\(\int \frac {x^5 (a+b \log (c (d+e x)^n))}{f+g x^2} \, dx\) [256]

Optimal result

Integrand size = 27, antiderivative size = 397 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=-\frac {b d f n x}{2 e g^2}+\frac {b d^3 n x}{4 e^3 g}+\frac {b f n x^2}{4 g^2}-\frac {b d^2 n x^2}{8 e^2 g}+\frac {b d n x^3}{12 e g}-\frac {b n x^4}{16 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3} \] Output:

-1/2*b*d*f*n*x/e/g^2+1/4*b*d^3*n*x/e^3/g+1/4*b*f*n*x^2/g^2-1/8*b*d^2*n*x^2 
/e^2/g+1/12*b*d*n*x^3/e/g-1/16*b*n*x^4/g+1/2*b*d^2*f*n*ln(e*x+d)/e^2/g^2-1 
/4*b*d^4*n*ln(e*x+d)/e^4/g-1/2*f*x^2*(a+b*ln(c*(e*x+d)^n))/g^2+1/4*x^4*(a+ 
b*ln(c*(e*x+d)^n))/g+1/2*f^2*(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+1/2*f^2*(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+1/2*b*f^2*n*polylog(2,-g^ 
(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/g^3+1/2*b*f^2*n*polylog(2,g^(1/2)* 
(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/g^3
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\frac {\frac {12 b f g n \left (e x (-2 d+e x)+2 d^2 \log (d+e x)\right )}{e^2}-\frac {b g^2 n \left (e x \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+12 d^4 \log (d+e x)\right )}{e^4}-24 f g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+12 g^2 x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+24 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+24 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )+24 b f^2 n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+24 b f^2 n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{48 g^3} \] Input:

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

Output:

((12*b*f*g*n*(e*x*(-2*d + e*x) + 2*d^2*Log[d + e*x]))/e^2 - (b*g^2*n*(e*x* 
(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 12*d^4*Log[d + e*x]))/e^ 
4 - 24*f*g*x^2*(a + b*Log[c*(d + e*x)^n]) + 12*g^2*x^4*(a + b*Log[c*(d + e 
*x)^n]) + 24*f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x)) 
/(e*Sqrt[-f] + d*Sqrt[g])] + 24*f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqr 
t[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])] + 24*b*f^2*n*PolyLog[2, -((S 
qrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + 24*b*f^2*n*PolyLog[2, (Sqrt 
[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(48*g^3)
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {f^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3}+\frac {f^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^3}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}+\frac {b d^3 n x}{4 e^3 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^2 n x^2}{8 e^2 g}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 g^3}-\frac {b d f n x}{2 e g^2}+\frac {b d n x^3}{12 e g}+\frac {b f n x^2}{4 g^2}-\frac {b n x^4}{16 g}\)

Input:

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

Output:

-1/2*(b*d*f*n*x)/(e*g^2) + (b*d^3*n*x)/(4*e^3*g) + (b*f*n*x^2)/(4*g^2) - ( 
b*d^2*n*x^2)/(8*e^2*g) + (b*d*n*x^3)/(12*e*g) - (b*n*x^4)/(16*g) + (b*d^2* 
f*n*Log[d + e*x])/(2*e^2*g^2) - (b*d^4*n*Log[d + e*x])/(4*e^4*g) - (f*x^2* 
(a + b*Log[c*(d + e*x)^n]))/(2*g^2) + (x^4*(a + b*Log[c*(d + e*x)^n]))/(4* 
g) + (f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqr 
t[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqr 
t[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3) + (b*f^2*n*PolyLog[ 
2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^3) + (b*f^2*n*Po 
lyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3)
 

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 [C] (warning: unable to verify)

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

Time = 3.62 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.38

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{4}}{4 g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f \,x^{2}}{2 g^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}-\frac {b n \,f^{2} \ln \left (e x +d \right ) \ln \left (g \,x^{2}+f \right )}{2 g^{3}}+\frac {b n \,f^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-g f}-g \left (e x +d \right )+d g}{e \sqrt {-g f}+d g}\right )}{2 g^{3}}+\frac {b n \,f^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-g f}+g \left (e x +d \right )-d g}{e \sqrt {-g f}-d g}\right )}{2 g^{3}}+\frac {b n \,f^{2} \operatorname {dilog}\left (\frac {e \sqrt {-g f}-g \left (e x +d \right )+d g}{e \sqrt {-g f}+d g}\right )}{2 g^{3}}+\frac {b n \,f^{2} \operatorname {dilog}\left (\frac {e \sqrt {-g f}+g \left (e x +d \right )-d g}{e \sqrt {-g f}-d g}\right )}{2 g^{3}}-\frac {b n \,x^{4}}{16 g}+\frac {b d n \,x^{3}}{12 e g}-\frac {b \,d^{2} n \,x^{2}}{8 e^{2} g}+\frac {b f n \,x^{2}}{4 g^{2}}+\frac {b \,d^{3} n x}{4 e^{3} g}-\frac {b d f n x}{2 e \,g^{2}}-\frac {b \,d^{4} n \ln \left (e x +d \right )}{4 e^{4} g}+\frac {b \,d^{2} f n \ln \left (e x +d \right )}{2 e^{2} g^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} g \,x^{4}-f \,x^{2}}{2 g^{2}}+\frac {f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}\right )\)	\(549\)

