\(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^8} \, dx\) [208]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x} \] Output:

-1/49*b*d^3*n/x^7-3/25*b*d^2*e*n/x^5-1/3*b*d*e^2*n/x^3-b*e^3*n/x-1/7*d^3*( 
a+b*ln(c*x^n))/x^7-3/5*d^2*e*(a+b*ln(c*x^n))/x^5-d*e^2*(a+b*ln(c*x^n))/x^3 
-e^3*(a+b*ln(c*x^n))/x
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n}{25 x^5}-\frac {b d e^2 n}{3 x^3}-\frac {b e^3 n}{x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x} \] Input:

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^8,x]
 

Output:

-1/49*(b*d^3*n)/x^7 - (3*b*d^2*e*n)/(25*x^5) - (b*d*e^2*n)/(3*x^3) - (b*e^ 
3*n)/x - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (3*d^2*e*(a + b*Log[c*x^n]))/( 
5*x^5) - (d*e^2*(a + b*Log[c*x^n]))/x^3 - (e^3*(a + b*Log[c*x^n]))/x
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx\)

\(\Big \downarrow \) 2772

\(\displaystyle -b n \int -\frac {35 e^3 x^6+35 d e^2 x^4+21 d^2 e x^2+5 d^3}{35 x^8}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{35} b n \int \frac {35 e^3 x^6+35 d e^2 x^4+21 d^2 e x^2+5 d^3}{x^8}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {1}{35} b n \int \left (\frac {5 d^3}{x^8}+\frac {21 e d^2}{x^6}+\frac {35 e^2 d}{x^4}+\frac {35 e^3}{x^2}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {d e^2 \left (a+b \log \left (c x^n\right )\right )}{x^3}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {1}{35} b n \left (-\frac {5 d^3}{7 x^7}-\frac {21 d^2 e}{5 x^5}-\frac {35 d e^2}{3 x^3}-\frac {35 e^3}{x}\right )\)

Input:

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^8,x]
 

Output:

(b*n*((-5*d^3)/(7*x^7) - (21*d^2*e)/(5*x^5) - (35*d*e^2)/(3*x^3) - (35*e^3 
)/x))/35 - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (3*d^2*e*(a + b*Log[c*x^n])) 
/(5*x^5) - (d*e^2*(a + b*Log[c*x^n]))/x^3 - (e^3*(a + b*Log[c*x^n]))/x
 

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 = 1.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.10

method	result	size
parallelrisch	\(-\frac {3675 b \ln \left (c \,x^{n}\right ) e^{3} x^{6}+3675 b \,e^{3} n \,x^{6}+3675 x^{6} a \,e^{3}+3675 b \ln \left (c \,x^{n}\right ) e^{2} d \,x^{4}+1225 b d \,e^{2} n \,x^{4}+3675 x^{4} a \,e^{2} d +2205 b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{2}+441 b \,d^{2} e n \,x^{2}+2205 a \,d^{2} e \,x^{2}+525 b \ln \left (c \,x^{n}\right ) d^{3}+75 b \,d^{3} n +525 a \,d^{3}}{3675 x^{7}}\)	\(140\)
risch	\(-\frac {b \left (35 e^{3} x^{6}+35 e^{2} d \,x^{4}+21 d^{2} e \,x^{2}+5 d^{3}\right ) \ln \left (x^{n}\right )}{35 x^{7}}-\frac {1050 a \,d^{3}+525 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+7350 x^{6} a \,e^{3}+7350 x^{4} a \,e^{2} d +4410 a \,d^{2} e \,x^{2}+7350 \ln \left (c \right ) b d \,e^{2} x^{4}+1050 d^{3} b \ln \left (c \right )+4410 \ln \left (c \right ) b \,d^{2} x^{2} e -525 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2205 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+150 b \,d^{3} n -3675 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2205 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+3675 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3675 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-525 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 )-3675 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2205 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3675 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-3675 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-3675 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+525 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2450 b d \,e^{2} n \,x^{4}+882 b \,d^{2} e n \,x^{2}+7350 \ln \left (c \right ) b \,e^{3} x^{6}-2205 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+7350 b \,e^{3} n \,x^{6}+3675 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{7350 x^{7}}\)	\(587\)

Input:

int((e*x^2+d)^3*(a+b*ln(c*x^n))/x^8,x,method=_RETURNVERBOSE)
 

