\(\int \frac {x^3 (a+b \log (c (d+e x)^n))}{(f+g x)^2} \, dx\) [249]

Optimal result

Integrand size = 25, antiderivative size = 265 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=-\frac {2 a f x}{g^3}+\frac {2 b f n x}{g^3}+\frac {b d n x}{2 e g^2}-\frac {b n x^2}{4 g^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 g^2}-\frac {b e f^3 n \log (d+e x)}{g^4 (e f-d g)}-\frac {2 b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {f^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^4 (f+g x)}+\frac {b e f^3 n \log (f+g x)}{g^4 (e f-d g)}+\frac {3 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^4}+\frac {3 b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\frac {-8 a f g x+8 b f g n x-\frac {b g^2 n \left (e x (-2 d+e x)+2 d^2 \log (d+e x)\right )}{e^2}-\frac {8 b f g (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 g^2 x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {4 f^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}-\frac {4 b e f^3 n (\log (d+e x)-\log (f+g x))}{e f-d g}+12 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )+12 b f^2 n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{4 g^4} \] Input:

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

Output:

(-8*a*f*g*x + 8*b*f*g*n*x - (b*g^2*n*(e*x*(-2*d + e*x) + 2*d^2*Log[d + e*x 
]))/e^2 - (8*b*f*g*(d + e*x)*Log[c*(d + e*x)^n])/e + 2*g^2*x^2*(a + b*Log[ 
c*(d + e*x)^n]) + (4*f^3*(a + b*Log[c*(d + e*x)^n]))/(f + g*x) - (4*b*e*f^ 
3*n*(Log[d + e*x] - Log[f + g*x]))/(e*f - d*g) + 12*f^2*(a + b*Log[c*(d + 
e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)] + 12*b*f^2*n*PolyLog[2, (g*(d + e* 
x))/(-(e*f) + d*g)])/(4*g^4)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.89 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.08

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{2}}{2 g^{2}}-\frac {2 b \ln \left (\left (e x +d \right )^{n}\right ) x f}{g^{3}}+\frac {3 b \ln \left (\left (e x +d \right )^{n}\right ) f^{2} \ln \left (g x +f \right )}{g^{4}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f^{3}}{g^{4} \left (g x +f \right )}-\frac {3 b n \,f^{2} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{4}}-\frac {3 b n \,f^{2} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{4}}-\frac {b n \,x^{2}}{4 g^{2}}+\frac {2 b f n x}{g^{3}}+\frac {9 b n \,f^{2}}{4 g^{4}}+\frac {b d n x}{2 e \,g^{2}}+\frac {b n d f}{2 e \,g^{3}}-\frac {b e n \,f^{3} \ln \left (g x +f \right )}{g^{4} \left (d g -e f \right )}-\frac {b n \ln \left (\left (g x +f \right ) e +d g -e f \right ) d^{3}}{2 e^{2} g \left (d g -e f \right )}-\frac {3 b n \ln \left (\left (g x +f \right ) e +d g -e f \right ) d^{2} f}{2 e \,g^{2} \left (d g -e f \right )}+\frac {2 b n \ln \left (\left (g x +f \right ) e +d g -e f \right ) d \,f^{2}}{g^{3} \left (d g -e f \right )}+\frac {b e n \ln \left (\left (g x +f \right ) e +d g -e f \right ) f^{3}}{g^{4} \left (d g -e f \right )}+\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^{2}-2 f x}{g^{3}}+\frac {3 f^{2} \ln \left (g x +f \right )}{g^{4}}+\frac {f^{3}}{g^{4} \left (g x +f \right )}\right )\)	\(550\)

Input:

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

Output:

1/2*b*ln((e*x+d)^n)/g^2*x^2-2*b*ln((e*x+d)^n)/g^3*x*f+3*b*ln((e*x+d)^n)/g^ 
4*f^2*ln(g*x+f)+b*ln((e*x+d)^n)*f^3/g^4/(g*x+f)-3*b*n/g^4*f^2*dilog(((g*x+ 
f)*e+d*g-e*f)/(d*g-e*f))-3*b*n/g^4*f^2*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d 
*g-e*f))-1/4*b*n*x^2/g^2+2*b*f*n*x/g^3+9/4*b*n/g^4*f^2+1/2*b*d*n*x/e/g^2+1 
/2*b/e*n/g^3*d*f-b*e*n/g^4*f^3/(d*g-e*f)*ln(g*x+f)-1/2*b/e^2*n/g/(d*g-e*f) 
*ln((g*x+f)*e+d*g-e*f)*d^3-3/2*b/e*n/g^2/(d*g-e*f)*ln((g*x+f)*e+d*g-e*f)*d 
^2*f+2*b*n/g^3/(d*g-e*f)*ln((g*x+f)*e+d*g-e*f)*d*f^2+b*e*n/g^4/(d*g-e*f)*l 
n((g*x+f)*e+d*g-e*f)*f^3+(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*c 
sgn(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/g^3*(1/2*g*x^2-2*f*x)+3/g^4*f^2*ln(g*x+f)+f^3/g^4/(g*x+f))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(2*f^3/(g^5*x + f*g^4) + 6*f^2*log(g*x + f)/g^4 + (g*x^2 - 4*f*x)/g^3) 
*a + b*integrate((x^3*log((e*x + d)^n) + x^3*log(c))/(g^2*x^2 + 2*f*g*x + 
f^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(12*int((log((d + e*x)**n*c)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2 
*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b*d**3*e**2*f**3*g**4*n + 12*int((log( 
(d + e*x)**n*c)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g* 
x**2 + e*g**2*x**3),x)*b*d**3*e**2*f**2*g**5*n*x - 36*int((log((d + e*x)** 
n*c)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g* 
*2*x**3),x)*b*d**2*e**3*f**4*g**3*n - 36*int((log((d + e*x)**n*c)*x)/(d*f* 
*2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b 
*d**2*e**3*f**3*g**4*n*x + 36*int((log((d + e*x)**n*c)*x)/(d*f**2 + 2*d*f* 
g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b*d*e**4*f** 
5*g**2*n + 36*int((log((d + e*x)**n*c)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x** 
2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b*d*e**4*f**4*g**3*n*x - 12* 
int((log((d + e*x)**n*c)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 
 2*e*f*g*x**2 + e*g**2*x**3),x)*b*e**5*f**6*g*n - 12*int((log((d + e*x)**n 
*c)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g** 
2*x**3),x)*b*e**5*f**5*g**2*n*x - 2*log(d + e*x)*b*d**4*f*g**4*n**2 - 2*lo 
g(d + e*x)*b*d**4*g**5*n**2*x - 6*log(d + e*x)*b*d**3*e*f**2*g**3*n**2 - 6 
*log(d + e*x)*b*d**3*e*f*g**4*n**2*x + 12*log(d + e*x)*b*d*e**3*f**4*g*n** 
2 + 12*log(d + e*x)*b*d*e**3*f**3*g**2*n**2*x + 12*log(f + g*x)*a*d**2*e** 
2*f**3*g**2*n + 12*log(f + g*x)*a*d**2*e**2*f**2*g**3*n*x - 12*log(f + g*x 
)*a*d*e**3*f**4*g*n - 12*log(f + g*x)*a*d*e**3*f**3*g**2*n*x + 8*log(f ...