\(\int x^4 (d+e x^2)^3 (a+b \log (c x^n)) \, dx\) [202]

Optimal result

Integrand size = 23, antiderivative size = 100 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^3 n x^5-\frac {3}{49} b d^2 e n x^7-\frac {1}{27} b d e^2 n x^9-\frac {1}{121} b e^3 n x^{11}+\frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155} \] Output:

-1/25*b*d^3*n*x^5-3/49*b*d^2*e*n*x^7-1/27*b*d*e^2*n*x^9-1/121*b*e^3*n*x^11 
+1/1155*(105*e^3*x^11+385*d*e^2*x^9+495*d^2*e*x^7+231*d^3*x^5)*(a+b*ln(c*x 
^n))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.33 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^3 n x^5-\frac {3}{49} b d^2 e n x^7-\frac {1}{27} b d e^2 n x^9-\frac {1}{121} b e^3 n x^{11}+\frac {1}{5} d^3 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{7} d^2 e x^7 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} d e^2 x^9 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{11} e^3 x^{11} \left (a+b \log \left (c x^n\right )\right ) \] Input:

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

Output:

-1/25*(b*d^3*n*x^5) - (3*b*d^2*e*n*x^7)/49 - (b*d*e^2*n*x^9)/27 - (b*e^3*n 
*x^11)/121 + (d^3*x^5*(a + b*Log[c*x^n]))/5 + (3*d^2*e*x^7*(a + b*Log[c*x^ 
n]))/7 + (d*e^2*x^9*(a + b*Log[c*x^n]))/3 + (e^3*x^11*(a + b*Log[c*x^n]))/ 
11
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2771, 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.

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

\(\Big \downarrow \) 2771

\(\displaystyle \frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-b n \int \left (\frac {e^3 x^{10}}{11}+\frac {1}{3} d e^2 x^8+\frac {3}{7} d^2 e x^6+\frac {d^3 x^4}{5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-b n \left (\frac {d^3 x^5}{25}+\frac {3}{49} d^2 e x^7+\frac {1}{27} d e^2 x^9+\frac {e^3 x^{11}}{121}\right )\)

Input:

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

Output:

-(b*n*((d^3*x^5)/25 + (3*d^2*e*x^7)/49 + (d*e^2*x^9)/27 + (e^3*x^11)/121)) 
 + ((231*d^3*x^5 + 495*d^2*e*x^7 + 385*d*e^2*x^9 + 105*e^3*x^11)*(a + b*Lo 
g[c*x^n]))/1155
 

Defintions of rubi rules used

rule 2009

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

rule 2771

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[u*(a 
 + b*Log[c*x^n]), x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; 
 FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 3.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.44

method	result	size
parallelrisch	\(\frac {x^{11} b \ln \left (c \,x^{n}\right ) e^{3}}{11}-\frac {b \,e^{3} n \,x^{11}}{121}+\frac {x^{11} a \,e^{3}}{11}+\frac {x^{9} b \ln \left (c \,x^{n}\right ) e^{2} d}{3}-\frac {b d \,e^{2} n \,x^{9}}{27}+\frac {x^{9} a \,e^{2} d}{3}+\frac {3 x^{7} b \ln \left (c \,x^{n}\right ) d^{2} e}{7}-\frac {3 b \,d^{2} e n \,x^{7}}{49}+\frac {3 x^{7} a \,d^{2} e}{7}+\frac {x^{5} b \ln \left (c \,x^{n}\right ) d^{3}}{5}-\frac {b \,d^{3} n \,x^{5}}{25}+\frac {x^{5} a \,d^{3}}{5}\)	\(144\)
risch	\(\frac {\ln \left (c \right ) b \,d^{3} x^{5}}{5}+\frac {\ln \left (c \right ) b \,e^{3} x^{11}}{11}+\frac {x^{5} a \,d^{3}}{5}+\frac {x^{11} a \,e^{3}}{11}+\frac {x^{9} a \,e^{2} d}{3}+\frac {3 x^{7} a \,d^{2} e}{7}+\frac {b \,x^{5} \left (105 e^{3} x^{6}+385 e^{2} d \,x^{4}+495 d^{2} e \,x^{2}+231 d^{3}\right ) \ln \left (x^{n}\right )}{1155}+\frac {\ln \left (c \right ) b d \,e^{2} x^{9}}{3}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{7}}{7}-\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{10}+\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{14}+\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{22}+\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{10}+\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{22}+\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{22}+\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{14}+\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{6}-\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{6}-\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{14}-\frac {b \,d^{3} n \,x^{5}}{25}-\frac {b \,e^{3} n \,x^{11}}{121}-\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{22}-\frac {3 b \,d^{2} e n \,x^{7}}{49}-\frac {b d \,e^{2} n \,x^{9}}{27}\)	\(602\)

