\(\int \frac {x^5 (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [320]

Optimal result

Integrand size = 29, antiderivative size = 936 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 3.31 (sec) , antiderivative size = 1254, normalized size of antiderivative = 1.34 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[(x^5*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2)^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 3.12 (sec) , antiderivative size = 936, 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.

\(\displaystyle \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx\)

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^5*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2)^2,x]
 

Output:

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

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]

\[\int \frac {x^{5} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g \,x^{2}+f \right )^{2}}d x\]

Input:

int(x^5*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Output:

int(x^5*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Fricas [F]

\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="fricas")
 

Output:

integral((b^2*x^5*log((e*x + d)^n*c)^2 + 2*a*b*x^5*log((e*x + d)^n*c) + a^ 
2*x^5)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**5*(a+b*ln(c*(e*x+d)**n))**2/(g*x**2+f)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="maxima")
 

Output:

-1/2*a^2*(f^2/(g^4*x^2 + f*g^3) - x^2/g^2 + 2*f*log(g*x^2 + f)/g^3) + inte 
grate((b^2*x^5*log((e*x + d)^n)^2 + 2*(b^2*log(c) + a*b)*x^5*log((e*x + d) 
^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^5)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Giac [F]

\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="giac")
 

Output:

integrate((b*log((e*x + d)^n*c) + a)^2*x^5/(g*x^2 + f)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^5\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \] Input:

int((x^5*(a + b*log(c*(d + e*x)^n))^2)/(f + g*x^2)^2,x)
 

Output:

int((x^5*(a + b*log(c*(d + e*x)^n))^2)/(f + g*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {too large to display} \] Input:

int(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Output:

(24*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d**3*e*f*g*n**2 + 24 
*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d**3*e*g**2*n**2*x**2 - 
 12*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d*e**3*f**2*n**2 - 1 
2*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d*e**3*f*g*n**2*x**2 + 
 24*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d**3*e*f*g*n**3 + 2 
4*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d**3*e*g**2*n**3*x**2 
 - 66*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d*e**3*f**2*n**3 
- 66*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d*e**3*f*g*n**3*x* 
*2 - 36*int(log((d + e*x)**n*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + 
e*f**2*x + 2*e*f*g*x**3 + e*g**2*x**5),x)*b**2*d**4*e*f**3*g**2*n - 36*int 
(log((d + e*x)**n*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x + 
2*e*f*g*x**3 + e*g**2*x**5),x)*b**2*d**4*e*f**2*g**3*n*x**2 - 48*int(log(( 
d + e*x)**n*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f* 
g*x**3 + e*g**2*x**5),x)*b**2*d**2*e**3*f**4*g*n - 48*int(log((d + e*x)**n 
*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g*x**3 + e* 
g**2*x**5),x)*b**2*d**2*e**3*f**3*g**2*n*x**2 - 12*int(log((d + e*x)**n*c) 
**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g*x**3 + e*g** 
2*x**5),x)*b**2*e**5*f**5*n - 12*int(log((d + e*x)**n*c)**2/(d*f**2 + 2*d* 
f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g*x**3 + e*g**2*x**5),x)*b**2*e* 
*5*f**4*g*n*x**2 - 48*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x**2 + ...