3.1.23 \(\int \frac {(d+e x)^3 (a+b \log (c x^n))}{x} \, dx\) [23]

3.1.23.1 Optimal result

Integrand size = 21, antiderivative size = 122 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-3 b d^2 e n x-\frac {3}{4} b d e^2 n x^2-\frac {1}{9} b e^3 n x^3-\frac {1}{2} b d^3 n \log ^2(x)+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right ) \]

output

-3*b*d^2*e*n*x-3/4*b*d*e^2*n*x^2-1/9*b*e^3*n*x^3-1/2*b*d^3*n*ln(x)^2+3*d^2 
*e*x*(a+b*ln(c*x^n))+3/2*d*e^2*x^2*(a+b*ln(c*x^n))+1/3*e^3*x^3*(a+b*ln(c*x 
^n))+d^3*ln(x)*(a+b*ln(c*x^n))
 

3.1.23.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=3 a d^2 e x-3 b d^2 e n x-\frac {3}{4} b d e^2 n x^2-\frac {1}{9} b e^3 n x^3+3 b d^2 e x \log \left (c x^n\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

input

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

output

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

3.1.23.3 Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2772, 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.

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

\(\Big \downarrow \) 2772

\(\displaystyle -b n \int \left (\frac {\log (x) d^3}{x}+\frac {1}{6} e \left (18 d^2+9 e x d+2 e^2 x^2\right )\right )dx+d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle d^3 \log (x) \left (a+b \log \left (c x^n\right )\right )+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {1}{2} d^3 \log ^2(x)+3 d^2 e x+\frac {3}{4} d e^2 x^2+\frac {e^3 x^3}{9}\right )\)

input

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

output

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

3.1.23.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

3.1.23.4 Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(\frac {12 x^{3} \ln \left (c \,x^{n}\right ) b \,e^{3} n -4 x^{3} b \,e^{3} n^{2}+12 x^{3} a \,e^{3} n +54 x^{2} \ln \left (c \,x^{n}\right ) b d \,e^{2} n -27 x^{2} b d \,e^{2} n^{2}+54 x^{2} a d \,e^{2} n +108 x \ln \left (c \,x^{n}\right ) b \,d^{2} e n -108 x b \,d^{2} e \,n^{2}+36 \ln \left (x \right ) a \,d^{3} n +108 x a \,d^{2} e n +18 b \,d^{3} \ln \left (c \,x^{n}\right )^{2}}{36 n}\)	\(144\)
risch	\(\frac {\ln \left (c \right ) b \,e^{3} x^{3}}{3}-\frac {3 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{4}-\frac {3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) e x}{2}-\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}-\frac {i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{6}+\frac {3 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {3 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x}{2}+\frac {3 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x}{2}+\ln \left (x \right ) \ln \left (c \right ) b \,d^{3}+\frac {a \,e^{3} x^{3}}{3}+\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} e x}{2}+\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {3 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}+\left (\frac {x^{3} b \,e^{3}}{3}+\frac {3 x^{2} b d \,e^{2}}{2}+3 e \,d^{2} b x +b \,d^{3} \ln \left (x \right )\right ) \ln \left (x^{n}\right )+\ln \left (x \right ) a \,d^{3}+\frac {3 a d \,e^{2} x^{2}}{2}+3 a \,d^{2} e x +\frac {3 \ln \left (c \right ) b d \,e^{2} x^{2}}{2}+3 \ln \left (c \right ) b \,d^{2} e x -3 b \,d^{2} e n x -\frac {3 b d \,e^{2} n \,x^{2}}{4}-\frac {b \,e^{3} n \,x^{3}}{9}-\frac {b \,d^{3} n \ln \left (x \right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{6}\)	\(579\)

input

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

output

1/36*(12*x^3*ln(c*x^n)*b*e^3*n-4*x^3*b*e^3*n^2+12*x^3*a*e^3*n+54*x^2*ln(c* 
x^n)*b*d*e^2*n-27*x^2*b*d*e^2*n^2+54*x^2*a*d*e^2*n+108*x*ln(c*x^n)*b*d^2*e 
*n-108*x*b*d^2*e*n^2+36*ln(x)*a*d^3*n+108*x*a*d^2*e*n+18*b*d^3*ln(c*x^n)^2 
)/n
 

3.1.23.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac {1}{9} \, {\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3} - \frac {3}{4} \, {\left (b d e^{2} n - 2 \, a d e^{2}\right )} x^{2} - 3 \, {\left (b d^{2} e n - a d^{2} e\right )} x + \frac {1}{6} \, {\left (2 \, b e^{3} x^{3} + 9 \, b d e^{2} x^{2} + 18 \, b d^{2} e x\right )} \log \left (c\right ) + \frac {1}{6} \, {\left (2 \, b e^{3} n x^{3} + 9 \, b d e^{2} n x^{2} + 18 \, b d^{2} e n x + 6 \, b d^{3} \log \left (c\right ) + 6 \, a d^{3}\right )} \log \left (x\right ) \]

input

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

output

1/2*b*d^3*n*log(x)^2 - 1/9*(b*e^3*n - 3*a*e^3)*x^3 - 3/4*(b*d*e^2*n - 2*a* 
d*e^2)*x^2 - 3*(b*d^2*e*n - a*d^2*e)*x + 1/6*(2*b*e^3*x^3 + 9*b*d*e^2*x^2 
+ 18*b*d^2*e*x)*log(c) + 1/6*(2*b*e^3*n*x^3 + 9*b*d*e^2*n*x^2 + 18*b*d^2*e 
*n*x + 6*b*d^3*log(c) + 6*a*d^3)*log(x)
 

