*Log[c*x^n]^2 + 3*a^2*e*h*q*x*Log[g + h*x] + 6*a*b*e*h*n*q*x*Log[g + h*x] + 6*b^2*e*h*n^2*q*x*Log[g + h*x] - 6* a*b*e*h*n*q*x*Log[x]*Log[g + h*x] - 6*b^2*e*h*n^2*q*x*Log[x]*Log[g + h*x] + 3*b^2*e*h*n^2*q*x*Log[x]^2*Log[g + h*x] + 6*a*b*e*h*q*x*Log[c*x^n]*Log[g + h*x] + 6*b^2*e*h*n*q*x*Log[c*x^n]*Log[g + h*x] - 6*b^2*e*h*n*q*x*Log[x] *Log[c*x^n]*Log[g + h*x] + 3*b^2*e*h*q*x*Log[c*x^n]^2*Log[g + h*x] + 3*a^2 *e*g*Log[f*(g + h*x)^q] + 6*a*b*e*g*n*Log[f*(g + h*x)^q] + 6*b^2*e*g*n^2*L og[f*(g + h*x)^q] + 6*a*b*e*g*Log[c*x^n]*Log[f*(g + h*x)^q] + 6*b^2*e*g*n* Log[c*x^n]*Log[f*(g + h*x)^q] + 3*b^2*e*g*Log[c*x^n]^2*Log[f*(g + h*x)^q] + 6*a*b*e*h*n*q*x*Log[x]*Log[1 + (h*x)/g] + 6*b^2*e*h*n^2*q*x*Log[x]*Log[1 + (h*x)/g] - 3*b^2*e*h*n^2*q*x*Log[x]^2*Log[1 + (h*x)/g] + 6*b^2*e*h*n*q* x*Log[x]*Log[c*x^n]*Log[1 + (h*x)/g] + 6*b*e*h*n*q*x*(a + b*n + b*Log[c*x^ n])*PolyLog[2, -((h*x)/g)] - 6*b^2*e*h*n^2*q*x*PolyLog[3, -((h*x)/g)])/(g* x)
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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2891 |
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2}dx\) |
Input:
Int[((a + b*Log[c*x^n])^2*(d + e*Log[f*(g + h*x)^q]))/x^2,x]
Output:
$Aborted
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))^(q_.)*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^p*(f + g* Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 105.11 (sec) , antiderivative size = 4127, normalized size of antiderivative = 13.49
Input:
int((a+b*ln(c*x^n))^2*(d+e*ln(f*(h*x+g)^q))/x^2,x,method=_RETURNVERBOSE)
Output:
2*q*h*b^2*e*n^2/g*ln(h*x+g)*ln(-h*x/g)+1/4*q*h*e/g*ln(h*x+g)*Pi^2*csgn(I*c *x^n)^6*b^2-2*q*h*e/g*ln(h*x+g)*ln(c)*b^2*n+2*q*h*e/g*ln(x)*a*n*b-2*q*h*b* e*ln(x^n)/g*ln(h*x+g)*a+I*q*h*e/g*ln(x)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*b^2 *n+I*q*h*e/g*ln(x)*Pi*csgn(I*c*x^n)^2*csgn(I*c)*b^2*n+I*q*h*e/g*ln(x)*Pi*c sgn(I*x^n)*csgn(I*c*x^n)^2*a*b+I*q*h*e/g*ln(x)*Pi*csgn(I*c*x^n)^2*csgn(I*c )*a*b-1/2*I*q*h*b^2*e*n/g*ln(x)^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*q*h*e/g *ln(h*x+g)*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*b^2-1/2*I*q*h*b^2*e*n/g*ln (x)^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)-I*q*h*b^2*e*n/g*ln(h*x+g)*ln(-h*x/g)*Pi *csgn(I*c*x^n)^3-I*q*h*b^2*e*ln(x^n)/g*ln(h*x+g)*Pi*csgn(I*x^n)*csgn(I*c*x ^n)^2-I*q*h*b^2*e*ln(x^n)/g*ln(h*x+g)*Pi*csgn(I*c*x^n)^2*csgn(I*c)-I*q*h*e /g*ln(h*x+g)*ln(c)*Pi*csgn(I*c*x^n)^2*csgn(I*c)*b^2+1/2*I*q*h*b^2*e*n/g*ln (x)^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*q*h*e/g*ln(x)*Pi*csgn(I*x^n )*csgn(I*c*x^n)*csgn(I*c)*a*b-I*q*h*b^2*e*n/g*dilog(-h*x/g)*Pi*csgn(I*x^n) *csgn(I*c*x^n)*csgn(I*c)+(-b^2*e/x*ln(x^n)^2-(I*Pi*b^2*e*csgn(I*x^n)*csgn( I*c*x^n)^2-I*Pi*b^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b^2*e*csgn( I*c*x^n)^3+I*Pi*b^2*e*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)*b^2*e+2*n*b^2*e+2* e*a*b)/x*ln(x^n)-1/4*(8*a*b*ln(c)*e+4*a^2*e+8*a*n*b*e-Pi^2*b^2*e*csgn(I*c* x^n)^6-Pi^2*b^2*e*csgn(I*c*x^n)^4*csgn(I*c)^2-Pi^2*b^2*e*csgn(I*x^n)^2*csg n(I*c*x^n)^4+2*Pi^2*b^2*e*csgn(I*x^n)*csgn(I*c*x^n)^5+8*ln(c)*b^2*e*n+2*Pi ^2*b^2*e*csgn(I*c*x^n)^5*csgn(I*c)-4*I*Pi*ln(c)*b^2*e*csgn(I*x^n)*csgn(...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=\int { \frac {{\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*x^n))^2*(d+e*log(f*(h*x+g)^q))/x^2,x, algorithm="fric as")
Output:
integral((b^2*d*log(c*x^n)^2 + 2*a*b*d*log(c*x^n) + a^2*d + (b^2*e*log(c*x ^n)^2 + 2*a*b*e*log(c*x^n) + a^2*e)*log((h*x + g)^q*f))/x^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*x**n))**2*(d+e*ln(f*(h*x+g)**q))/x**2,x)
Output:
Timed out
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=\int { \frac {{\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*x^n))^2*(d+e*log(f*(h*x+g)^q))/x^2,x, algorithm="maxi ma")
Output:
-a^2*e*h*q*(log(h*x + g)/g - log(x)/g) - 2*b^2*d*(n^2/x + n*log(c*x^n)/x) - b^2*d*log(c*x^n)^2/x - 2*a*b*d*n/x - a^2*e*log((h*x + g)^q*f)/x - 2*a*b* d*log(c*x^n)/x - a^2*d/x - ((b^2*e*h*q*x*log(h*x + g) - b^2*e*h*q*x*log(x) + b^2*e*g*log(f))*log(x^n)^2 + (b^2*e*g*log(x^n)^2 + 2*(e*g*n + e*g*log(c ))*a*b + (2*e*g*n^2 + 2*e*g*n*log(c) + e*g*log(c)^2)*b^2 + 2*(a*b*e*g + (e *g*n + e*g*log(c))*b^2)*log(x^n))*log((h*x + g)^q))/(g*x) + integrate((b^2 *e*g^2*log(c)^2*log(f) + 2*a*b*e*g^2*log(c)*log(f) + (2*(e*g*h*n*q + (g*h* q + g*h*log(f))*e*log(c))*a*b + (2*e*g*h*n^2*q + 2*e*g*h*n*q*log(c) + (g*h *q + g*h*log(f))*e*log(c)^2)*b^2)*x + 2*(a*b*e*g^2*log(f) + (e*g^2*n*log(f ) + e*g^2*log(c)*log(f))*b^2 + ((g*h*q + g*h*log(f))*a*b*e + ((g*h*q + g*h *log(f))*e*log(c) + (g*h*n*q + g*h*n*log(f))*e)*b^2)*x + (b^2*e*h^2*n*q*x^ 2 + b^2*e*g*h*n*q*x)*log(h*x + g) - (b^2*e*h^2*n*q*x^2 + b^2*e*g*h*n*q*x)* log(x))*log(x^n))/(g*h*x^3 + g^2*x^2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=\int { \frac {{\left (e \log \left ({\left (h x + g\right )}^{q} f\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*x^n))^2*(d+e*log(f*(h*x+g)^q))/x^2,x, algorithm="giac ")
Output:
integrate((e*log((h*x + g)^q*f) + d)*(b*log(c*x^n) + a)^2/x^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=\int \frac {\left (d+e\,\ln \left (f\,{\left (g+h\,x\right )}^q\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \] Input:
int(((d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n))^2)/x^2,x)
