\(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^{10}} \, dx\) [209]

Optimal result

Integrand size = 23, antiderivative size = 133 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {b d^3 n}{81 x^9}-\frac {3 b d^2 e n}{49 x^7}-\frac {3 b d e^2 n}{25 x^5}-\frac {b e^3 n}{9 x^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \] Output:

-1/81*b*d^3*n/x^9-3/49*b*d^2*e*n/x^7-3/25*b*d*e^2*n/x^5-1/9*b*e^3*n/x^3-1/ 
9*d^3*(a+b*ln(c*x^n))/x^9-3/7*d^2*e*(a+b*ln(c*x^n))/x^7-3/5*d*e^2*(a+b*ln( 
c*x^n))/x^5-1/3*e^3*(a+b*ln(c*x^n))/x^3
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {b d^3 n}{81 x^9}-\frac {3 b d^2 e n}{49 x^7}-\frac {3 b d e^2 n}{25 x^5}-\frac {b e^3 n}{9 x^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \] Input:

Integrate[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^10,x]
 

Output:

-1/81*(b*d^3*n)/x^9 - (3*b*d^2*e*n)/(49*x^7) - (3*b*d*e^2*n)/(25*x^5) - (b 
*e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c 
*x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c 
*x^n]))/(3*x^3)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98, 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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx\)

\(\Big \downarrow \) 2772

\(\displaystyle -b n \int -\frac {105 e^3 x^6+189 d e^2 x^4+135 d^2 e x^2+35 d^3}{315 x^{10}}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{315} b n \int \frac {105 e^3 x^6+189 d e^2 x^4+135 d^2 e x^2+35 d^3}{x^{10}}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {1}{315} b n \int \left (\frac {35 d^3}{x^{10}}+\frac {135 e d^2}{x^8}+\frac {189 e^2 d}{x^6}+\frac {105 e^3}{x^4}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {1}{315} b n \left (-\frac {35 d^3}{9 x^9}-\frac {135 d^2 e}{7 x^7}-\frac {189 d e^2}{5 x^5}-\frac {35 e^3}{x^3}\right )\)

Input:

Int[((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^10,x]
 

Output:

(b*n*((-35*d^3)/(9*x^9) - (135*d^2*e)/(7*x^7) - (189*d*e^2)/(5*x^5) - (35* 
e^3)/x^3))/315 - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c* 
x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (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 = 1.15 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.05

method	result	size
parallelrisch	\(-\frac {33075 b \ln \left (c \,x^{n}\right ) e^{3} x^{6}+11025 b \,e^{3} n \,x^{6}+33075 x^{6} a \,e^{3}+59535 b \ln \left (c \,x^{n}\right ) e^{2} d \,x^{4}+11907 b d \,e^{2} n \,x^{4}+59535 x^{4} a \,e^{2} d +42525 b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{2}+6075 b \,d^{2} e n \,x^{2}+42525 a \,d^{2} e \,x^{2}+11025 b \ln \left (c \,x^{n}\right ) d^{3}+1225 b \,d^{3} n +11025 a \,d^{3}}{99225 x^{9}}\)	\(140\)
risch	\(-\frac {b \left (105 e^{3} x^{6}+189 e^{2} d \,x^{4}+135 d^{2} e \,x^{2}+35 d^{3}\right ) \ln \left (x^{n}\right )}{315 x^{9}}-\frac {22050 a \,d^{3}+11025 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+66150 x^{6} a \,e^{3}+119070 x^{4} a \,e^{2} d +85050 a \,d^{2} e \,x^{2}+119070 \ln \left (c \right ) b d \,e^{2} x^{4}+22050 d^{3} b \ln \left (c \right )+85050 \ln \left (c \right ) b \,d^{2} x^{2} e -11025 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+42525 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2450 b \,d^{3} n -59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+42525 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-11025 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 )-33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-42525 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-33075 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 )-59535 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 )+11025 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+23814 b d \,e^{2} n \,x^{4}+12150 b \,d^{2} e n \,x^{2}+66150 \ln \left (c \right ) b \,e^{3} x^{6}-42525 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 )+22050 b \,e^{3} n \,x^{6}+33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{198450 x^{9}}\)	\(587\)

Input:

int((e*x^2+d)^3*(a+b*ln(c*x^n))/x^10,x,method=_RETURNVERBOSE)
 

Output:

-1/99225/x^9*(33075*b*ln(c*x^n)*e^3*x^6+11025*b*e^3*n*x^6+33075*x^6*a*e^3+ 
59535*b*ln(c*x^n)*e^2*d*x^4+11907*b*d*e^2*n*x^4+59535*x^4*a*e^2*d+42525*b* 
ln(c*x^n)*d^2*e*x^2+6075*b*d^2*e*n*x^2+42525*a*d^2*e*x^2+11025*b*ln(c*x^n) 
*d^3+1225*b*d^3*n+11025*a*d^3)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {11025 \, {\left (b e^{3} n + 3 \, a e^{3}\right )} x^{6} + 1225 \, b d^{3} n + 11907 \, {\left (b d e^{2} n + 5 \, a d e^{2}\right )} x^{4} + 11025 \, a d^{3} + 6075 \, {\left (b d^{2} e n + 7 \, a d^{2} e\right )} x^{2} + 315 \, {\left (105 \, b e^{3} x^{6} + 189 \, b d e^{2} x^{4} + 135 \, b d^{2} e x^{2} + 35 \, b d^{3}\right )} \log \left (c\right ) + 315 \, {\left (105 \, b e^{3} n x^{6} + 189 \, b d e^{2} n x^{4} + 135 \, b d^{2} e n x^{2} + 35 \, b d^{3} n\right )} \log \left (x\right )}{99225 \, x^{9}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^10,x, algorithm="fricas")
 

Output:

-1/99225*(11025*(b*e^3*n + 3*a*e^3)*x^6 + 1225*b*d^3*n + 11907*(b*d*e^2*n 
+ 5*a*d*e^2)*x^4 + 11025*a*d^3 + 6075*(b*d^2*e*n + 7*a*d^2*e)*x^2 + 315*(1 
05*b*e^3*x^6 + 189*b*d*e^2*x^4 + 135*b*d^2*e*x^2 + 35*b*d^3)*log(c) + 315* 
(105*b*e^3*n*x^6 + 189*b*d*e^2*n*x^4 + 135*b*d^2*e*n*x^2 + 35*b*d^3*n)*log 
(x))/x^9
 

Sympy [A] (verification not implemented)

Time = 1.87 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.33 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=- \frac {a d^{3}}{9 x^{9}} - \frac {3 a d^{2} e}{7 x^{7}} - \frac {3 a d e^{2}}{5 x^{5}} - \frac {a e^{3}}{3 x^{3}} - \frac {b d^{3} n}{81 x^{9}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{9 x^{9}} - \frac {3 b d^{2} e n}{49 x^{7}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{7 x^{7}} - \frac {3 b d e^{2} n}{25 x^{5}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b e^{3} n}{9 x^{3}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} \] Input:

integrate((e*x**2+d)**3*(a+b*ln(c*x**n))/x**10,x)
 

Output:

-a*d**3/(9*x**9) - 3*a*d**2*e/(7*x**7) - 3*a*d*e**2/(5*x**5) - a*e**3/(3*x 
**3) - b*d**3*n/(81*x**9) - b*d**3*log(c*x**n)/(9*x**9) - 3*b*d**2*e*n/(49 
*x**7) - 3*b*d**2*e*log(c*x**n)/(7*x**7) - 3*b*d*e**2*n/(25*x**5) - 3*b*d* 
e**2*log(c*x**n)/(5*x**5) - 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 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {b e^{3} n}{9 \, x^{3}} - \frac {b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e^{3}}{3 \, x^{3}} - \frac {3 \, b d e^{2} n}{25 \, x^{5}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {3 \, a d e^{2}}{5 \, x^{5}} - \frac {3 \, b d^{2} e n}{49 \, x^{7}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {3 \, a d^{2} e}{7 \, x^{7}} - \frac {b d^{3} n}{81 \, x^{9}} - \frac {b d^{3} \log \left (c x^{n}\right )}{9 \, x^{9}} - \frac {a d^{3}}{9 \, x^{9}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^10,x, algorithm="maxima")
 

Output:

-1/9*b*e^3*n/x^3 - 1/3*b*e^3*log(c*x^n)/x^3 - 1/3*a*e^3/x^3 - 3/25*b*d*e^2 
*n/x^5 - 3/5*b*d*e^2*log(c*x^n)/x^5 - 3/5*a*d*e^2/x^5 - 3/49*b*d^2*e*n/x^7 
 - 3/7*b*d^2*e*log(c*x^n)/x^7 - 3/7*a*d^2*e/x^7 - 1/81*b*d^3*n/x^9 - 1/9*b 
*d^3*log(c*x^n)/x^9 - 1/9*a*d^3/x^9
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {{\left (105 \, b e^{3} n x^{6} + 189 \, b d e^{2} n x^{4} + 135 \, b d^{2} e n x^{2} + 35 \, b d^{3} n\right )} \log \left (x\right )}{315 \, x^{9}} - \frac {11025 \, b e^{3} n x^{6} + 33075 \, b e^{3} x^{6} \log \left (c\right ) + 33075 \, a e^{3} x^{6} + 11907 \, b d e^{2} n x^{4} + 59535 \, b d e^{2} x^{4} \log \left (c\right ) + 59535 \, a d e^{2} x^{4} + 6075 \, b d^{2} e n x^{2} + 42525 \, b d^{2} e x^{2} \log \left (c\right ) + 42525 \, a d^{2} e x^{2} + 1225 \, b d^{3} n + 11025 \, b d^{3} \log \left (c\right ) + 11025 \, a d^{3}}{99225 \, x^{9}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^10,x, algorithm="giac")
 

Output:

-1/315*(105*b*e^3*n*x^6 + 189*b*d*e^2*n*x^4 + 135*b*d^2*e*n*x^2 + 35*b*d^3 
*n)*log(x)/x^9 - 1/99225*(11025*b*e^3*n*x^6 + 33075*b*e^3*x^6*log(c) + 330 
75*a*e^3*x^6 + 11907*b*d*e^2*n*x^4 + 59535*b*d*e^2*x^4*log(c) + 59535*a*d* 
e^2*x^4 + 6075*b*d^2*e*n*x^2 + 42525*b*d^2*e*x^2*log(c) + 42525*a*d^2*e*x^ 
2 + 1225*b*d^3*n + 11025*b*d^3*log(c) + 11025*a*d^3)/x^9
 

Mupad [B] (verification not implemented)

Time = 25.82 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {x^6\,\left (105\,a\,e^3+35\,b\,e^3\,n\right )+35\,a\,d^3+x^2\,\left (135\,a\,d^2\,e+\frac {135\,b\,d^2\,e\,n}{7}\right )+x^4\,\left (189\,a\,d\,e^2+\frac {189\,b\,d\,e^2\,n}{5}\right )+\frac {35\,b\,d^3\,n}{9}}{315\,x^9}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{9}+\frac {3\,b\,d^2\,e\,x^2}{7}+\frac {3\,b\,d\,e^2\,x^4}{5}+\frac {b\,e^3\,x^6}{3}\right )}{x^9} \] Input:

int(((d + e*x^2)^3*(a + b*log(c*x^n)))/x^10,x)
 

Output:

- (x^6*(105*a*e^3 + 35*b*e^3*n) + 35*a*d^3 + x^2*(135*a*d^2*e + (135*b*d^2 
*e*n)/7) + x^4*(189*a*d*e^2 + (189*b*d*e^2*n)/5) + (35*b*d^3*n)/9)/(315*x^ 
9) - (log(c*x^n)*((b*d^3)/9 + (b*e^3*x^6)/3 + (3*b*d^2*e*x^2)/7 + (3*b*d*e 
^2*x^4)/5))/x^9
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=\frac {-11025 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}-42525 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e \,x^{2}-59535 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{4}-33075 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{6}-11025 a \,d^{3}-42525 a \,d^{2} e \,x^{2}-59535 a d \,e^{2} x^{4}-33075 a \,e^{3} x^{6}-1225 b \,d^{3} n -6075 b \,d^{2} e n \,x^{2}-11907 b d \,e^{2} n \,x^{4}-11025 b \,e^{3} n \,x^{6}}{99225 x^{9}} \] Input:

int((e*x^2+d)^3*(a+b*log(c*x^n))/x^10,x)
 

Output:

( - 11025*log(x**n*c)*b*d**3 - 42525*log(x**n*c)*b*d**2*e*x**2 - 59535*log 
(x**n*c)*b*d*e**2*x**4 - 33075*log(x**n*c)*b*e**3*x**6 - 11025*a*d**3 - 42 
525*a*d**2*e*x**2 - 59535*a*d*e**2*x**4 - 33075*a*e**3*x**6 - 1225*b*d**3* 
n - 6075*b*d**2*e*n*x**2 - 11907*b*d*e**2*n*x**4 - 11025*b*e**3*n*x**6)/(9 
9225*x**9)