3.1.23.6 Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \frac {a d^{3} \log {\left (c x^{n} \right )}}{n} + 3 a d^{2} e x + \frac {3 a d e^{2} x^{2}}{2} + \frac {a e^{3} x^{3}}{3} + \frac {b d^{3} \log {\left (c x^{n} \right )}^{2}}{2 n} - 3 b d^{2} e n x + 3 b d^{2} e x \log {\left (c x^{n} \right )} - \frac {3 b d e^{2} n x^{2}}{4} + \frac {3 b d e^{2} x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {b e^{3} n x^{3}}{9} + \frac {b e^{3} x^{3} \log {\left (c x^{n} \right )}}{3} & \text {for}\: n \neq 0 \\\left (a + b \log {\left (c \right )}\right ) \left (d^{3} \log {\left (x \right )} + 3 d^{2} e x + \frac {3 d e^{2} x^{2}}{2} + \frac {e^{3} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

input

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

output

Piecewise((a*d**3*log(c*x**n)/n + 3*a*d**2*e*x + 3*a*d*e**2*x**2/2 + a*e** 
3*x**3/3 + b*d**3*log(c*x**n)**2/(2*n) - 3*b*d**2*e*n*x + 3*b*d**2*e*x*log 
(c*x**n) - 3*b*d*e**2*n*x**2/4 + 3*b*d*e**2*x**2*log(c*x**n)/2 - b*e**3*n* 
x**3/9 + b*e**3*x**3*log(c*x**n)/3, Ne(n, 0)), ((a + b*log(c))*(d**3*log(x 
) + 3*d**2*e*x + 3*d*e**2*x**2/2 + e**3*x**3/3), True))
 

3.1.23.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {1}{9} \, b e^{3} n x^{3} + \frac {1}{3} \, b e^{3} x^{3} \log \left (c x^{n}\right ) - \frac {3}{4} \, b d e^{2} n x^{2} + \frac {1}{3} \, a e^{3} x^{3} + \frac {3}{2} \, b d e^{2} x^{2} \log \left (c x^{n}\right ) - 3 \, b d^{2} e n x + \frac {3}{2} \, a d e^{2} x^{2} + 3 \, b d^{2} e x \log \left (c x^{n}\right ) + 3 \, a d^{2} e x + \frac {b d^{3} \log \left (c x^{n}\right )^{2}}{2 \, n} + a d^{3} \log \left (x\right ) \]

input

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

output

-1/9*b*e^3*n*x^3 + 1/3*b*e^3*x^3*log(c*x^n) - 3/4*b*d*e^2*n*x^2 + 1/3*a*e^ 
3*x^3 + 3/2*b*d*e^2*x^2*log(c*x^n) - 3*b*d^2*e*n*x + 3/2*a*d*e^2*x^2 + 3*b 
*d^2*e*x*log(c*x^n) + 3*a*d^2*e*x + 1/2*b*d^3*log(c*x^n)^2/n + a*d^3*log(x 
)
 

3.1.23.8 Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {1}{2} \, b d^{3} n \log \left (x\right )^{2} - \frac {1}{9} \, {\left (b e^{3} n - 3 \, b e^{3} \log \left (c\right ) - 3 \, a e^{3}\right )} x^{3} - \frac {3}{4} \, {\left (b d e^{2} n - 2 \, b d e^{2} \log \left (c\right ) - 2 \, a d e^{2}\right )} x^{2} - 3 \, {\left (b d^{2} e n - b d^{2} e \log \left (c\right ) - a d^{2} e\right )} x + \frac {1}{6} \, {\left (2 \, b e^{3} n x^{3} + 9 \, b d e^{2} n x^{2} + 18 \, b d^{2} e n x\right )} \log \left (x\right ) + {\left (b d^{3} \log \left (c\right ) + a d^{3}\right )} \log \left (x\right ) \]

input

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

output

1/2*b*d^3*n*log(x)^2 - 1/9*(b*e^3*n - 3*b*e^3*log(c) - 3*a*e^3)*x^3 - 3/4* 
(b*d*e^2*n - 2*b*d*e^2*log(c) - 2*a*d*e^2)*x^2 - 3*(b*d^2*e*n - b*d^2*e*lo 
g(c) - a*d^2*e)*x + 1/6*(2*b*e^3*n*x^3 + 9*b*d*e^2*n*x^2 + 18*b*d^2*e*n*x) 
*log(x) + (b*d^3*log(c) + a*d^3)*log(x)
 

3.1.23.9 Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\ln \left (c\,x^n\right )\,\left (3\,b\,d^2\,e\,x+\frac {3\,b\,d\,e^2\,x^2}{2}+\frac {b\,e^3\,x^3}{3}\right )+\frac {e^3\,x^3\,\left (3\,a-b\,n\right )}{9}+a\,d^3\,\ln \left (x\right )+\frac {b\,d^3\,{\ln \left (c\,x^n\right )}^2}{2\,n}+\frac {3\,d\,e^2\,x^2\,\left (2\,a-b\,n\right )}{4}+3\,d^2\,e\,x\,\left (a-b\,n\right ) \]

input

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

output

log(c*x^n)*((b*e^3*x^3)/3 + 3*b*d^2*e*x + (3*b*d*e^2*x^2)/2) + (e^3*x^3*(3 
*a - b*n))/9 + a*d^3*log(x) + (b*d^3*log(c*x^n)^2)/(2*n) + (3*d*e^2*x^2*(2 
*a - b*n))/4 + 3*d^2*e*x*(a - b*n)