\(\int \frac {(d+e x)^3 (a+b \log (c x^n))}{x^2} \, dx\) [24]

Optimal result

Integrand size = 21, antiderivative size = 119 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b d^3 n}{x}-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2-\frac {3}{2} b d^2 e n \log ^2(x)-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+3 d^2 e \log (x) \left (a+b \log \left (c x^n\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b d^3 n}{x}+3 a d e^2 x-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2+3 b d e^2 x \log \left (c x^n\right )-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97, 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.

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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])
 

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.22

method	result	size
parallelrisch	\(\frac {2 x^{3} \ln \left (c \,x^{n}\right ) b \,e^{3} n -x^{3} b \,e^{3} n^{2}+2 x^{3} a \,e^{3} n +12 x^{2} \ln \left (c \,x^{n}\right ) b d \,e^{2} n -12 x^{2} b d \,e^{2} n^{2}+12 \ln \left (x \right ) x a \,d^{2} e n +12 x^{2} a d \,e^{2} n +6 e \,d^{2} b \ln \left (c \,x^{n}\right )^{2} x -4 \ln \left (c \,x^{n}\right ) b \,d^{3} n -4 b \,d^{3} n^{2}-4 a \,d^{3} n}{4 x n}\)	\(145\)
risch	\(-\frac {b \left (-e^{3} x^{3}-6 e \,d^{2} \ln \left (x \right ) x -6 d \,e^{2} x^{2}+2 d^{3}\right ) \ln \left (x^{n}\right )}{2 x}-\frac {-i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-12 a \,e^{2} x^{2} d +4 a \,d^{3}+i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+6 i \ln \left (x \right ) \pi b \,d^{2} e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x -12 \ln \left (x \right ) e \,d^{2} a x +2 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-2 \ln \left (c \right ) b \,e^{3} x^{3}-12 \ln \left (c \right ) b d \,e^{2} x^{2}+i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-6 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+4 d^{3} b \ln \left (c \right )-2 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-12 \ln \left (x \right ) \ln \left (c \right ) b \,d^{2} e x +6 e \,d^{2} b n \ln \left (x \right )^{2} x -2 a \,e^{3} x^{3}+4 b \,d^{3} n +6 i \ln \left (x \right ) \pi b \,d^{2} e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) x -2 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+6 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-6 i \ln \left (x \right ) \pi b \,d^{2} e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x +6 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-6 i \ln \left (x \right ) \pi b \,d^{2} e \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) x +12 b d \,e^{2} n \,x^{2}+b \,e^{3} n \,x^{3}}{4 x}\)	\(588\)

Input:

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

Output:

1/4/x*(2*x^3*ln(c*x^n)*b*e^3*n-x^3*b*e^3*n^2+2*x^3*a*e^3*n+12*x^2*ln(c*x^n 
)*b*d*e^2*n-12*x^2*b*d*e^2*n^2+12*ln(x)*x*a*d^2*e*n+12*x^2*a*d*e^2*n+6*e*d 
^2*b*ln(c*x^n)^2*x-4*ln(c*x^n)*b*d^3*n-4*b*d^3*n^2-4*a*d^3*n)/n
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {6 \, b d^{2} e n x \log \left (x\right )^{2} - 4 \, b d^{3} n - 4 \, a d^{3} - {\left (b e^{3} n - 2 \, a e^{3}\right )} x^{3} - 12 \, {\left (b d e^{2} n - a d e^{2}\right )} x^{2} + 2 \, {\left (b e^{3} x^{3} + 6 \, b d e^{2} x^{2} - 2 \, b d^{3}\right )} \log \left (c\right ) + 2 \, {\left (b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 6 \, b d^{2} e x \log \left (c\right ) - 2 \, b d^{3} n + 6 \, a d^{2} e x\right )} \log \left (x\right )}{4 \, x} \] Input:

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

Output:

1/4*(6*b*d^2*e*n*x*log(x)^2 - 4*b*d^3*n - 4*a*d^3 - (b*e^3*n - 2*a*e^3)*x^ 
3 - 12*(b*d*e^2*n - a*d*e^2)*x^2 + 2*(b*e^3*x^3 + 6*b*d*e^2*x^2 - 2*b*d^3) 
*log(c) + 2*(b*e^3*n*x^3 + 6*b*d*e^2*n*x^2 + 6*b*d^2*e*x*log(c) - 2*b*d^3* 
n + 6*a*d^2*e*x)*log(x))/x
 

