\(\int \frac {(d+e x)^3 (a+b \log (c x^n))}{x^7} \, dx\) [29]

Optimal result

Integrand size = 21, antiderivative size = 133 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=-\frac {b d^3 n}{36 x^6}-\frac {3 b d^2 e n}{25 x^5}-\frac {3 b d e^2 n}{16 x^4}-\frac {b e^3 n}{9 x^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \] Output:

-1/36*b*d^3*n/x^6-3/25*b*d^2*e*n/x^5-3/16*b*d*e^2*n/x^4-1/9*b*e^3*n/x^3-1/ 
6*d^3*(a+b*ln(c*x^n))/x^6-3/5*d^2*e*(a+b*ln(c*x^n))/x^5-3/4*d*e^2*(a+b*ln( 
c*x^n))/x^4-1/3*e^3*(a+b*ln(c*x^n))/x^3
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=-\frac {60 a \left (10 d^3+36 d^2 e x+45 d e^2 x^2+20 e^3 x^3\right )+b n \left (100 d^3+432 d^2 e x+675 d e^2 x^2+400 e^3 x^3\right )+60 b \left (10 d^3+36 d^2 e x+45 d e^2 x^2+20 e^3 x^3\right ) \log \left (c x^n\right )}{3600 x^6} \] Input:

Integrate[((d + e*x)^3*(a + b*Log[c*x^n]))/x^7,x]
 

Output:

-1/3600*(60*a*(10*d^3 + 36*d^2*e*x + 45*d*e^2*x^2 + 20*e^3*x^3) + b*n*(100 
*d^3 + 432*d^2*e*x + 675*d*e^2*x^2 + 400*e^3*x^3) + 60*b*(10*d^3 + 36*d^2* 
e*x + 45*d*e^2*x^2 + 20*e^3*x^3)*Log[c*x^n])/x^6
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2772, 27, 2010, 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.

\(\displaystyle \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx\)

\(\Big \downarrow \) 2772

\(\displaystyle -b n \int -\frac {10 d^3+36 e x d^2+45 e^2 x^2 d+20 e^3 x^3}{60 x^7}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{60} b n \int \frac {10 d^3+36 e x d^2+45 e^2 x^2 d+20 e^3 x^3}{x^7}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {1}{60} b n \int \left (\frac {10 d^3}{x^7}+\frac {36 e d^2}{x^6}+\frac {45 e^2 d}{x^5}+\frac {20 e^3}{x^4}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {1}{60} b n \left (-\frac {5 d^3}{3 x^6}-\frac {36 d^2 e}{5 x^5}-\frac {45 d e^2}{4 x^4}-\frac {20 e^3}{3 x^3}\right )\)

Input:

Int[((d + e*x)^3*(a + b*Log[c*x^n]))/x^7,x]
 

Output:

(b*n*((-5*d^3)/(3*x^6) - (36*d^2*e)/(5*x^5) - (45*d*e^2)/(4*x^4) - (20*e^3 
)/(3*x^3)))/60 - (d^3*(a + b*Log[c*x^n]))/(6*x^6) - (3*d^2*e*(a + b*Log[c* 
x^n]))/(5*x^5) - (3*d*e^2*(a + b*Log[c*x^n]))/(4*x^4) - (e^3*(a + b*Log[c* 
x^n]))/(3*x^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

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 = 0.53 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01

method	result	size
parallelrisch	\(-\frac {1200 b \ln \left (c \,x^{n}\right ) e^{3} x^{3}+400 b \,e^{3} n \,x^{3}+1200 a \,e^{3} x^{3}+2700 b \ln \left (c \,x^{n}\right ) d \,e^{2} x^{2}+675 b d \,e^{2} n \,x^{2}+2700 a \,e^{2} x^{2} d +2160 b \ln \left (c \,x^{n}\right ) d^{2} e x +432 b \,d^{2} e n x +2160 a \,d^{2} e x +600 b \ln \left (c \,x^{n}\right ) d^{3}+100 b \,d^{3} n +600 a \,d^{3}}{3600 x^{6}}\)	\(134\)
risch	\(-\frac {b \left (20 e^{3} x^{3}+45 d \,e^{2} x^{2}+36 d^{2} e x +10 d^{3}\right ) \ln \left (x^{n}\right )}{60 x^{6}}-\frac {600 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2700 a \,e^{2} x^{2} d +2160 a \,d^{2} e x +600 a \,d^{3}-600 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+1200 \ln \left (c \right ) b \,e^{3} x^{3}+2700 \ln \left (c \right ) b d \,e^{2} x^{2}+2160 \ln \left (c \right ) b \,d^{2} e x -600 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 )+1080 i \pi b \,d^{2} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+1080 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) e x +1350 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+1350 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+600 d^{3} b \ln \left (c \right )-300 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+1200 a \,e^{3} x^{3}+100 b \,d^{3} n -300 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 )+600 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-1350 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-1080 i \pi b \,d^{2} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-1080 i \pi b \,d^{2} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-1350 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 )+432 b \,d^{2} e n x +675 b d \,e^{2} n \,x^{2}+400 b \,e^{3} n \,x^{3}}{3600 x^{6}}\)	\(571\)