Input:

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

Output:

1/4*b*ln((e*x+d)^n)/g*x^4-1/2*b*ln((e*x+d)^n)/g^2*f*x^2+1/2*b*ln((e*x+d)^n 
)*f^2/g^3*ln(g*x^2+f)-1/2*b*n*f^2/g^3*ln(e*x+d)*ln(g*x^2+f)+1/2*b*n*f^2/g^ 
3*ln(e*x+d)*ln((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g))+1/2*b* 
n*f^2/g^3*ln(e*x+d)*ln((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g*f)^(1/2)-d*g) 
)+1/2*b*n*f^2/g^3*dilog((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g 
))+1/2*b*n*f^2/g^3*dilog((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g*f)^(1/2)-d* 
g))-1/16*b*n*x^4/g+1/12*b*d*n*x^3/e/g-1/8*b*d^2*n*x^2/e^2/g+1/4*b*f*n*x^2/ 
g^2+1/4*b*d^3*n*x/e^3/g-1/2*b*d*f*n*x/e/g^2-1/4*b*d^4*n*ln(e*x+d)/e^4/g+1/ 
2*b*d^2*f*n*ln(e*x+d)/e^2/g^2+(1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+ 
d)^n)^2-1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-1/2*I*b 
*Pi*csgn(I*c*(e*x+d)^n)^3+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+b*ln( 
c)+a)*(1/2/g^2*(1/2*g*x^4-f*x^2)+1/2*f^2/g^3*ln(g*x^2+f))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\frac {-48 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b \,e^{5} f^{3} n +48 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) x}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b d \,e^{4} f^{2} g n +24 \,\mathrm {log}\left (g \,x^{2}+f \right ) a \,e^{4} f^{2} n +24 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b \,e^{4} f^{2}-12 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,d^{4} g^{2} n +24 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,d^{2} e^{2} f g n -24 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{4} f g n \,x^{2}+12 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{4} g^{2} n \,x^{4}-24 a \,e^{4} f g n \,x^{2}+12 a \,e^{4} g^{2} n \,x^{4}+12 b \,d^{3} e \,g^{2} n^{2} x -6 b \,d^{2} e^{2} g^{2} n^{2} x^{2}-24 b d \,e^{3} f g \,n^{2} x +4 b d \,e^{3} g^{2} n^{2} x^{3}+12 b \,e^{4} f g \,n^{2} x^{2}-3 b \,e^{4} g^{2} n^{2} x^{4}}{48 e^{4} g^{3} n} \] Input:

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

Output:

( - 48*int(log((d + e*x)**n*c)/(d*f + d*g*x**2 + e*f*x + e*g*x**3),x)*b*e* 
*5*f**3*n + 48*int((log((d + e*x)**n*c)*x)/(d*f + d*g*x**2 + e*f*x + e*g*x 
**3),x)*b*d*e**4*f**2*g*n + 24*log(f + g*x**2)*a*e**4*f**2*n + 24*log((d + 
 e*x)**n*c)**2*b*e**4*f**2 - 12*log((d + e*x)**n*c)*b*d**4*g**2*n + 24*log 
((d + e*x)**n*c)*b*d**2*e**2*f*g*n - 24*log((d + e*x)**n*c)*b*e**4*f*g*n*x 
**2 + 12*log((d + e*x)**n*c)*b*e**4*g**2*n*x**4 - 24*a*e**4*f*g*n*x**2 + 1 
2*a*e**4*g**2*n*x**4 + 12*b*d**3*e*g**2*n**2*x - 6*b*d**2*e**2*g**2*n**2*x 
**2 - 24*b*d*e**3*f*g*n**2*x + 4*b*d*e**3*g**2*n**2*x**3 + 12*b*e**4*f*g*n 
**2*x**2 - 3*b*e**4*g**2*n**2*x**4)/(48*e**4*g**3*n)