\(\int \frac {x^3 (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [321]

Optimal result

Integrand size = 29, antiderivative size = 739 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=-\frac {e^2 f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^2 \left (f+g x^2\right )}+\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\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 )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2}+\frac {b e \left (e f-d \sqrt {-f} \sqrt {g}\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 )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}-\frac {b^2 e \sqrt {-f} \left (e \sqrt {-f}+d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {b 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 )}{g^2}+\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {b 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 )}{g^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^2} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.61 (sec) , antiderivative size = 1103, normalized size of antiderivative = 1.49 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((2*f*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + 2*(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])*((Sqrt[f]*((-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)) + (Sqrt[f]*(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)) + 2*(Log[d + e*x]*Lo 
g[(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])]) + 2*(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*(2*Log[d + e*x]^2*Log[ 
1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]^2*L 
og[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + (Sqrt[f]*(Log[d + 
e*x]*(I*Sqrt[g]*(d + e*x)*Log[d + e*x] + 2*e*(Sqrt[f] - I*Sqrt[g]*x)*Log[( 
e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])]) + 2*e*(Sqrt[f] - I* 
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)) + 4*Log[d + e*x]*PolyLo 
g[2, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + 4*Log[d + e*x]*Po 
lyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + (Sqrt[f]*(Log[d 
+ e*x]*((-I)*Sqrt[g]*(d + e*x)*Log[d + e*x] + 2*e*(Sqrt[f] + I*Sqrt[g]*...
 

Rubi [A] (verified)

Time = 2.54 (sec) , antiderivative size = 739, 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.

\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx\)

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(e^2*f*(a + b*Log[c*(d + e*x)^n])^2)/(g^2*(e^2*f + d^2*g)) + (f*(a + 
b*Log[c*(d + e*x)^n])^2)/(2*g^2*(f + g*x^2)) + (b*e*(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^2*(e^2*f + d^2*g)) + ((a + b*Log[c*(d + e*x)^n])^2* 
Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2) + (b*e*( 
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^2*(e^2*f + d^2*g)) + ((a + b*L 
og[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g 
])])/(2*g^2) - (b^2*e*Sqrt[-f]*(e*Sqrt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -(( 
Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^2*(e^2*f + d^2*g)) + ( 
b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f 
] - d*Sqrt[g]))])/g^2 + (b^2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n^2*PolyLog[2, ( 
Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2*(e^2*f + d^2*g)) + (b 
*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + 
 d*Sqrt[g])])/g^2 - (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] 
- d*Sqrt[g]))])/g^2 - (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] 
+ d*Sqrt[g])])/g^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]

\[\int \frac {x^{3} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g \,x^{2}+f \right )^{2}}d x\]

Input:

int(x^3*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Output:

int(x^3*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Fricas [F]

\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{3}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^3*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="fricas")
 

Output:

integral((b^2*x^3*log((e*x + d)^n*c)^2 + 2*a*b*x^3*log((e*x + d)^n*c) + a^ 
2*x^3)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**3*(a+b*ln(c*(e*x+d)**n))**2/(g*x**2+f)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{3}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^3*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{3}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \] Input:

integrate(x^3*(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^3/(g*x^2 + f)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^3\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \] Input:

int((x^3*(a + b*log(c*(d + e*x)^n))^2)/(f + g*x^2)^2,x)
 

Output:

int((x^3*(a + b*log(c*(d + e*x)^n))^2)/(f + g*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {too large to display} \] Input:

int(x^3*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x)
 

Output:

( - 12*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d**3*e*f*g*n**2 - 
 12*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)))*b**2*d**3*e*f*g*n**3 
- 12*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d**3*e*g**2*n**3*x 
**2 + 30*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d*e**3*f**2*n* 
*3 + 30*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b**2*d*e**3*f*g*n**3 
*x**2 + 18*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 + 18* 
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 + 24*int(lo 
g((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 + 24*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 + 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*e**5*f**5*n + 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*e 
**5*f**4*g*n*x**2 + 24*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**4*e*f**3*g**2 
*n + 24*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + ...