\(\int (d+e x^2)^3 (a+b \log (c x^n)) \, dx\) [204]

Optimal result

Integrand size = 20, antiderivative size = 121 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^3 n x-\frac {1}{3} b d^2 e n x^3-\frac {3}{25} b d e^2 n x^5-\frac {1}{49} b e^3 n x^7+d^3 x \left (a+b \log \left (c x^n\right )\right )+d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=a d^3 x-b d^3 n x-\frac {1}{3} b d^2 e n x^3-\frac {3}{25} b d e^2 n x^5-\frac {1}{49} b e^3 n x^7+b d^3 x \log \left (c x^n\right )+d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2750, 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.

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

\(\Big \downarrow \) 2750

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

\(\Big \downarrow \) 2009

\(\displaystyle d^3 x \left (a+b \log \left (c x^n\right )\right )+d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right )-b n \left (d^3 x+\frac {1}{3} d^2 e x^3+\frac {3}{25} d e^2 x^5+\frac {e^3 x^7}{49}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2750

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), 
x_Symbol] :> With[{u = IntHide[(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]
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.11

method	result	size
parallelrisch	\(\frac {b \ln \left (c \,x^{n}\right ) e^{3} x^{7}}{7}-\frac {b \,e^{3} n \,x^{7}}{49}+\frac {a \,e^{3} x^{7}}{7}+\frac {3 b \ln \left (c \,x^{n}\right ) e^{2} d \,x^{5}}{5}-\frac {3 b d \,e^{2} n \,x^{5}}{25}+\frac {3 x^{5} a \,e^{2} d}{5}+b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{3}-\frac {b \,d^{2} e n \,x^{3}}{3}+x^{3} a \,d^{2} e +x b \ln \left (c \,x^{n}\right ) d^{3}-b \,d^{3} n x +x a \,d^{3}\)	\(134\)
risch	\(\frac {3 x^{5} a \,e^{2} d}{5}+x a \,d^{3}+\frac {\ln \left (c \right ) b \,e^{3} x^{7}}{7}+x^{3} a \,d^{2} e +\ln \left (c \right ) b \,d^{2} e \,x^{3}+x \ln \left (c \right ) b \,d^{3}+\frac {a \,e^{3} x^{7}}{7}+\frac {b x \left (5 e^{3} x^{6}+21 e^{2} d \,x^{4}+35 d^{2} e \,x^{2}+35 d^{3}\right ) \ln \left (x^{n}\right )}{35}-\frac {3 i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3} b d \,e^{2} x^{5}}{10}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3} b \,d^{2} e \,x^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) b \,d^{3} x}{2}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right ) b \,d^{3} x}{2}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) b \,e^{3} x^{7}}{14}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3} b \,e^{3} x^{7}}{14}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3} b \,d^{3} x}{2}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right ) b \,e^{3} x^{7}}{14}-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) b \,d^{2} e \,x^{3}}{2}-\frac {3 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) b d \,e^{2} x^{5}}{10}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) b \,d^{2} e \,x^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) b \,e^{3} x^{7}}{14}+\frac {3 i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right ) b d \,e^{2} x^{5}}{10}+\frac {3 i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right ) b d \,e^{2} x^{5}}{10}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right ) b \,d^{2} e \,x^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) b \,d^{3} x}{2}+\frac {3 \ln \left (c \right ) b d \,e^{2} x^{5}}{5}-\frac {3 b d \,e^{2} n \,x^{5}}{25}-\frac {b \,d^{2} e n \,x^{3}}{3}-\frac {b \,e^{3} n \,x^{7}}{49}-b \,d^{3} n x\)	\(582\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.33 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{49} \, {\left (b e^{3} n - 7 \, a e^{3}\right )} x^{7} - \frac {3}{25} \, {\left (b d e^{2} n - 5 \, a d e^{2}\right )} x^{5} - \frac {1}{3} \, {\left (b d^{2} e n - 3 \, a d^{2} e\right )} x^{3} - {\left (b d^{3} n - a d^{3}\right )} x + \frac {1}{35} \, {\left (5 \, b e^{3} x^{7} + 21 \, b d e^{2} x^{5} + 35 \, b d^{2} e x^{3} + 35 \, b d^{3} x\right )} \log \left (c\right ) + \frac {1}{35} \, {\left (5 \, b e^{3} n x^{7} + 21 \, b d e^{2} n x^{5} + 35 \, b d^{2} e n x^{3} + 35 \, b d^{3} n x\right )} \log \left (x\right ) \] Input:

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

Output:

-1/49*(b*e^3*n - 7*a*e^3)*x^7 - 3/25*(b*d*e^2*n - 5*a*d*e^2)*x^5 - 1/3*(b* 
d^2*e*n - 3*a*d^2*e)*x^3 - (b*d^3*n - a*d^3)*x + 1/35*(5*b*e^3*x^7 + 21*b* 
d*e^2*x^5 + 35*b*d^2*e*x^3 + 35*b*d^3*x)*log(c) + 1/35*(5*b*e^3*n*x^7 + 21 
*b*d*e^2*n*x^5 + 35*b*d^2*e*n*x^3 + 35*b*d^3*n*x)*log(x)
 

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.29 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} - b d^{3} n x + b d^{3} x \log {\left (c x^{n} \right )} - \frac {b d^{2} e n x^{3}}{3} + b d^{2} e x^{3} \log {\left (c x^{n} \right )} - \frac {3 b d e^{2} n x^{5}}{25} + \frac {3 b d e^{2} x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {b e^{3} n x^{7}}{49} + \frac {b e^{3} x^{7} \log {\left (c x^{n} \right )}}{7} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{49} \, b e^{3} n x^{7} + \frac {1}{7} \, b e^{3} x^{7} \log \left (c x^{n}\right ) + \frac {1}{7} \, a e^{3} x^{7} - \frac {3}{25} \, b d e^{2} n x^{5} + \frac {3}{5} \, b d e^{2} x^{5} \log \left (c x^{n}\right ) + \frac {3}{5} \, a d e^{2} x^{5} - \frac {1}{3} \, b d^{2} e n x^{3} + b d^{2} e x^{3} \log \left (c x^{n}\right ) + a d^{2} e x^{3} - b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.35 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{7} \, b e^{3} n x^{7} \log \left (x\right ) - \frac {1}{49} \, b e^{3} n x^{7} + \frac {1}{7} \, b e^{3} x^{7} \log \left (c\right ) + \frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, b d e^{2} n x^{5} \log \left (x\right ) - \frac {3}{25} \, b d e^{2} n x^{5} + \frac {3}{5} \, b d e^{2} x^{5} \log \left (c\right ) + \frac {3}{5} \, a d e^{2} x^{5} + b d^{2} e n x^{3} \log \left (x\right ) - \frac {1}{3} \, b d^{2} e n x^{3} + b d^{2} e x^{3} \log \left (c\right ) + a d^{2} e x^{3} + b d^{3} n x \log \left (x\right ) - b d^{3} n x + b d^{3} x \log \left (c\right ) + a d^{3} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 25.99 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (b\,d^3\,x+b\,d^2\,e\,x^3+\frac {3\,b\,d\,e^2\,x^5}{5}+\frac {b\,e^3\,x^7}{7}\right )+\frac {e^3\,x^7\,\left (7\,a-b\,n\right )}{49}+d^3\,x\,\left (a-b\,n\right )+\frac {d^2\,e\,x^3\,\left (3\,a-b\,n\right )}{3}+\frac {3\,d\,e^2\,x^5\,\left (5\,a-b\,n\right )}{25} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.13 \[ \int \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (3675 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}+3675 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e \,x^{2}+2205 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{4}+525 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{6}+3675 a \,d^{3}+3675 a \,d^{2} e \,x^{2}+2205 a d \,e^{2} x^{4}+525 a \,e^{3} x^{6}-3675 b \,d^{3} n -1225 b \,d^{2} e n \,x^{2}-441 b d \,e^{2} n \,x^{4}-75 b \,e^{3} n \,x^{6}\right )}{3675} \] Input:

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

Output:

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