\(\int \frac {x^3 (a+b \log (c x^n))}{(d+e x)^4} \, dx\) [55]

Optimal result

Integrand size = 21, antiderivative size = 141 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x \left (6 a+5 b n+6 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {\left (6 a+11 b n+6 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{6 e^4}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4} \] Output:

-1/3*x^3*(a+b*ln(c*x^n))/e/(e*x+d)^3-1/6*x^2*(3*a+b*n+3*b*ln(c*x^n))/e^2/( 
e*x+d)^2-1/6*x*(6*a+5*b*n+6*b*ln(c*x^n))/e^3/(e*x+d)+1/6*(6*a+11*b*n+6*b*l 
n(c*x^n))*ln(1+e*x/d)/e^4+b*n*polylog(2,-e*x/d)/e^4
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.27 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac {9 d^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {18 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}-b n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}+2 \log (x)-2 \log (d+e x)\right )-18 b n (\log (x)-\log (d+e x))+9 b n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+6 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+6 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 e^4} \] Input:

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^4,x]
 

Output:

((2*d^3*(a + b*Log[c*x^n]))/(d + e*x)^3 - (9*d^2*(a + b*Log[c*x^n]))/(d + 
e*x)^2 + (18*d*(a + b*Log[c*x^n]))/(d + e*x) - b*n*((d*(3*d + 2*e*x))/(d + 
 e*x)^2 + 2*Log[x] - 2*Log[d + e*x]) - 18*b*n*(Log[x] - Log[d + e*x]) + 9* 
b*n*(d/(d + e*x) + Log[x] - Log[d + e*x]) + 6*(a + b*Log[c*x^n])*Log[1 + ( 
e*x)/d] + 6*b*n*PolyLog[2, -((e*x)/d)])/(6*e^4)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2784, 2784, 2784, 2754, 2838}

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.

\(\displaystyle \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2784

\(\displaystyle \frac {\frac {\int \frac {x \left (6 a+5 b n+6 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{2 e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 2784

\(\displaystyle \frac {\frac {\frac {\int \frac {6 a+11 b n+6 b \log \left (c x^n\right )}{d+e x}dx}{e}-\frac {x \left (6 a+6 b \log \left (c x^n\right )+5 b n\right )}{e (d+e x)}}{2 e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 2754

\(\displaystyle \frac {\frac {\frac {\frac {\log \left (\frac {e x}{d}+1\right ) \left (6 a+6 b \log \left (c x^n\right )+11 b n\right )}{e}-\frac {6 b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}}{e}-\frac {x \left (6 a+6 b \log \left (c x^n\right )+5 b n\right )}{e (d+e x)}}{2 e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {\frac {\frac {\log \left (\frac {e x}{d}+1\right ) \left (6 a+6 b \log \left (c x^n\right )+11 b n\right )}{e}+\frac {6 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}}{e}-\frac {x \left (6 a+6 b \log \left (c x^n\right )+5 b n\right )}{e (d+e x)}}{2 e}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}\)

Input:

Int[(x^3*(a + b*Log[c*x^n]))/(d + e*x)^4,x]
 

Output:

-1/3*(x^3*(a + b*Log[c*x^n]))/(e*(d + e*x)^3) + (-1/2*(x^2*(3*a + b*n + 3* 
b*Log[c*x^n]))/(e*(d + e*x)^2) + (-((x*(6*a + 5*b*n + 6*b*Log[c*x^n]))/(e* 
(d + e*x))) + (((6*a + 11*b*n + 6*b*Log[c*x^n])*Log[1 + (e*x)/d])/e + (6*b 
*n*PolyLog[2, -((e*x)/d)])/e)/e)/(2*e))/(3*e)
 

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

Maple [C] (warning: unable to verify)

