\(\int \frac {(d+e x^2)^3 (a+b \log (c x^n))}{x^4} \, dx\) [206]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n}{x}-3 b d e^2 n x-\frac {1}{9} b e^3 n x^3-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} e^3 x^3 \left (a+b \log \left (c x^n\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {3 a \left (d^3+9 d^2 e x^2-9 d e^2 x^4-e^3 x^6\right )+b n \left (d^3+27 d^2 e x^2+27 d e^2 x^4+e^3 x^6\right )+3 b \left (d^3+9 d^2 e x^2-9 d e^2 x^4-e^3 x^6\right ) \log \left (c x^n\right )}{9 x^3} \] Input:

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

Output:

-1/9*(3*a*(d^3 + 9*d^2*e*x^2 - 9*d*e^2*x^4 - e^3*x^6) + b*n*(d^3 + 27*d^2* 
e*x^2 + 27*d*e^2*x^4 + e^3*x^6) + 3*b*(d^3 + 9*d^2*e*x^2 - 9*d*e^2*x^4 - e 
^3*x^6)*Log[c*x^n])/x^3
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2772, 27, 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.

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*(b*n*(d^3/(3*x^3) + (9*d^2*e)/x + 9*d*e^2*x + (e^3*x^3)/3)) - (d^3*(a 
 + b*Log[c*x^n]))/(3*x^3) - (3*d^2*e*(a + b*Log[c*x^n]))/x + 3*d*e^2*x*(a 
+ b*Log[c*x^n]) + (e^3*x^3*(a + b*Log[c*x^n]))/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 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 = 138, normalized size of antiderivative = 1.14

method	result	size
parallelrisch	\(-\frac {-3 b \ln \left (c \,x^{n}\right ) e^{3} x^{6}+b \,e^{3} n \,x^{6}-3 x^{6} a \,e^{3}-27 b \ln \left (c \,x^{n}\right ) e^{2} d \,x^{4}+27 b d \,e^{2} n \,x^{4}-27 x^{4} a \,e^{2} d +27 b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{2}+27 b \,d^{2} e n \,x^{2}+27 a \,d^{2} e \,x^{2}+3 b \ln \left (c \,x^{n}\right ) d^{3}+b \,d^{3} n +3 a \,d^{3}}{9 x^{3}}\)	\(138\)
risch	\(-\frac {b \left (-e^{3} x^{6}-9 e^{2} d \,x^{4}+9 d^{2} e \,x^{2}+d^{3}\right ) \ln \left (x^{n}\right )}{3 x^{3}}-\frac {6 a \,d^{3}+3 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-6 x^{6} a \,e^{3}-54 x^{4} a \,e^{2} d +54 a \,d^{2} e \,x^{2}-54 \ln \left (c \right ) b d \,e^{2} x^{4}+6 d^{3} b \ln \left (c \right )+54 \ln \left (c \right ) b \,d^{2} x^{2} e -3 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+27 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 b \,d^{3} n +27 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+27 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-27 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-3 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 )+3 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-27 i \pi b \,d^{2} e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-27 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+3 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 )+27 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 )+3 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+54 b d \,e^{2} n \,x^{4}+54 b \,d^{2} e n \,x^{2}-6 \ln \left (c \right ) b \,e^{3} x^{6}-27 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 )+2 b \,e^{3} n \,x^{6}-3 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{18 x^{3}}\)	\(585\)

Input:

int((e*x^2+d)^3*(a+b*ln(c*x^n))/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/9/x^3*(-3*b*ln(c*x^n)*e^3*x^6+b*e^3*n*x^6-3*x^6*a*e^3-27*b*ln(c*x^n)*e^ 
2*d*x^4+27*b*d*e^2*n*x^4-27*x^4*a*e^2*d+27*b*ln(c*x^n)*d^2*e*x^2+27*b*d^2* 
e*n*x^2+27*a*d^2*e*x^2+3*b*ln(c*x^n)*d^3+b*d^3*n+3*a*d^3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {{\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} + b d^{3} n + 27 \, {\left (b d e^{2} n - a d e^{2}\right )} x^{4} + 3 \, a d^{3} + 27 \, {\left (b d^{2} e n + a d^{2} e\right )} x^{2} - 3 \, {\left (b e^{3} x^{6} + 9 \, b d e^{2} x^{4} - 9 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) - 3 \, {\left (b e^{3} n x^{6} + 9 \, b d e^{2} n x^{4} - 9 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )}{9 \, x^{3}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")
 

Output:

-1/9*((b*e^3*n - 3*a*e^3)*x^6 + b*d^3*n + 27*(b*d*e^2*n - a*d*e^2)*x^4 + 3 
*a*d^3 + 27*(b*d^2*e*n + a*d^2*e)*x^2 - 3*(b*e^3*x^6 + 9*b*d*e^2*x^4 - 9*b 
*d^2*e*x^2 - b*d^3)*log(c) - 3*(b*e^3*n*x^6 + 9*b*d*e^2*n*x^4 - 9*b*d^2*e* 
n*x^2 - b*d^3*n)*log(x))/x^3
 