Input:

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

Output:

1/11*x^11*b*ln(c*x^n)*e^3-1/121*b*e^3*n*x^11+1/11*x^11*a*e^3+1/3*x^9*b*ln( 
c*x^n)*e^2*d-1/27*b*d*e^2*n*x^9+1/3*x^9*a*e^2*d+3/7*x^7*b*ln(c*x^n)*d^2*e- 
3/49*b*d^2*e*n*x^7+3/7*x^7*a*d^2*e+1/5*x^5*b*ln(c*x^n)*d^3-1/25*b*d^3*n*x^ 
5+1/5*x^5*a*d^3
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.67 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{121} \, {\left (b e^{3} n - 11 \, a e^{3}\right )} x^{11} - \frac {1}{27} \, {\left (b d e^{2} n - 9 \, a d e^{2}\right )} x^{9} - \frac {3}{49} \, {\left (b d^{2} e n - 7 \, a d^{2} e\right )} x^{7} - \frac {1}{25} \, {\left (b d^{3} n - 5 \, a d^{3}\right )} x^{5} + \frac {1}{1155} \, {\left (105 \, b e^{3} x^{11} + 385 \, b d e^{2} x^{9} + 495 \, b d^{2} e x^{7} + 231 \, b d^{3} x^{5}\right )} \log \left (c\right ) + \frac {1}{1155} \, {\left (105 \, b e^{3} n x^{11} + 385 \, b d e^{2} n x^{9} + 495 \, b d^{2} e n x^{7} + 231 \, b d^{3} n x^{5}\right )} \log \left (x\right ) \] Input:

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

Output:

-1/121*(b*e^3*n - 11*a*e^3)*x^11 - 1/27*(b*d*e^2*n - 9*a*d*e^2)*x^9 - 3/49 
*(b*d^2*e*n - 7*a*d^2*e)*x^7 - 1/25*(b*d^3*n - 5*a*d^3)*x^5 + 1/1155*(105* 
b*e^3*x^11 + 385*b*d*e^2*x^9 + 495*b*d^2*e*x^7 + 231*b*d^3*x^5)*log(c) + 1 
/1155*(105*b*e^3*n*x^11 + 385*b*d*e^2*n*x^9 + 495*b*d^2*e*n*x^7 + 231*b*d^ 
3*n*x^5)*log(x)
 

Sympy [A] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.70 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{3} x^{5}}{5} + \frac {3 a d^{2} e x^{7}}{7} + \frac {a d e^{2} x^{9}}{3} + \frac {a e^{3} x^{11}}{11} - \frac {b d^{3} n x^{5}}{25} + \frac {b d^{3} x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {3 b d^{2} e n x^{7}}{49} + \frac {3 b d^{2} e x^{7} \log {\left (c x^{n} \right )}}{7} - \frac {b d e^{2} n x^{9}}{27} + \frac {b d e^{2} x^{9} \log {\left (c x^{n} \right )}}{3} - \frac {b e^{3} n x^{11}}{121} + \frac {b e^{3} x^{11} \log {\left (c x^{n} \right )}}{11} \] Input:

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

Output:

a*d**3*x**5/5 + 3*a*d**2*e*x**7/7 + a*d*e**2*x**9/3 + a*e**3*x**11/11 - b* 
d**3*n*x**5/25 + b*d**3*x**5*log(c*x**n)/5 - 3*b*d**2*e*n*x**7/49 + 3*b*d* 
*2*e*x**7*log(c*x**n)/7 - b*d*e**2*n*x**9/27 + b*d*e**2*x**9*log(c*x**n)/3 
 - b*e**3*n*x**11/121 + b*e**3*x**11*log(c*x**n)/11
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.43 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{121} \, b e^{3} n x^{11} + \frac {1}{11} \, b e^{3} x^{11} \log \left (c x^{n}\right ) + \frac {1}{11} \, a e^{3} x^{11} - \frac {1}{27} \, b d e^{2} n x^{9} + \frac {1}{3} \, b d e^{2} x^{9} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d e^{2} x^{9} - \frac {3}{49} \, b d^{2} e n x^{7} + \frac {3}{7} \, b d^{2} e x^{7} \log \left (c x^{n}\right ) + \frac {3}{7} \, a d^{2} e x^{7} - \frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{3} x^{5} \] Input:

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

Output:

-1/121*b*e^3*n*x^11 + 1/11*b*e^3*x^11*log(c*x^n) + 1/11*a*e^3*x^11 - 1/27* 
b*d*e^2*n*x^9 + 1/3*b*d*e^2*x^9*log(c*x^n) + 1/3*a*d*e^2*x^9 - 3/49*b*d^2* 
e*n*x^7 + 3/7*b*d^2*e*x^7*log(c*x^n) + 3/7*a*d^2*e*x^7 - 1/25*b*d^3*n*x^5 
+ 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.77 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{11} \, b e^{3} n x^{11} \log \left (x\right ) - \frac {1}{121} \, b e^{3} n x^{11} + \frac {1}{11} \, b e^{3} x^{11} \log \left (c\right ) + \frac {1}{11} \, a e^{3} x^{11} + \frac {1}{3} \, b d e^{2} n x^{9} \log \left (x\right ) - \frac {1}{27} \, b d e^{2} n x^{9} + \frac {1}{3} \, b d e^{2} x^{9} \log \left (c\right ) + \frac {1}{3} \, a d e^{2} x^{9} + \frac {3}{7} \, b d^{2} e n x^{7} \log \left (x\right ) - \frac {3}{49} \, b d^{2} e n x^{7} + \frac {3}{7} \, b d^{2} e x^{7} \log \left (c\right ) + \frac {3}{7} \, a d^{2} e x^{7} + \frac {1}{5} \, b d^{3} n x^{5} \log \left (x\right ) - \frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c\right ) + \frac {1}{5} \, a d^{3} x^{5} \] Input:

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

Output:

1/11*b*e^3*n*x^11*log(x) - 1/121*b*e^3*n*x^11 + 1/11*b*e^3*x^11*log(c) + 1 
/11*a*e^3*x^11 + 1/3*b*d*e^2*n*x^9*log(x) - 1/27*b*d*e^2*n*x^9 + 1/3*b*d*e 
^2*x^9*log(c) + 1/3*a*d*e^2*x^9 + 3/7*b*d^2*e*n*x^7*log(x) - 3/49*b*d^2*e* 
n*x^7 + 3/7*b*d^2*e*x^7*log(c) + 3/7*a*d^2*e*x^7 + 1/5*b*d^3*n*x^5*log(x) 
- 1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5*log(c) + 1/5*a*d^3*x^5
 

Mupad [B] (verification not implemented)

Time = 25.84 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.13 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3\,x^5}{5}+\frac {3\,b\,d^2\,e\,x^7}{7}+\frac {b\,d\,e^2\,x^9}{3}+\frac {b\,e^3\,x^{11}}{11}\right )+\frac {d^3\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {e^3\,x^{11}\,\left (11\,a-b\,n\right )}{121}+\frac {3\,d^2\,e\,x^7\,\left (7\,a-b\,n\right )}{49}+\frac {d\,e^2\,x^9\,\left (9\,a-b\,n\right )}{27} \] Input:

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

Output:

log(c*x^n)*((b*d^3*x^5)/5 + (b*e^3*x^11)/11 + (3*b*d^2*e*x^7)/7 + (b*d*e^2 
*x^9)/3) + (d^3*x^5*(5*a - b*n))/25 + (e^3*x^11*(11*a - b*n))/121 + (3*d^2 
*e*x^7*(7*a - b*n))/49 + (d*e^2*x^9*(9*a - b*n))/27
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.39 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{5} \left (800415 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3}+1715175 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e \,x^{2}+1334025 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{4}+363825 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{3} x^{6}+800415 a \,d^{3}+1715175 a \,d^{2} e \,x^{2}+1334025 a d \,e^{2} x^{4}+363825 a \,e^{3} x^{6}-160083 b \,d^{3} n -245025 b \,d^{2} e n \,x^{2}-148225 b d \,e^{2} n \,x^{4}-33075 b \,e^{3} n \,x^{6}\right )}{4002075} \] Input:

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

Output:

(x**5*(800415*log(x**n*c)*b*d**3 + 1715175*log(x**n*c)*b*d**2*e*x**2 + 133 
4025*log(x**n*c)*b*d*e**2*x**4 + 363825*log(x**n*c)*b*e**3*x**6 + 800415*a 
*d**3 + 1715175*a*d**2*e*x**2 + 1334025*a*d*e**2*x**4 + 363825*a*e**3*x**6 
 - 160083*b*d**3*n - 245025*b*d**2*e*n*x**2 - 148225*b*d*e**2*n*x**4 - 330 
75*b*e**3*n*x**6))/4002075