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

Time = 0.89 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.20

method	result	size
risch	\(\frac {3 b \ln \left (x^{n}\right ) d}{e^{4} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{4}}-\frac {3 b \ln \left (x^{n}\right ) d^{2}}{2 e^{4} \left (e x +d \right )^{2}}+\frac {b \ln \left (x^{n}\right ) d^{3}}{3 e^{4} \left (e x +d \right )^{3}}+\frac {7 b n d}{6 e^{4} \left (e x +d \right )}+\frac {11 b n \ln \left (e x +d \right )}{6 e^{4}}-\frac {b n \,d^{2}}{6 e^{4} \left (e x +d \right )^{2}}-\frac {11 b n \ln \left (e x \right )}{6 e^{4}}-\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {3 d}{e^{4} \left (e x +d \right )}+\frac {\ln \left (e x +d \right )}{e^{4}}-\frac {3 d^{2}}{2 e^{4} \left (e x +d \right )^{2}}+\frac {d^{3}}{3 e^{4} \left (e x +d \right )^{3}}\right )\)	\(310\)

Input:

int(x^3*(a+b*ln(c*x^n))/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

3*b*ln(x^n)/e^4*d/(e*x+d)+b*ln(x^n)/e^4*ln(e*x+d)-3/2*b*ln(x^n)/e^4*d^2/(e 
*x+d)^2+1/3*b*ln(x^n)/e^4*d^3/(e*x+d)^3+7/6*b*n/e^4*d/(e*x+d)+11/6*b*n/e^4 
*ln(e*x+d)-1/6*b*n/e^4*d^2/(e*x+d)^2-11/6*b*n/e^4*ln(e*x)-b*n/e^4*ln(e*x+d 
)*ln(-e*x/d)-b*n/e^4*dilog(-e*x/d)+(1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2 
-1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*csgn(I*c*x^n)^3 
+1/2*I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)*(3/e^4*d/(e*x+d)+1/e^4*ln 
(e*x+d)-3/2/e^4*d^2/(e*x+d)^2+1/3/e^4*d^3/(e*x+d)^3)
 

Fricas [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{4}} \,d x } \] Input:

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="fricas")
 

Output:

integral((b*x^3*log(c*x^n) + a*x^3)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 
 + 4*d^3*e*x + d^4), x)
 

Sympy [A] (verification not implemented)

Time = 35.38 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.67 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate(x**3*(a+b*ln(c*x**n))/(e*x+d)**4,x)
 

Output:

-a*d**3*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**3 
+ 3*a*d**2*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e* 
*3 - 3*a*d*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e**3 + 
 a*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**3 + b*d**3*n*Piec 
ewise((x/d**4, Eq(e, 0)), (-3*d/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x 
**2) - 2*e*x/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - log(x)/(3*d* 
*3*e) + log(d/e + x)/(3*d**3*e), True))/e**3 - b*d**3*Piecewise((x/d**4, E 
q(e, 0)), (-1/(3*e*(d + e*x)**3), True))*log(c*x**n)/e**3 - 3*b*d**2*n*Pie 
cewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log(x)/(2*d**2*e) 
 + log(d/e + x)/(2*d**2*e), True))/e**3 + 3*b*d**2*Piecewise((x/d**3, Eq(e 
, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e**3 + 3*b*d*n*Piecewise 
((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/e**3 - 3* 
b*d*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e 
**3 - b*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_polar 
(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x 
*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_p 
olar(I*pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log 
(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_p 
olar(I*pi)/d), True))/e, True))/e**3 + b*Piecewise((x/d, Eq(e, 0)), (log(d 
 + e*x)/e, True))*log(c*x**n)/e**3
 

Maxima [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{4}} \,d x } \] Input:

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/6*a*((18*d*e^2*x^2 + 27*d^2*e*x + 11*d^3)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2 
*e^5*x + d^3*e^4) + 6*log(e*x + d)/e^4) + b*integrate((x^3*log(c) + x^3*lo 
g(x^n))/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)
 

Giac [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{4}} \,d x } \] Input:

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)*x^3/(e*x + d)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \] Input:

int((x^3*(a + b*log(c*x^n)))/(d + e*x)^4,x)
 

Output:

int((x^3*(a + b*log(c*x^n)))/(d + e*x)^4, x)
 

