\(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x) (h+i x)} \, dx\) [227]

Optimal result

Integrand size = 31, antiderivative size = 264 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g h-f i}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (h+i x)}{e h-d i}\right )}{g h-f i}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \] Output:

(a+b*ln(c*(e*x+d)^n))^2*ln(e*(g*x+f)/(-d*g+e*f))/(-f*i+g*h)-(a+b*ln(c*(e*x 
+d)^n))^2*ln(e*(i*x+h)/(-d*i+e*h))/(-f*i+g*h)+2*b*n*(a+b*ln(c*(e*x+d)^n))* 
polylog(2,-g*(e*x+d)/(-d*g+e*f))/(-f*i+g*h)-2*b*n*(a+b*ln(c*(e*x+d)^n))*po 
lylog(2,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)-2*b^2*n^2*polylog(3,-g*(e*x+d)/( 
-d*g+e*f))/(-f*i+g*h)+2*b^2*n^2*polylog(3,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h 
)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)-\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (h+i x)+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-\operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )\right )+b^2 n^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\log ^2(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+2 \operatorname {PolyLog}\left (3,\frac {i (d+e x)}{-e h+d i}\right )\right )}{g h-f i} \] Input:

Integrate[(a + b*Log[c*(d + e*x)^n])^2/((f + g*x)*(h + i*x)),x]
 

Output:

((a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x] - (a - b*n*L 
og[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[h + i*x] + 2*b*n*(a - b*n*Log[d 
+ e*x] + b*Log[c*(d + e*x)^n])*(Log[d + e*x]*(Log[(e*(f + g*x))/(e*f - d*g 
)] - Log[(e*(h + i*x))/(e*h - d*i)]) + PolyLog[2, (g*(d + e*x))/(-(e*f) + 
d*g)] - PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)]) + b^2*n^2*(Log[d + e*x]^ 
2*Log[(e*(f + g*x))/(e*f - d*g)] - Log[d + e*x]^2*Log[(e*(h + i*x))/(e*h - 
 d*i)] + 2*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*Log[d 
 + e*x]*PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)] - 2*PolyLog[3, (g*(d + e* 
x))/(-(e*f) + d*g)] + 2*PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)]))/(g*h - 
f*i)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2865, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx\)

\(\Big \downarrow \) 2865

\(\displaystyle \int \left (\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (g h-f i)}-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(h+i x) (g h-f i)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}\)

Input:

Int[(a + b*Log[c*(d + e*x)^n])^2/((f + g*x)*(h + i*x)),x]
 

Output:

((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i) 
- ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i 
) + (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d* 
g))])/(g*h - f*i) - (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((i*(d + 
 e*x))/(e*h - d*i))])/(g*h - f*i) - (2*b^2*n^2*PolyLog[3, -((g*(d + e*x))/ 
(e*f - d*g))])/(g*h - f*i) + (2*b^2*n^2*PolyLog[3, -((i*(d + e*x))/(e*h - 
d*i))])/(g*h - f*i)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 7.83 (sec) , antiderivative size = 1427, normalized size of antiderivative = 5.41

method	result	size
risch	\(\text {Expression too large to display}\)	\(1427\)

Input:

int((a+b*ln(c*(e*x+d)^n))^2/(g*x+f)/(i*x+h),x,method=_RETURNVERBOSE)
 

Output:

b^2/(f*i-g*h)*ln((e*x+d)*i-d*i+e*h)*ln(e*x+d)^2*n^2-2*b^2/(f*i-g*h)*ln((e* 
x+d)*i-d*i+e*h)*ln((e*x+d)^n)*ln(e*x+d)*n+b^2/(f*i-g*h)*ln((e*x+d)*i-d*i+e 
*h)*ln((e*x+d)^n)^2-b^2/(f*i-g*h)*ln(g*(e*x+d)-d*g+e*f)*ln(e*x+d)^2*n^2+2* 
b^2/(f*i-g*h)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)*ln(e*x+d)*n-b^2/(f*i-g*h 
)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)^2+b^2*n^2/(f*i-g*h)*ln(e*x+d)^2*ln(1 
+i*(e*x+d)/(-d*i+e*h))+2*b^2*n^2/(f*i-g*h)*ln(e*x+d)*polylog(2,-i*(e*x+d)/ 
(-d*i+e*h))-2*b^2*n^2/(f*i-g*h)*polylog(3,-i*(e*x+d)/(-d*i+e*h))-b^2*n^2/( 
f*i-g*h)*ln(e*x+d)^2*ln(1+g*(e*x+d)/(-d*g+e*f))-2*b^2*n^2/(f*i-g*h)*ln(e*x 
+d)*polylog(2,-g*(e*x+d)/(-d*g+e*f))+2*b^2*n^2/(f*i-g*h)*polylog(3,-g*(e*x 
+d)/(-d*g+e*f))-2*b^2*n^2/(f*i-g*h)*dilog(((e*x+d)*i-d*i+e*h)/(-d*i+e*h))* 
ln(e*x+d)+2*b^2*n/(f*i-g*h)*dilog(((e*x+d)*i-d*i+e*h)/(-d*i+e*h))*ln((e*x+ 
d)^n)-2*b^2*n^2/(f*i-g*h)*ln(e*x+d)^2*ln(((e*x+d)*i-d*i+e*h)/(-d*i+e*h))+2 
*b^2*n/(f*i-g*h)*ln(e*x+d)*ln(((e*x+d)*i-d*i+e*h)/(-d*i+e*h))*ln((e*x+d)^n 
)+2*b^2*n^2/(f*i-g*h)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln(e*x+d)-2*b^ 
2*n/(f*i-g*h)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln((e*x+d)^n)+2*b^2*n^ 
2/(f*i-g*h)*ln(e*x+d)^2*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))-2*b^2*n/(f*i-g* 
h)*ln(e*x+d)*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln((e*x+d)^n)+(I*b*Pi*csgn 
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x 
+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2 
*csgn(I*c)+2*b*ln(c)+2*a)*b*(ln((e*x+d)^n)/(f*i-g*h)*ln(i*x+h)-ln((e*x+...
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )} {\left (i x + h\right )}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))^2/(g*x+f)/(i*x+h),x, algorithm="fricas")
 

Output:

integral((b^2*log((e*x + d)^n*c)^2 + 2*a*b*log((e*x + d)^n*c) + a^2)/(g*i* 
x^2 + f*h + (g*h + f*i)*x), x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}{\left (f + g x\right ) \left (h + i x\right )}\, dx \] Input:

integrate((a+b*ln(c*(e*x+d)**n))**2/(g*x+f)/(i*x+h),x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))**2/((f + g*x)*(h + i*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )} {\left (i x + h\right )}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))^2/(g*x+f)/(i*x+h),x, algorithm="maxima")
 

Output:

a^2*(log(g*x + f)/(g*h - f*i) - log(i*x + h)/(g*h - f*i)) + integrate((b^2 
*log((e*x + d)^n)^2 + b^2*log(c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*l 
og((e*x + d)^n))/(g*i*x^2 + f*h + (g*h + f*i)*x), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )} {\left (i x + h\right )}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))^2/(g*x+f)/(i*x+h),x, algorithm="giac")
 

Output:

integrate((b*log((e*x + d)^n*c) + a)^2/((g*x + f)*(i*x + h)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{\left (f+g\,x\right )\,\left (h+i\,x\right )} \,d x \] Input:

int((a + b*log(c*(d + e*x)^n))^2/((f + g*x)*(h + i*x)),x)
 

Output:

int((a + b*log(c*(d + e*x)^n))^2/((f + g*x)*(h + i*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x) (h+i x)} \, dx=\int \frac {{\left (\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b +a \right )}^{2}}{\left (g x +f \right ) \left (i x +h \right )}d x \] Input:

int((a+b*log(c*(e*x+d)^n))^2/(g*x+f)/(i*x+h),x)
 

Output:

int((a+b*log(c*(e*x+d)^n))^2/(g*x+f)/(i*x+h),x)