\(\int \frac {x^4 (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [325]

Optimal result

Integrand size = 29, antiderivative size = 897 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.85 (sec) , antiderivative size = 1237, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[(x^4*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2)^2,x]
 

Output:

(4*Sqrt[g]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + (2*f*Sqrt[g 
]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) - 6*Sqrt[ 
f]*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[g]*(d 
+ e*x)*(-1 + Log[d + e*x]))/e + (f*(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)) + (f*(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*Sqr 
t[g])*(Sqrt[f] - I*Sqrt[g]*x)) + (3*I)*Sqrt[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])]) - (3*I)*Sqrt[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*((4*Sqrt[g]*(2*e*x - 
2*(d + e*x)*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2))/e - (f*(-(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] + Sqr 
t[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*(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*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I)*e*(Sqrt[f] + I*Sqr...
 

Rubi [A] (verified)

Time = 3.46 (sec) , antiderivative size = 897, 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.

\(\displaystyle \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx\)

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^4*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2)^2,x]
 

Output:

(-2*a*b*n*x)/g^2 + (2*b^2*n^2*x)/g^2 - (2*b^2*n*(d + e*x)*Log[c*(d + e*x)^ 
n])/(e*g^2) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) - (f*(d + e 
*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^2*(Sqrt[-f 
] - Sqrt[g]*x)) - (f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f 
] - d*Sqrt[g])*g^2*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*f*n*(a + b*Log[c*(d + e* 
x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqr 
t[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])^2*Log 
[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b*e* 
f*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] 
- d*Sqrt[g])])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*Sqrt[-f]*(a + b*L 
og[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g 
])])/(4*g^(5/2)) + (b^2*e*f*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[- 
f] - d*Sqrt[g]))])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*b*Sqrt[-f]*n* 
(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - 
d*Sqrt[g]))])/(2*g^(5/2)) - (b^2*e*f*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e 
*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*b*Sqrt[ 
-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f 
] + d*Sqrt[g])])/(2*g^(5/2)) + (3*b^2*Sqrt[-f]*n^2*PolyLog[3, -((Sqrt[g]*( 
d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(5/2)) - (3*b^2*Sqrt[-f]*n^2*Po 
lyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2))
 

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]

\[\int \frac {x^{4} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g \,x^{2}+f \right )^{2}}d x\]

Input:

int(x^4*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Output:

int(x^4*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Fricas [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^4*(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^4/(g*x^2 + f)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^4\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {too large to display} \] Input:

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

Output:

( - 3*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d**2*e*f*g - 3*sq 
rt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d**2*e*g**2*x**2 - 3*sqrt 
(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*e**3*f**2 - 3*sqrt(g)*sqrt( 
f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*e**3*f*g*x**2 - 8*sqrt(g)*sqrt(f)*at 
an((g*x)/(sqrt(g)*sqrt(f)))*a*b*e**3*f**2*n - 8*sqrt(g)*sqrt(f)*atan((g*x) 
/(sqrt(g)*sqrt(f)))*a*b*e**3*f*g*n*x**2 + 4*sqrt(g)*sqrt(f)*atan((g*x)/(sq 
rt(g)*sqrt(f)))*b**2*d**2*e*f*g*n**2 + 4*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt( 
g)*sqrt(f)))*b**2*d**2*e*g**2*n**2*x**2 - 8*sqrt(g)*sqrt(f)*atan((g*x)/(sq 
rt(g)*sqrt(f)))*b**2*e**3*f**2*n**2 - 8*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g 
)*sqrt(f)))*b**2*e**3*f*g*n**2*x**2 - 6*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**3*e*f**3*g**2 - 6*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**3*e*f* 
*2*g**3*x**2 - 6*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*e**3*f**4*g - 6* 
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*e**3*f**3*g**2*x**2 - 12*int(log( 
(d + e*x)**n*c)/(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)*a*b*d**3*e*f**3*g**2 - 12*int(log((d + e*x)**n*c)/( 
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*...