Sympy [A] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=- \frac {a d^{3}}{3 x^{3}} - \frac {3 a d^{2} e}{x} + 3 a d e^{2} x + \frac {a e^{3} x^{3}}{3} - \frac {b d^{3} n}{9 x^{3}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {3 b d^{2} e n}{x} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{x} - 3 b d e^{2} n x + 3 b d e^{2} x \log {\left (c x^{n} \right )} - \frac {b e^{3} n x^{3}}{9} + \frac {b e^{3} x^{3} \log {\left (c x^{n} \right )}}{3} \] Input:

integrate((e*x**2+d)**3*(a+b*ln(c*x**n))/x**4,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {1}{9} \, b e^{3} n x^{3} + \frac {1}{3} \, b e^{3} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a e^{3} x^{3} - 3 \, b d e^{2} n x + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x - \frac {3 \, b d^{2} e n}{x} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{x} - \frac {3 \, a d^{2} e}{x} - \frac {b d^{3} n}{9 \, x^{3}} - \frac {b d^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d^{3}}{3 \, x^{3}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {1}{9} \, {\left (b e^{3} n - 3 \, b e^{3} \log \left (c\right ) - 3 \, a e^{3}\right )} x^{3} - 3 \, {\left (b d e^{2} n - b d e^{2} \log \left (c\right ) - a d e^{2}\right )} x + \frac {1}{3} \, {\left (b e^{3} n x^{3} + 9 \, b d e^{2} n x - \frac {9 \, b d^{2} e n x^{2} + b d^{3} n}{x^{3}}\right )} \log \left (x\right ) - \frac {27 \, b d^{2} e n x^{2} + 27 \, b d^{2} e x^{2} \log \left (c\right ) + 27 \, a d^{2} e x^{2} + b d^{3} n + 3 \, b d^{3} \log \left (c\right ) + 3 \, a d^{3}}{9 \, x^{3}} \] Input:

integrate((e*x^2+d)^3*(a+b*log(c*x^n))/x^4,x, algorithm="giac")
 

Output:

-1/9*(b*e^3*n - 3*b*e^3*log(c) - 3*a*e^3)*x^3 - 3*(b*d*e^2*n - b*d*e^2*log 
(c) - a*d*e^2)*x + 1/3*(b*e^3*n*x^3 + 9*b*d*e^2*n*x - (9*b*d^2*e*n*x^2 + b 
*d^3*n)/x^3)*log(x) - 1/9*(27*b*d^2*e*n*x^2 + 27*b*d^2*e*x^2*log(c) + 27*a 
*d^2*e*x^2 + b*d^3*n + 3*b*d^3*log(c) + 3*a*d^3)/x^3
 

Mupad [B] (verification not implemented)

Time = 25.77 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\ln \left (c\,x^n\right )\,\left (\frac {\frac {8\,b\,e^3\,x^6}{3}+8\,b\,d\,e^2\,x^4}{x^3}-\frac {\frac {b\,d^3}{3}+3\,b\,d^2\,e\,x^2+5\,b\,d\,e^2\,x^4+\frac {7\,b\,e^3\,x^6}{3}}{x^3}\right )-\frac {a\,d^3+x^2\,\left (9\,a\,d^2\,e+9\,b\,d^2\,e\,n\right )+\frac {b\,d^3\,n}{3}}{3\,x^3}+\frac {e^3\,x^3\,\left (3\,a-b\,n\right )}{9}+3\,d\,e^2\,x\,\left (a-b\,n\right ) \] Input:

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

Output:

log(c*x^n)*(((8*b*e^3*x^6)/3 + 8*b*d*e^2*x^4)/x^3 - ((b*d^3)/3 + (7*b*e^3* 
x^6)/3 + 3*b*d^2*e*x^2 + 5*b*d*e^2*x^4)/x^3) - (a*d^3 + x^2*(9*a*d^2*e + 9 
*b*d^2*e*n) + (b*d^3*n)/3)/(3*x^3) + (e^3*x^3*(3*a - b*n))/9 + 3*d*e^2*x*( 
a - b*n)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {-3 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}-27 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e \,x^{2}+27 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{4}+3 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{6}-3 a \,d^{3}-27 a \,d^{2} e \,x^{2}+27 a d \,e^{2} x^{4}+3 a \,e^{3} x^{6}-b \,d^{3} n -27 b \,d^{2} e n \,x^{2}-27 b d \,e^{2} n \,x^{4}-b \,e^{3} n \,x^{6}}{9 x^{3}} \] Input:

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

Output:

( - 3*log(x**n*c)*b*d**3 - 27*log(x**n*c)*b*d**2*e*x**2 + 27*log(x**n*c)*b 
*d*e**2*x**4 + 3*log(x**n*c)*b*e**3*x**6 - 3*a*d**3 - 27*a*d**2*e*x**2 + 2 
7*a*d*e**2*x**4 + 3*a*e**3*x**6 - b*d**3*n - 27*b*d**2*e*n*x**2 - 27*b*d*e 
**2*n*x**4 - b*e**3*n*x**6)/(9*x**3)