\(\int \frac {(a+b \log (c (d+e x)^n))^2}{x^2 (f+g x^2)^2} \, dx\) [328]

Optimal result

Integrand size = 29, antiderivative size = 919 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 \left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Output:

2*b*e*n*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))/d/f^2-(e*x+d)*(a+b*ln(c*(e*x+d)^n 
))^2/d/f^2/x+1/4*g*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/f^2/(e*(-f)^(1/2)+d*g^( 
1/2))/((-f)^(1/2)-g^(1/2)*x)+1/4*g*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/f^2/(e* 
(-f)^(1/2)-d*g^(1/2))/((-f)^(1/2)+g^(1/2)*x)+1/2*b*e*g^(1/2)*n*(a+b*ln(c*( 
e*x+d)^n))*ln(e*((-f)^(1/2)-g^(1/2)*x)/(e*(-f)^(1/2)+d*g^(1/2)))/f^2/(e*(- 
f)^(1/2)+d*g^(1/2))-3/4*g^(1/2)*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)-g 
^(1/2)*x)/(e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(5/2)+1/2*b*e*g^(1/2)*n*(a+b*ln(c 
*(e*x+d)^n))*ln(e*((-f)^(1/2)+g^(1/2)*x)/(e*(-f)^(1/2)-d*g^(1/2)))/f/(e*(- 
f)^(3/2)+d*f*g^(1/2))+3/4*g^(1/2)*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2) 
+g^(1/2)*x)/(e*(-f)^(1/2)-d*g^(1/2)))/(-f)^(5/2)+1/2*b^2*e*g^(1/2)*n^2*pol 
ylog(2,-g^(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/f/(e*(-f)^(3/2)+d*f*g^(1 
/2))+3/2*b*g^(1/2)*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-g^(1/2)*(e*x+d)/(e*( 
-f)^(1/2)-d*g^(1/2)))/(-f)^(5/2)+1/2*b^2*e*g^(1/2)*n^2*polylog(2,g^(1/2)*( 
e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/f^2/(e*(-f)^(1/2)+d*g^(1/2))-3/2*b*g^(1/2 
)*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2 
)))/(-f)^(5/2)+2*b^2*e*n^2*polylog(2,1+e*x/d)/d/f^2-3/2*b^2*g^(1/2)*n^2*po 
lylog(3,-g^(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/(-f)^(5/2)+3/2*b^2*g^(1 
/2)*n^2*polylog(3,g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(5/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.10 (sec) , antiderivative size = 1304, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 \left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((-4*Sqrt[f]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/x - (2*Sqrt[ 
f]*g*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) - 6*Sq 
rt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x 
)^n])^2 + 2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((4*Sqrt[f]* 
(e*x*Log[x] - (d + e*x)*Log[d + e*x]))/(d*x) - (Sqrt[f]*Sqrt[g]*(Sqrt[g]*( 
d + e*x)*Log[d + e*x] + I*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[I*Sqrt[f] - Sqrt[g 
]*x]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + (Sqrt[f]*Sqrt 
[g]*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]) + e*(I*Sqrt[f] + Sqrt[g]*x)*Log[I*S 
qrt[f] + Sqrt[g]*x]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) 
+ (3*I)*Sqrt[g]*(Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - 
 I*d*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(d + e*x))/(e*Sqrt[f] - I*d*Sqrt 
[g])]) - (3*I)*Sqrt[g]*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sq 
rt[f] + I*d*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d* 
Sqrt[g])])) + b^2*n^2*((Sqrt[f]*Sqrt[g]*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]^ 
2) + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]* 
x))/(e*Sqrt[f] + I*d*Sqrt[g])] + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I 
*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]))/((e*Sqrt[f] + I*d*Sqrt[g] 
)*(Sqrt[f] - I*Sqrt[g]*x)) - (Sqrt[f]*Sqrt[g]*(Log[d + e*x]*(Sqrt[g]*(d + 
e*x)*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f] + I*Sq 
rt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)...
 

