\(\int \frac {x^4 (a+b \log (c (d+e x)^n))}{(f+g x^2)^2} \, dx\) [271]

Optimal result

Integrand size = 27, antiderivative size = 534 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {a x}{g^2}-\frac {b n x}{g^2}-\frac {b e f n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}+\frac {b e f n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e f n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {b e f n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}-\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {4 a \sqrt {g} x-4 b \sqrt {g} n x+\frac {4 b \sqrt {g} (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}-\sqrt {g} x}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {b e f n \left (\log (d+e x)-\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{e \sqrt {-f}+d \sqrt {g}}+3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+\frac {b e f n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{-e \sqrt {-f}+d \sqrt {g}}-3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )-3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}} \] Input:

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

Output:

(4*a*Sqrt[g]*x - 4*b*Sqrt[g]*n*x + (4*b*Sqrt[g]*(d + e*x)*Log[c*(d + e*x)^ 
n])/e - (f*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[-f] - Sqrt[g]*x) + (f*(a + b* 
Log[c*(d + e*x)^n]))/(Sqrt[-f] + Sqrt[g]*x) + (b*e*f*n*(Log[d + e*x] - Log 
[Sqrt[-f] - Sqrt[g]*x]))/(e*Sqrt[-f] + d*Sqrt[g]) + 3*Sqrt[-f]*(a + b*Log[ 
c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])] + 
 (b*e*f*n*(Log[d + e*x] - Log[Sqrt[-f] + Sqrt[g]*x]))/(-(e*Sqrt[-f]) + d*S 
qrt[g]) - 3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g] 
*x))/(e*Sqrt[-f] - d*Sqrt[g])] - 3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + 
 e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + 3*b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*(d 
+ e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))
 

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 534, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a*x)/g^2 - (b*n*x)/g^2 - (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g 
])*g^(5/2)) + (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) 
+ (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (f*(a + b*Log[c*(d + e*x)^n]) 
)/(4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*(a + b*Log[c*(d + e*x)^n]))/(4*g 
^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*f*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e 
*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*L 
og[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b* 
e*f*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3 
*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqr 
t[-f] - d*Sqrt[g])])/(4*g^(5/2)) - (3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*( 
d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*g^(5/2)) + (3*b*Sqrt[-f]*n*PolyLo 
g[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*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 [C] (warning: unable to verify)

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

Time = 2.60 (sec) , antiderivative size = 1619, normalized size of antiderivative = 3.03

method	result	size
risch	\(\text {Expression too large to display}\)	\(1619\)

Input:

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

Output:

-1/4*b*e*n/g^2*f/(d^2*g+e^2*f)*d*ln(g*(e*x+d)^2-2*(e*x+d)*d*g+d^2*g+f*e^2) 
-1/2*b*e^2*n/g^2*f^2/(d^2*g+e^2*f)/(g*f)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d 
*g)/e/(g*f)^(1/2))-1/2*b*e^2/g^2*f*x/(e^2*g*x^2+e^2*f)*n*ln(e*x+d)-3/2*b/g 
^2*f/(g*f)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(g*f)^(1/2))*ln((e*x+d)^ 
n)-b*n/g^2*f*ln(e*x+d)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*( 
-g*f)^(1/2)+d*g))+b*n/g^2*f*ln(e*x+d)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2)+g*(e 
*x+d)-d*g)/(e*(-g*f)^(1/2)-d*g))+b/e/g^2*d*ln((e*x+d)^n)+b*ln((e*x+d)^n)/g 
^2*x+1/2*b*e^2/g^2*f*x/(e^2*g*x^2+e^2*f)*ln((e*x+d)^n)+3/2*b/g^2*f/(g*f)^( 
1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(g*f)^(1/2))*n*ln(e*x+d)-3/4*b*n/g^2 
*f/(-g*f)^(1/2)*dilog((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g)) 
+3/4*b*n/g^2*f/(-g*f)^(1/2)*dilog((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g*f) 
^(1/2)-d*g))+(1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*b*P 
i*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)*(x/g^2-1/g^ 
2*f*(-1/2*x/(g*x^2+f)+3/2/(g*f)^(1/2)*arctan(g*x/(g*f)^(1/2))))-b*d*n/e/g^ 
2-1/4*b*e^2*n/g*f^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-g*f)^(1/2) 
*ln((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g*f)^(1/2)-d*g))*d^2+1/4*b*e^2*n*f 
*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2) 
-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g))*x^2*d^2+1/4*b*e^4*n/g*f^2*ln(e*x+d)/ 
(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2)-g*(e*x+...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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