Output:
int(((d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n))^2)/x^2, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f (g+h x)^q\right )\right )}{x^2} \, dx=\frac {-\mathrm {log}\left (x^{n} c \right )^{2} b^{2} e g q +\mathrm {log}\left (x \right ) a^{2} e h q x -2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) a b e g -2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) b^{2} e g n -2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a b e g n -2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b^{2} e h \,n^{2} x -2 \,\mathrm {log}\left (x^{n} c \right ) a b e g q -4 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e g n q -2 a b e g n q -2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a b e h n x +2 \,\mathrm {log}\left (x \right ) b^{2} e h \,n^{2} q x -\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a^{2} e g -\mathrm {log}\left (x^{n} c \right )^{2} b^{2} d g -a^{2} d g -2 b^{2} d g \,n^{2}-2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b^{2} e g \,n^{2}-2 \,\mathrm {log}\left (x^{n} c \right ) a b d g -2 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d g n -2 a b d g n -4 b^{2} e g \,n^{2} q -2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{3}+g \,x^{2}}d x \right ) a b e \,g^{2} q x -2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{3}+g \,x^{2}}d x \right ) b^{2} e \,g^{2} n q x -\left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{h \,x^{3}+g \,x^{2}}d x \right ) b^{2} e \,g^{2} q x +2 \,\mathrm {log}\left (x \right ) a b e h n q x -\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e g -\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a^{2} e h x}{g x} \] Input:
int((a+b*log(c*x^n))^2*(d+e*log(f*(h*x+g)^q))/x^2,x)
Output:
( - int(log(x**n*c)**2/(g*x**2 + h*x**3),x)*b**2*e*g**2*q*x - 2*int(log(x* *n*c)/(g*x**2 + h*x**3),x)*a*b*e*g**2*q*x - 2*int(log(x**n*c)/(g*x**2 + h* x**3),x)*b**2*e*g**2*n*q*x - log((g + h*x)**q*f)*log(x**n*c)**2*b**2*e*g - 2*log((g + h*x)**q*f)*log(x**n*c)*a*b*e*g - 2*log((g + h*x)**q*f)*log(x** n*c)*b**2*e*g*n - log((g + h*x)**q*f)*a**2*e*g - log((g + h*x)**q*f)*a**2* e*h*x - 2*log((g + h*x)**q*f)*a*b*e*g*n - 2*log((g + h*x)**q*f)*a*b*e*h*n* x - 2*log((g + h*x)**q*f)*b**2*e*g*n**2 - 2*log((g + h*x)**q*f)*b**2*e*h*n **2*x - log(x**n*c)**2*b**2*d*g - log(x**n*c)**2*b**2*e*g*q - 2*log(x**n*c )*a*b*d*g - 2*log(x**n*c)*a*b*e*g*q - 2*log(x**n*c)*b**2*d*g*n - 4*log(x** n*c)*b**2*e*g*n*q + log(x)*a**2*e*h*q*x + 2*log(x)*a*b*e*h*n*q*x + 2*log(x )*b**2*e*h*n**2*q*x - a**2*d*g - 2*a*b*d*g*n - 2*a*b*e*g*n*q - 2*b**2*d*g* n**2 - 4*b**2*e*g*n**2*q)/(g*x)