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\begin {cases} - \frac {a d^{3}}{x} + \frac {3 a d^{2} e \log {\left (c x^{n} \right )}}{n} + 3 a d e^{2} x + \frac {a e^{3} x^{2}}{2} - \frac {b d^{3} n}{x} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{x} + \frac {3 b d^{2} e \log {\left (c x^{n} \right )}^{2}}{2 n} - 3 b d e^{2} n x + 3 b d e^{2} x \log {\left (c x^{n} \right )} - \frac {b e^{3} n x^{2}}{4} + \frac {b e^{3} x^{2} \log {\left (c x^{n} \right )}}{2} & \text {for}\: n \neq 0 \\\left (a + b \log {\left (c \right )}\right ) \left (- \frac {d^{3}}{x} + 3 d^{2} e \log {\left (x \right )} + 3 d e^{2} x + \frac {e^{3} x^{2}}{2}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {1}{4} \, b e^{3} n x^{2} + \frac {1}{2} \, b e^{3} x^{2} \log \left (c x^{n}\right ) - 3 \, b d e^{2} n x + \frac {1}{2} \, a e^{3} x^{2} + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x + \frac {3 \, b d^{2} e \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d^{2} e \log \left (x\right ) - \frac {b d^{3} n}{x} - \frac {b d^{3} \log \left (c x^{n}\right )}{x} - \frac {a d^{3}}{x} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {1}{2} \, b e^{3} x^{2} \log \left (c\right ) + \frac {3}{2} \, b d^{2} e n \log \left (x\right )^{2} + 3 \, {\left (x \log \left (x\right ) - x\right )} b d e^{2} n + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} b e^{3} n + \frac {1}{2} \, a e^{3} x^{2} - b d^{3} n {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} + 3 \, b d e^{2} x \log \left (c\right ) + 3 \, b d^{2} e \log \left (c\right ) \log \left ({\left | x \right |}\right ) + 3 \, a d e^{2} x + 3 \, a d^{2} e \log \left ({\left | x \right |}\right ) - \frac {b d^{3} \log \left (c\right )}{x} - \frac {a d^{3}}{x} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 27.44 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\ln \left (x\right )\,\left (3\,a\,d^2\,e+3\,b\,d^2\,e\,n\right )-\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3+3\,b\,d^2\,e\,x+3\,b\,d\,e^2\,x^2+b\,e^3\,x^3}{x}-\frac {\frac {3\,b\,e^3\,x^3}{2}+6\,b\,d\,e^2\,x^2}{x}\right )-\frac {a\,d^3+b\,d^3\,n}{x}+\frac {e^3\,x^2\,\left (2\,a-b\,n\right )}{4}+3\,d\,e^2\,x\,\left (a-b\,n\right )+\frac {3\,b\,d^2\,e\,{\ln \left (c\,x^n\right )}^2}{2\,n} \] Input:

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

Output:

log(x)*(3*a*d^2*e + 3*b*d^2*e*n) - log(c*x^n)*((b*d^3 + b*e^3*x^3 + 3*b*d^ 
2*e*x + 3*b*d*e^2*x^2)/x - ((3*b*e^3*x^3)/2 + 6*b*d*e^2*x^2)/x) - (a*d^3 + 
 b*d^3*n)/x + (e^3*x^2*(2*a - b*n))/4 + 3*d*e^2*x*(a - b*n) + (3*b*d^2*e*l 
og(c*x^n)^2)/(2*n)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {6 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{2} e x -4 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3} n +12 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} n \,x^{2}+2 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} n \,x^{3}+12 \,\mathrm {log}\left (x \right ) a \,d^{2} e n x -4 a \,d^{3} n +12 a d \,e^{2} n \,x^{2}+2 a \,e^{3} n \,x^{3}-4 b \,d^{3} n^{2}-12 b d \,e^{2} n^{2} x^{2}-b \,e^{3} n^{2} x^{3}}{4 n x} \] Input:

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

Output:

(6*log(x**n*c)**2*b*d**2*e*x - 4*log(x**n*c)*b*d**3*n + 12*log(x**n*c)*b*d 
*e**2*n*x**2 + 2*log(x**n*c)*b*e**3*n*x**3 + 12*log(x)*a*d**2*e*n*x - 4*a* 
d**3*n + 12*a*d*e**2*n*x**2 + 2*a*e**3*n*x**3 - 4*b*d**3*n**2 - 12*b*d*e** 
2*n**2*x**2 - b*e**3*n**2*x**3)/(4*n*x)