Input:

int((e*x+d)^3*(a+b*ln(c*x^n))/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/3600/x^6*(1200*b*ln(c*x^n)*e^3*x^3+400*b*e^3*n*x^3+1200*a*e^3*x^3+2700* 
b*ln(c*x^n)*d*e^2*x^2+675*b*d*e^2*n*x^2+2700*a*e^2*x^2*d+2160*b*ln(c*x^n)* 
d^2*e*x+432*b*d^2*e*n*x+2160*a*d^2*e*x+600*b*ln(c*x^n)*d^3+100*b*d^3*n+600 
*a*d^3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=-\frac {100 \, b d^{3} n + 600 \, a d^{3} + 400 \, {\left (b e^{3} n + 3 \, a e^{3}\right )} x^{3} + 675 \, {\left (b d e^{2} n + 4 \, a d e^{2}\right )} x^{2} + 432 \, {\left (b d^{2} e n + 5 \, a d^{2} e\right )} x + 60 \, {\left (20 \, b e^{3} x^{3} + 45 \, b d e^{2} x^{2} + 36 \, b d^{2} e x + 10 \, b d^{3}\right )} \log \left (c\right ) + 60 \, {\left (20 \, b e^{3} n x^{3} + 45 \, b d e^{2} n x^{2} + 36 \, b d^{2} e n x + 10 \, b d^{3} n\right )} \log \left (x\right )}{3600 \, x^{6}} \] Input:

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^7,x, algorithm="fricas")
 

Output:

-1/3600*(100*b*d^3*n + 600*a*d^3 + 400*(b*e^3*n + 3*a*e^3)*x^3 + 675*(b*d* 
e^2*n + 4*a*d*e^2)*x^2 + 432*(b*d^2*e*n + 5*a*d^2*e)*x + 60*(20*b*e^3*x^3 
+ 45*b*d*e^2*x^2 + 36*b*d^2*e*x + 10*b*d^3)*log(c) + 60*(20*b*e^3*n*x^3 + 
45*b*d*e^2*n*x^2 + 36*b*d^2*e*n*x + 10*b*d^3*n)*log(x))/x^6
 

Sympy [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=- \frac {a d^{3}}{6 x^{6}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {3 a d e^{2}}{4 x^{4}} - \frac {a e^{3}}{3 x^{3}} - \frac {b d^{3} n}{36 x^{6}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{6 x^{6}} - \frac {3 b d^{2} e n}{25 x^{5}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {3 b d e^{2} n}{16 x^{4}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{4 x^{4}} - \frac {b e^{3} n}{9 x^{3}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} \] Input:

integrate((e*x+d)**3*(a+b*ln(c*x**n))/x**7,x)
 

Output:

-a*d**3/(6*x**6) - 3*a*d**2*e/(5*x**5) - 3*a*d*e**2/(4*x**4) - a*e**3/(3*x 
**3) - b*d**3*n/(36*x**6) - b*d**3*log(c*x**n)/(6*x**6) - 3*b*d**2*e*n/(25 
*x**5) - 3*b*d**2*e*log(c*x**n)/(5*x**5) - 3*b*d*e**2*n/(16*x**4) - 3*b*d* 
e**2*log(c*x**n)/(4*x**4) - b*e**3*n/(9*x**3) - b*e**3*log(c*x**n)/(3*x**3 
)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=-\frac {b e^{3} n}{9 \, x^{3}} - \frac {b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {3 \, b d e^{2} n}{16 \, x^{4}} - \frac {a e^{3}}{3 \, x^{3}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {3 \, b d^{2} e n}{25 \, x^{5}} - \frac {3 \, a d e^{2}}{4 \, x^{4}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {b d^{3} n}{36 \, x^{6}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {b d^{3} \log \left (c x^{n}\right )}{6 \, x^{6}} - \frac {a d^{3}}{6 \, x^{6}} \] Input:

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^7,x, algorithm="maxima")
 

Output:

-1/9*b*e^3*n/x^3 - 1/3*b*e^3*log(c*x^n)/x^3 - 3/16*b*d*e^2*n/x^4 - 1/3*a*e 
^3/x^3 - 3/4*b*d*e^2*log(c*x^n)/x^4 - 3/25*b*d^2*e*n/x^5 - 3/4*a*d*e^2/x^4 
 - 3/5*b*d^2*e*log(c*x^n)/x^5 - 1/36*b*d^3*n/x^6 - 3/5*a*d^2*e/x^5 - 1/6*b 
*d^3*log(c*x^n)/x^6 - 1/6*a*d^3/x^6
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=-\frac {{\left (20 \, b e^{3} n x^{3} + 45 \, b d e^{2} n x^{2} + 36 \, b d^{2} e n x + 10 \, b d^{3} n\right )} \log \left (x\right )}{60 \, x^{6}} - \frac {400 \, b e^{3} n x^{3} + 1200 \, b e^{3} x^{3} \log \left (c\right ) + 675 \, b d e^{2} n x^{2} + 1200 \, a e^{3} x^{3} + 2700 \, b d e^{2} x^{2} \log \left (c\right ) + 432 \, b d^{2} e n x + 2700 \, a d e^{2} x^{2} + 2160 \, b d^{2} e x \log \left (c\right ) + 100 \, b d^{3} n + 2160 \, a d^{2} e x + 600 \, b d^{3} \log \left (c\right ) + 600 \, a d^{3}}{3600 \, x^{6}} \] Input:

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^7,x, algorithm="giac")
 

Output:

-1/60*(20*b*e^3*n*x^3 + 45*b*d*e^2*n*x^2 + 36*b*d^2*e*n*x + 10*b*d^3*n)*lo 
g(x)/x^6 - 1/3600*(400*b*e^3*n*x^3 + 1200*b*e^3*x^3*log(c) + 675*b*d*e^2*n 
*x^2 + 1200*a*e^3*x^3 + 2700*b*d*e^2*x^2*log(c) + 432*b*d^2*e*n*x + 2700*a 
*d*e^2*x^2 + 2160*b*d^2*e*x*log(c) + 100*b*d^3*n + 2160*a*d^2*e*x + 600*b* 
d^3*log(c) + 600*a*d^3)/x^6
 

Mupad [B] (verification not implemented)

Time = 27.77 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=-\frac {x^3\,\left (20\,a\,e^3+\frac {20\,b\,e^3\,n}{3}\right )+x\,\left (36\,a\,d^2\,e+\frac {36\,b\,d^2\,e\,n}{5}\right )+10\,a\,d^3+x^2\,\left (45\,a\,d\,e^2+\frac {45\,b\,d\,e^2\,n}{4}\right )+\frac {5\,b\,d^3\,n}{3}}{60\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{6}+\frac {3\,b\,d^2\,e\,x}{5}+\frac {3\,b\,d\,e^2\,x^2}{4}+\frac {b\,e^3\,x^3}{3}\right )}{x^6} \] Input:

int(((a + b*log(c*x^n))*(d + e*x)^3)/x^7,x)
 

Output:

- (x^3*(20*a*e^3 + (20*b*e^3*n)/3) + x*(36*a*d^2*e + (36*b*d^2*e*n)/5) + 1 
0*a*d^3 + x^2*(45*a*d*e^2 + (45*b*d*e^2*n)/4) + (5*b*d^3*n)/3)/(60*x^6) - 
(log(c*x^n)*((b*d^3)/6 + (b*e^3*x^3)/3 + (3*b*d^2*e*x)/5 + (3*b*d*e^2*x^2) 
/4))/x^6
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx=\frac {-600 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}-2160 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e x -2700 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{2}-1200 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{3}-600 a \,d^{3}-2160 a \,d^{2} e x -2700 a d \,e^{2} x^{2}-1200 a \,e^{3} x^{3}-100 b \,d^{3} n -432 b \,d^{2} e n x -675 b d \,e^{2} n \,x^{2}-400 b \,e^{3} n \,x^{3}}{3600 x^{6}} \] Input:

int((e*x+d)^3*(a+b*log(c*x^n))/x^7,x)
 

Output:

( - 600*log(x**n*c)*b*d**3 - 2160*log(x**n*c)*b*d**2*e*x - 2700*log(x**n*c 
)*b*d*e**2*x**2 - 1200*log(x**n*c)*b*e**3*x**3 - 600*a*d**3 - 2160*a*d**2* 
e*x - 2700*a*d*e**2*x**2 - 1200*a*e**3*x**3 - 100*b*d**3*n - 432*b*d**2*e* 
n*x - 675*b*d*e**2*n*x**2 - 400*b*e**3*n*x**3)/(3600*x**6)