Output:

-1/3675/x^7*(3675*b*ln(c*x^n)*e^3*x^6+3675*b*e^3*n*x^6+3675*x^6*a*e^3+3675 
*b*ln(c*x^n)*e^2*d*x^4+1225*b*d*e^2*n*x^4+3675*x^4*a*e^2*d+2205*b*ln(c*x^n 
)*d^2*e*x^2+441*b*d^2*e*n*x^2+2205*a*d^2*e*x^2+525*b*ln(c*x^n)*d^3+75*b*d^ 
3*n+525*a*d^3)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {3675 \, {\left (b e^{3} n + a e^{3}\right )} x^{6} + 75 \, b d^{3} n + 1225 \, {\left (b d e^{2} n + 3 \, a d e^{2}\right )} x^{4} + 525 \, a d^{3} + 441 \, {\left (b d^{2} e n + 5 \, a d^{2} e\right )} x^{2} + 105 \, {\left (35 \, b e^{3} x^{6} + 35 \, b d e^{2} x^{4} + 21 \, b d^{2} e x^{2} + 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \, {\left (35 \, b e^{3} n x^{6} + 35 \, b d e^{2} n x^{4} + 21 \, b d^{2} e n x^{2} + 5 \, b d^{3} n\right )} \log \left (x\right )}{3675 \, x^{7}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="fricas")
 

Output:

-1/3675*(3675*(b*e^3*n + a*e^3)*x^6 + 75*b*d^3*n + 1225*(b*d*e^2*n + 3*a*d 
*e^2)*x^4 + 525*a*d^3 + 441*(b*d^2*e*n + 5*a*d^2*e)*x^2 + 105*(35*b*e^3*x^ 
6 + 35*b*d*e^2*x^4 + 21*b*d^2*e*x^2 + 5*b*d^3)*log(c) + 105*(35*b*e^3*n*x^ 
6 + 35*b*d*e^2*n*x^4 + 21*b*d^2*e*n*x^2 + 5*b*d^3*n)*log(x))/x^7
 

Sympy [A] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=- \frac {a d^{3}}{7 x^{7}} - \frac {3 a d^{2} e}{5 x^{5}} - \frac {a d e^{2}}{x^{3}} - \frac {a e^{3}}{x} - \frac {b d^{3} n}{49 x^{7}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{7 x^{7}} - \frac {3 b d^{2} e n}{25 x^{5}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b d e^{2} n}{3 x^{3}} - \frac {b d e^{2} \log {\left (c x^{n} \right )}}{x^{3}} - \frac {b e^{3} n}{x} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{x} \] Input:

integrate((e*x**2+d)**3*(a+b*ln(c*x**n))/x**8,x)
 

Output:

-a*d**3/(7*x**7) - 3*a*d**2*e/(5*x**5) - a*d*e**2/x**3 - a*e**3/x - b*d**3 
*n/(49*x**7) - b*d**3*log(c*x**n)/(7*x**7) - 3*b*d**2*e*n/(25*x**5) - 3*b* 
d**2*e*log(c*x**n)/(5*x**5) - b*d*e**2*n/(3*x**3) - b*d*e**2*log(c*x**n)/x 
**3 - b*e**3*n/x - b*e**3*log(c*x**n)/x
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b e^{3} n}{x} - \frac {b e^{3} \log \left (c x^{n}\right )}{x} - \frac {a e^{3}}{x} - \frac {b d e^{2} n}{3 \, x^{3}} - \frac {b d e^{2} \log \left (c x^{n}\right )}{x^{3}} - \frac {a d e^{2}}{x^{3}} - \frac {3 \, b d^{2} e n}{25 \, x^{5}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {3 \, a d^{2} e}{5 \, x^{5}} - \frac {b d^{3} n}{49 \, x^{7}} - \frac {b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {a d^{3}}{7 \, x^{7}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="maxima")
 

Output:

-b*e^3*n/x - b*e^3*log(c*x^n)/x - a*e^3/x - 1/3*b*d*e^2*n/x^3 - b*d*e^2*lo 
g(c*x^n)/x^3 - a*d*e^2/x^3 - 3/25*b*d^2*e*n/x^5 - 3/5*b*d^2*e*log(c*x^n)/x 
^5 - 3/5*a*d^2*e/x^5 - 1/49*b*d^3*n/x^7 - 1/7*b*d^3*log(c*x^n)/x^7 - 1/7*a 
*d^3/x^7
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.35 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {{\left (35 \, b e^{3} n x^{6} + 35 \, b d e^{2} n x^{4} + 21 \, b d^{2} e n x^{2} + 5 \, b d^{3} n\right )} \log \left (x\right )}{35 \, x^{7}} - \frac {3675 \, b e^{3} n x^{6} + 3675 \, b e^{3} x^{6} \log \left (c\right ) + 3675 \, a e^{3} x^{6} + 1225 \, b d e^{2} n x^{4} + 3675 \, b d e^{2} x^{4} \log \left (c\right ) + 3675 \, a d e^{2} x^{4} + 441 \, b d^{2} e n x^{2} + 2205 \, b d^{2} e x^{2} \log \left (c\right ) + 2205 \, a d^{2} e x^{2} + 75 \, b d^{3} n + 525 \, b d^{3} \log \left (c\right ) + 525 \, a d^{3}}{3675 \, x^{7}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="giac")
 

Output:

-1/35*(35*b*e^3*n*x^6 + 35*b*d*e^2*n*x^4 + 21*b*d^2*e*n*x^2 + 5*b*d^3*n)*l 
og(x)/x^7 - 1/3675*(3675*b*e^3*n*x^6 + 3675*b*e^3*x^6*log(c) + 3675*a*e^3* 
x^6 + 1225*b*d*e^2*n*x^4 + 3675*b*d*e^2*x^4*log(c) + 3675*a*d*e^2*x^4 + 44 
1*b*d^2*e*n*x^2 + 2205*b*d^2*e*x^2*log(c) + 2205*a*d^2*e*x^2 + 75*b*d^3*n 
+ 525*b*d^3*log(c) + 525*a*d^3)/x^7
 

Mupad [B] (verification not implemented)

Time = 25.69 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {x^6\,\left (35\,a\,e^3+35\,b\,e^3\,n\right )+5\,a\,d^3+x^2\,\left (21\,a\,d^2\,e+\frac {21\,b\,d^2\,e\,n}{5}\right )+x^4\,\left (35\,a\,d\,e^2+\frac {35\,b\,d\,e^2\,n}{3}\right )+\frac {5\,b\,d^3\,n}{7}}{35\,x^7}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{7}+\frac {3\,b\,d^2\,e\,x^2}{5}+b\,d\,e^2\,x^4+b\,e^3\,x^6\right )}{x^7} \] Input:

int(((d + e*x^2)^3*(a + b*log(c*x^n)))/x^8,x)
 

Output:

- (x^6*(35*a*e^3 + 35*b*e^3*n) + 5*a*d^3 + x^2*(21*a*d^2*e + (21*b*d^2*e*n 
)/5) + x^4*(35*a*d*e^2 + (35*b*d*e^2*n)/3) + (5*b*d^3*n)/7)/(35*x^7) - (lo 
g(c*x^n)*((b*d^3)/7 + b*e^3*x^6 + (3*b*d^2*e*x^2)/5 + b*d*e^2*x^4))/x^7
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {-525 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}-2205 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e \,x^{2}-3675 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{4}-3675 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{6}-525 a \,d^{3}-2205 a \,d^{2} e \,x^{2}-3675 a d \,e^{2} x^{4}-3675 a \,e^{3} x^{6}-75 b \,d^{3} n -441 b \,d^{2} e n \,x^{2}-1225 b d \,e^{2} n \,x^{4}-3675 b \,e^{3} n \,x^{6}}{3675 x^{7}} \] Input:

int((e*x^2+d)^3*(a+b*log(c*x^n))/x^8,x)
 

Output:

( - 525*log(x**n*c)*b*d**3 - 2205*log(x**n*c)*b*d**2*e*x**2 - 3675*log(x** 
n*c)*b*d*e**2*x**4 - 3675*log(x**n*c)*b*e**3*x**6 - 525*a*d**3 - 2205*a*d* 
*2*e*x**2 - 3675*a*d*e**2*x**4 - 3675*a*e**3*x**6 - 75*b*d**3*n - 441*b*d* 
*2*e*n*x**2 - 1225*b*d*e**2*n*x**4 - 3675*b*e**3*n*x**6)/(3675*x**7)