Reduce [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {27 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{2} e x +15 a \,d^{3} n -2 b \,d^{3} n^{2}+9 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{3}+18 \,\mathrm {log}\left (e x +d \right ) a \,e^{3} n \,x^{3}+66 \,\mathrm {log}\left (e x +d \right ) b \,e^{3} n^{2} x^{3}-42 \,\mathrm {log}\left (x \right ) b \,e^{3} n^{2} x^{3}-24 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} n \,x^{3}+27 a \,d^{2} e n x -18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{7} n +27 \mathrm {log}\left (x^{n} c \right )^{2} b d \,e^{2} x^{2}+54 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e n x +54 \,\mathrm {log}\left (e x +d \right ) a d \,e^{2} n \,x^{2}+198 \,\mathrm {log}\left (e x +d \right ) b d \,e^{2} n^{2} x^{2}-126 \,\mathrm {log}\left (x \right ) b \,d^{2} e \,n^{2} x -126 \,\mathrm {log}\left (x \right ) b d \,e^{2} n^{2} x^{2}+54 \,\mathrm {log}\left (e x +d \right ) a \,d^{2} e n x +198 \,\mathrm {log}\left (e x +d \right ) b \,d^{2} e \,n^{2} x +42 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3} n -18 a \,e^{3} n \,x^{3}-2 b \,e^{3} n^{2} x^{3}+18 \,\mathrm {log}\left (e x +d \right ) a \,d^{3} n -42 \,\mathrm {log}\left (x \right ) b \,d^{3} n^{2}+66 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} n^{2}+9 \mathrm {log}\left (x^{n} c \right )^{2} b \,e^{3} x^{3}-54 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{6} e n x -54 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{5} e^{2} n \,x^{2}-18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{5}+4 d \,e^{3} x^{4}+6 d^{2} e^{2} x^{3}+4 d^{3} e \,x^{2}+d^{4} x}d x \right ) b \,d^{4} e^{3} n \,x^{3}}{18 e^{4} n \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int(x^3*(a+b*log(c*x^n))/(e*x+d)^4,x)
 

Output:

( - 18*int(log(x**n*c)/(d**4*x + 4*d**3*e*x**2 + 6*d**2*e**2*x**3 + 4*d*e* 
*3*x**4 + e**4*x**5),x)*b*d**7*n - 54*int(log(x**n*c)/(d**4*x + 4*d**3*e*x 
**2 + 6*d**2*e**2*x**3 + 4*d*e**3*x**4 + e**4*x**5),x)*b*d**6*e*n*x - 54*i 
nt(log(x**n*c)/(d**4*x + 4*d**3*e*x**2 + 6*d**2*e**2*x**3 + 4*d*e**3*x**4 
+ e**4*x**5),x)*b*d**5*e**2*n*x**2 - 18*int(log(x**n*c)/(d**4*x + 4*d**3*e 
*x**2 + 6*d**2*e**2*x**3 + 4*d*e**3*x**4 + e**4*x**5),x)*b*d**4*e**3*n*x** 
3 + 18*log(d + e*x)*a*d**3*n + 54*log(d + e*x)*a*d**2*e*n*x + 54*log(d + e 
*x)*a*d*e**2*n*x**2 + 18*log(d + e*x)*a*e**3*n*x**3 + 66*log(d + e*x)*b*d* 
*3*n**2 + 198*log(d + e*x)*b*d**2*e*n**2*x + 198*log(d + e*x)*b*d*e**2*n** 
2*x**2 + 66*log(d + e*x)*b*e**3*n**2*x**3 + 9*log(x**n*c)**2*b*d**3 + 27*l 
og(x**n*c)**2*b*d**2*e*x + 27*log(x**n*c)**2*b*d*e**2*x**2 + 9*log(x**n*c) 
**2*b*e**3*x**3 + 42*log(x**n*c)*b*d**3*n + 54*log(x**n*c)*b*d**2*e*n*x - 
24*log(x**n*c)*b*e**3*n*x**3 - 42*log(x)*b*d**3*n**2 - 126*log(x)*b*d**2*e 
*n**2*x - 126*log(x)*b*d*e**2*n**2*x**2 - 42*log(x)*b*e**3*n**2*x**3 + 15* 
a*d**3*n + 27*a*d**2*e*n*x - 18*a*e**3*n*x**3 - 2*b*d**3*n**2 - 2*b*e**3*n 
**2*x**3)/(18*e**4*n*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))