Rubi [A] (verified)

Time = 3.31 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2863, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 e \sqrt {g} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {b^2 e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {2 b^2 e \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) n^2}{d f^2}-\frac {3 b^2 \sqrt {g} \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{5/2}}+\frac {3 b^2 \sqrt {g} \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{5/2}}+\frac {2 b e \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) n}{d f^2}+\frac {b e \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right )}+\frac {b e \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 b \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 (-f)^{5/2}}-\frac {3 b \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 (-f)^{5/2}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 (-f)^{5/2}}+\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}\)

Input:

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

Output:

(2*b*e*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f^2) - ((d + e*x)* 
(a + b*Log[c*(d + e*x)^n])^2)/(d*f^2*x) + (g*(d + e*x)*(a + b*Log[c*(d + e 
*x)^n])^2)/(4*f^2*(e*Sqrt[-f] + d*Sqrt[g])*(Sqrt[-f] - Sqrt[g]*x)) + (g*(d 
 + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f^2*(e*Sqrt[-f] - d*Sqrt[g])*(Sqr 
t[-f] + Sqrt[g]*x)) + (b*e*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sq 
rt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e*Sqrt[-f] + d*Sqr 
t[g])) - (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g 
]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(5/2)) + (b*e*Sqrt[g]*n*(a + b*Lo 
g[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])] 
)/(2*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n 
])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*(-f)^(5/ 
2)) + (b^2*e*Sqrt[g]*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d* 
Sqrt[g]))])/(2*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*b*Sqrt[g]*n*(a + b*Log 
[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]) 
)])/(2*(-f)^(5/2)) + (b^2*e*Sqrt[g]*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e* 
Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (3*b*Sqrt[g]*n* 
(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d* 
Sqrt[g])])/(2*(-f)^(5/2)) + (2*b^2*e*n^2*PolyLog[2, 1 + (e*x)/d])/(d*f^2) 
- (3*b^2*Sqrt[g]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt 
[g]))])/(2*(-f)^(5/2)) + (3*b^2*Sqrt[g]*n^2*PolyLog[3, (Sqrt[g]*(d + e*...
 

Defintions of rubi rules used

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{2} \left (g \,x^{2}+f \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*log(c*(e*x+d)^n))^2/x^2/(g*x^2+f)^2,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^2* 
x^6 + 2*f*g*x^4 + f^2*x^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*a^2*((3*g*x^2 + 2*f)/(f^2*g*x^3 + f^3*x) + 3*g*arctan(g*x/sqrt(f*g))/ 
(sqrt(f*g)*f^2)) + 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)*log((e*x + d)^n))/(g^2*x^6 + 2*f*g*x^4 + f 
^2*x^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d**3*f*g*x - 3*sq 
rt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d**3*g**2*x**3 - 3*sqrt(g 
)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d*e**2*f**2*x - 3*sqrt(g)*sqr 
t(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d*e**2*f*g*x**3 - 4*sqrt(g)*sqrt(f 
)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d*e**2*f**2*n*x - 4*sqrt(g)*sqrt(f)*at 
an((g*x)/(sqrt(g)*sqrt(f)))*a*b*d*e**2*f*g*n*x**3 + 2*int(log((d + e*x)**n 
*c)**2/(f**2*x**2 + 2*f*g*x**4 + g**2*x**6),x)*b**2*d**3*f**4*g*x + 2*int( 
log((d + e*x)**n*c)**2/(f**2*x**2 + 2*f*g*x**4 + g**2*x**6),x)*b**2*d**3*f 
**3*g**2*x**3 + 2*int(log((d + e*x)**n*c)**2/(f**2*x**2 + 2*f*g*x**4 + g** 
2*x**6),x)*b**2*d*e**2*f**5*x + 2*int(log((d + e*x)**n*c)**2/(f**2*x**2 + 
2*f*g*x**4 + g**2*x**6),x)*b**2*d*e**2*f**4*g*x**3 - 12*int(log((d + e*x)* 
*n*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x)*a*b*d**3*f**3*g**2*x - 12*int(log 
((d + e*x)**n*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x)*a*b*d**3*f**2*g**3*x** 
3 - 12*int(log((d + e*x)**n*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x)*a*b*d*e* 
*2*f**4*g*x - 12*int(log((d + e*x)**n*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x 
)*a*b*d*e**2*f**3*g**2*x**3 - 4*log(d + e*x)*a*b*e**3*f**3*n*x - 4*log(d + 
 e*x)*a*b*e**3*f**2*g*n*x**3 - 2*log(f + g*x**2)*a*b*d**2*e*f**2*g*n*x - 2 
*log(f + g*x**2)*a*b*d**2*e*f*g**2*n*x**3 - 4*log((d + e*x)**n*c)*a*b*d**3 
*f**2*g - 4*log((d + e*x)**n*c)*a*b*d*e**2*f**3 + 4*log(x)*a*b*d**2*e*f**2 
*g*n*x + 4*log(x)*a*b*d**2*e*f*g**2*n*x**3 + 4*log(x)*a*b*e**3*f**3*n*x...