\(\int (d+e x)^4 \log (d (a+b x+c x^2)^n) \, dx\) [82]

Optimal result

Integrand size = 23, antiderivative size = 485 \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=-\frac {\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac {e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac {e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac {e^3 (10 c d-b e) n x^4}{20 c}-\frac {2}{25} e^4 n x^5+\frac {\sqrt {b^2-4 a c} \left (5 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+a e)-b^2 c e^3 (5 b d+3 a e)+c^2 e^2 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right )\right ) n \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{5 c^5}-\frac {(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e} \] Output:

-1/5*(10*c^4*d^4+b^4*e^4-10*c^3*d^2*e*(2*a*e+b*d)-b^2*c*e^3*(4*a*e+5*b*d)+ 
c^2*e^2*(2*a^2*e^2+15*a*b*d*e+10*b^2*d^2))*n*x/c^4-1/10*e*(20*c^3*d^3-b^3* 
e^3-10*c^2*d*e*(a*e+b*d)+b*c*e^2*(3*a*e+5*b*d))*n*x^2/c^3-1/15*e^2*(20*c^2 
*d^2+b^2*e^2-c*e*(2*a*e+5*b*d))*n*x^3/c^2-1/20*e^3*(-b*e+10*c*d)*n*x^4/c-2 
/25*e^4*n*x^5+1/5*(-4*a*c+b^2)^(1/2)*(5*c^4*d^4+b^4*e^4-10*c^3*d^2*e*(a*e+ 
b*d)-b^2*c*e^3*(3*a*e+5*b*d)+c^2*e^2*(a^2*e^2+10*a*b*d*e+10*b^2*d^2))*n*ar 
ctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^5-1/10*(-b*e+2*c*d)*(c^4*d^4+b^4*e^4 
-2*c^3*d^2*e*(5*a*e+b*d)-b^2*c*e^3*(5*a*e+3*b*d)+c^2*e^2*(5*a^2*e^2+10*a*b 
*d*e+4*b^2*d^2))*n*ln(c*x^2+b*x+a)/c^5/e+1/5*(e*x+d)^5*ln(d*(c*x^2+b*x+a)^ 
n)/e
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.96 \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\frac {-\frac {n \left (60 c e \left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) x+30 c^2 e^2 \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) x^2+20 c^3 e^3 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) x^3+15 c^4 e^4 (10 c d-b e) x^4+24 c^5 e^5 x^5-60 \sqrt {b^2-4 a c} e \left (5 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+a e)-b^2 c e^3 (5 b d+3 a e)+c^2 e^2 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+30 (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) \log (a+x (b+c x))\right )}{60 c^5}+(d+e x)^5 \log \left (d (a+x (b+c x))^n\right )}{5 e} \] Input:

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

Output:

(-1/60*(n*(60*c*e*(10*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + 2*a*e) - b^2 
*c*e^3*(5*b*d + 4*a*e) + c^2*e^2*(10*b^2*d^2 + 15*a*b*d*e + 2*a^2*e^2))*x 
+ 30*c^2*e^2*(20*c^3*d^3 - b^3*e^3 - 10*c^2*d*e*(b*d + a*e) + b*c*e^2*(5*b 
*d + 3*a*e))*x^2 + 20*c^3*e^3*(20*c^2*d^2 + b^2*e^2 - c*e*(5*b*d + 2*a*e)) 
*x^3 + 15*c^4*e^4*(10*c*d - b*e)*x^4 + 24*c^5*e^5*x^5 - 60*Sqrt[b^2 - 4*a* 
c]*e*(5*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + a*e) - b^2*c*e^3*(5*b*d + 
3*a*e) + c^2*e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/ 
Sqrt[b^2 - 4*a*c]] + 30*(2*c*d - b*e)*(c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(b* 
d + 5*a*e) - b^2*c*e^3*(3*b*d + 5*a*e) + c^2*e^2*(4*b^2*d^2 + 10*a*b*d*e + 
 5*a^2*e^2))*Log[a + x*(b + c*x)]))/c^5 + (d + e*x)^5*Log[d*(a + x*(b + c* 
x))^n])/(5*e)
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3005, 1200, 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.

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

\(\Big \downarrow \) 3005

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

\(\Big \downarrow \) 1200

\(\displaystyle \frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {n \int \left (2 x^4 e^5+\frac {(10 c d-b e) x^3 e^4}{c}+\frac {\left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) x^2 e^3}{c^2}+\frac {\left (20 c^3 d^3-10 c^2 e (b d+a e) d-b^3 e^3+b c e^2 (5 b d+3 a e)\right ) x e^2}{c^3}+\frac {\left (10 c^4 d^4-10 c^3 e (b d+2 a e) d^2+b^4 e^4-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b e d+2 a^2 e^2\right )\right ) e}{c^4}+\frac {-a b^4 e^5+5 a b^3 c d e^4-2 a b^2 c \left (5 c d^2-2 a e^2\right ) e^3-2 a c^2 \left (5 c^2 d^4-10 a c e^2 d^2+a^2 e^4\right ) e+b c^2 d \left (c^2 d^4+10 a c e^2 d^2-15 a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4-2 c^3 e (b d+5 a e) d^2+b^4 e^4-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b e d+5 a^2 e^2\right )\right ) x}{c^4 \left (c x^2+b x+a\right )}\right )dx}{5 e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {n \left (-\frac {e \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right )}{c^5}+\frac {e x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{c^4}+\frac {(2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{2 c^5}+\frac {e^2 x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{2 c^3}+\frac {e^3 x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{3 c^2}+\frac {e^4 x^4 (10 c d-b e)}{4 c}+\frac {2 e^5 x^5}{5}\right )}{5 e}\)

Input:

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

Output:

-1/5*(n*((e*(10*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + 2*a*e) - b^2*c*e^3 
*(5*b*d + 4*a*e) + c^2*e^2*(10*b^2*d^2 + 15*a*b*d*e + 2*a^2*e^2))*x)/c^4 + 
 (e^2*(20*c^3*d^3 - b^3*e^3 - 10*c^2*d*e*(b*d + a*e) + b*c*e^2*(5*b*d + 3* 
a*e))*x^2)/(2*c^3) + (e^3*(20*c^2*d^2 + b^2*e^2 - c*e*(5*b*d + 2*a*e))*x^3 
)/(3*c^2) + (e^4*(10*c*d - b*e)*x^4)/(4*c) + (2*e^5*x^5)/5 - (Sqrt[b^2 - 4 
*a*c]*e*(5*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + a*e) - b^2*c*e^3*(5*b*d 
 + 3*a*e) + c^2*e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c* 
x)/Sqrt[b^2 - 4*a*c]])/c^5 + ((2*c*d - b*e)*(c^4*d^4 + b^4*e^4 - 2*c^3*d^2 
*e*(b*d + 5*a*e) - b^2*c*e^3*(3*b*d + 5*a*e) + c^2*e^2*(4*b^2*d^2 + 10*a*b 
*d*e + 5*a^2*e^2))*Log[a + b*x + c*x^2])/(2*c^5)))/e + ((d + e*x)^5*Log[d* 
(a + b*x + c*x^2)^n])/(5*e)
 

Defintions of rubi rules used

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 2009

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

rule 3005

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_. 
), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))) 
, x] - Simp[b*n*(p/(e*(m + 1)))   Int[SimplifyIntegrand[(d + e*x)^(m + 1)*( 
a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, 
d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || 
 IntegerQ[m]) && NeQ[m, -1]
 

Maple [A] (verified)

Time = 3.44 (sec) , antiderivative size = 876, normalized size of antiderivative = 1.81

method	result	size
parts	\(\frac {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) e^{4} x^{5}}{5}+\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) e^{3} d \,x^{4}+2 \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) e^{2} d^{2} x^{3}+2 \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) e \,d^{3} x^{2}+\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) d^{4} x +\frac {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) d^{5}}{5 e}-\frac {n \left (\frac {e \left (\frac {2}{5} c^{4} e^{4} x^{5}-\frac {1}{4} b \,c^{3} e^{4} x^{4}+\frac {5}{2} c^{4} d \,e^{3} x^{4}-\frac {2}{3} a \,c^{3} e^{4} x^{3}+\frac {1}{3} b^{2} c^{2} e^{4} x^{3}-\frac {5}{3} b \,c^{3} d \,e^{3} x^{3}+\frac {20}{3} c^{4} d^{2} e^{2} x^{3}+\frac {3}{2} a b \,c^{2} e^{4} x^{2}-5 a \,c^{3} d \,e^{3} x^{2}-\frac {1}{2} b^{3} c \,e^{4} x^{2}+\frac {5}{2} b^{2} c^{2} d \,e^{3} x^{2}-5 b \,c^{3} d^{2} e^{2} x^{2}+10 c^{4} d^{3} e \,x^{2}+2 a^{2} x \,c^{2} e^{4}-4 a \,b^{2} x c \,e^{4}+15 a b x \,c^{2} d \,e^{3}-20 a \,c^{3} x \,d^{2} e^{2}+b^{4} x \,e^{4}-5 b^{3} x c d \,e^{3}+10 b^{2} x \,c^{2} d^{2} e^{2}-10 x b \,c^{3} d^{3} e +10 c^{4} x \,d^{4}\right )}{c^{4}}+\frac {\frac {\left (-5 a^{2} b \,c^{2} e^{5}+10 a^{2} c^{3} d \,e^{4}+5 a \,b^{3} c \,e^{5}-20 a \,b^{2} c^{2} d \,e^{4}+30 a b \,c^{3} d^{2} e^{3}-20 a \,c^{4} d^{3} e^{2}-b^{5} e^{5}+5 b^{4} c d \,e^{4}-10 b^{3} c^{2} d^{2} e^{3}+10 b^{2} c^{3} d^{3} e^{2}-5 b \,c^{4} d^{4} e +2 c^{5} d^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-2 a^{3} c^{2} e^{5}+4 a^{2} b^{2} c \,e^{5}-15 a^{2} b \,c^{2} d \,e^{4}+20 a^{2} c^{3} d^{2} e^{3}-a \,b^{4} e^{5}+5 a \,b^{3} c d \,e^{4}-10 a \,b^{2} c^{2} d^{2} e^{3}+10 a b \,c^{3} d^{3} e^{2}-10 a \,c^{4} d^{4} e +b \,c^{4} d^{5}-\frac {\left (-5 a^{2} b \,c^{2} e^{5}+10 a^{2} c^{3} d \,e^{4}+5 a \,b^{3} c \,e^{5}-20 a \,b^{2} c^{2} d \,e^{4}+30 a b \,c^{3} d^{2} e^{3}-20 a \,c^{4} d^{3} e^{2}-b^{5} e^{5}+5 b^{4} c d \,e^{4}-10 b^{3} c^{2} d^{2} e^{3}+10 b^{2} c^{3} d^{3} e^{2}-5 b \,c^{4} d^{4} e +2 c^{5} d^{5}\right ) b}{2 c}\right ) \arctan \left (\frac {2 x c +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{4}}\right )}{5 e}\)	\(876\)
risch	\(\text {Expression too large to display}\)	\(31895\)

Input:

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

Output:

1/5*ln(d*(c*x^2+b*x+a)^n)*e^4*x^5+ln(d*(c*x^2+b*x+a)^n)*e^3*d*x^4+2*ln(d*( 
c*x^2+b*x+a)^n)*e^2*d^2*x^3+2*ln(d*(c*x^2+b*x+a)^n)*e*d^3*x^2+ln(d*(c*x^2+ 
b*x+a)^n)*d^4*x+1/5*ln(d*(c*x^2+b*x+a)^n)/e*d^5-1/5/e*n*(e/c^4*(2/5*c^4*e^ 
4*x^5-1/4*b*c^3*e^4*x^4+5/2*c^4*d*e^3*x^4-2/3*a*c^3*e^4*x^3+1/3*b^2*c^2*e^ 
4*x^3-5/3*b*c^3*d*e^3*x^3+20/3*c^4*d^2*e^2*x^3+3/2*a*b*c^2*e^4*x^2-5*a*c^3 
*d*e^3*x^2-1/2*b^3*c*e^4*x^2+5/2*b^2*c^2*d*e^3*x^2-5*b*c^3*d^2*e^2*x^2+10* 
c^4*d^3*e*x^2+2*a^2*x*c^2*e^4-4*a*b^2*x*c*e^4+15*a*b*x*c^2*d*e^3-20*a*c^3* 
x*d^2*e^2+b^4*x*e^4-5*b^3*x*c*d*e^3+10*b^2*x*c^2*d^2*e^2-10*x*b*c^3*d^3*e+ 
10*c^4*x*d^4)+1/c^4*(1/2*(-5*a^2*b*c^2*e^5+10*a^2*c^3*d*e^4+5*a*b^3*c*e^5- 
20*a*b^2*c^2*d*e^4+30*a*b*c^3*d^2*e^3-20*a*c^4*d^3*e^2-b^5*e^5+5*b^4*c*d*e 
^4-10*b^3*c^2*d^2*e^3+10*b^2*c^3*d^3*e^2-5*b*c^4*d^4*e+2*c^5*d^5)/c*ln(c*x 
^2+b*x+a)+2*(-2*a^3*c^2*e^5+4*a^2*b^2*c*e^5-15*a^2*b*c^2*d*e^4+20*a^2*c^3* 
d^2*e^3-a*b^4*e^5+5*a*b^3*c*d*e^4-10*a*b^2*c^2*d^2*e^3+10*a*b*c^3*d^3*e^2- 
10*a*c^4*d^4*e+b*c^4*d^5-1/2*(-5*a^2*b*c^2*e^5+10*a^2*c^3*d*e^4+5*a*b^3*c* 
e^5-20*a*b^2*c^2*d*e^4+30*a*b*c^3*d^2*e^3-20*a*c^4*d^3*e^2-b^5*e^5+5*b^4*c 
*d*e^4-10*b^3*c^2*d^2*e^3+10*b^2*c^3*d^3*e^2-5*b*c^4*d^4*e+2*c^5*d^5)*b/c) 
/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 1270, normalized size of antiderivative = 2.62 \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/300*(24*c^5*e^4*n*x^5 + 15*(10*c^5*d*e^3 - b*c^4*e^4)*n*x^4 + 20*(20*c 
^5*d^2*e^2 - 5*b*c^4*d*e^3 + (b^2*c^3 - 2*a*c^4)*e^4)*n*x^3 + 30*(20*c^5*d 
^3*e - 10*b*c^4*d^2*e^2 + 5*(b^2*c^3 - 2*a*c^4)*d*e^3 - (b^3*c^2 - 3*a*b*c 
^3)*e^4)*n*x^2 - 30*(5*c^4*d^4 - 10*b*c^3*d^3*e + 10*(b^2*c^2 - a*c^3)*d^2 
*e^2 - 5*(b^3*c - 2*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + a^2*c^2)*e^4)*sqrt 
(b^2 - 4*a*c)*n*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c) 
*(2*c*x + b))/(c*x^2 + b*x + a)) + 60*(10*c^5*d^4 - 10*b*c^4*d^3*e + 10*(b 
^2*c^3 - 2*a*c^4)*d^2*e^2 - 5*(b^3*c^2 - 3*a*b*c^3)*d*e^3 + (b^4*c - 4*a*b 
^2*c^2 + 2*a^2*c^3)*e^4)*n*x - 30*(2*c^5*e^4*n*x^5 + 10*c^5*d*e^3*n*x^4 + 
20*c^5*d^2*e^2*n*x^3 + 20*c^5*d^3*e*n*x^2 + 10*c^5*d^4*n*x + (5*b*c^4*d^4 
- 10*(b^2*c^3 - 2*a*c^4)*d^3*e + 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 5*(b^4 
*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4) 
*n)*log(c*x^2 + b*x + a) - 60*(c^5*e^4*x^5 + 5*c^5*d*e^3*x^4 + 10*c^5*d^2* 
e^2*x^3 + 10*c^5*d^3*e*x^2 + 5*c^5*d^4*x)*log(d))/c^5, -1/300*(24*c^5*e^4* 
n*x^5 + 15*(10*c^5*d*e^3 - b*c^4*e^4)*n*x^4 + 20*(20*c^5*d^2*e^2 - 5*b*c^4 
*d*e^3 + (b^2*c^3 - 2*a*c^4)*e^4)*n*x^3 + 30*(20*c^5*d^3*e - 10*b*c^4*d^2* 
e^2 + 5*(b^2*c^3 - 2*a*c^4)*d*e^3 - (b^3*c^2 - 3*a*b*c^3)*e^4)*n*x^2 - 60* 
(5*c^4*d^4 - 10*b*c^3*d^3*e + 10*(b^2*c^2 - a*c^3)*d^2*e^2 - 5*(b^3*c - 2* 
a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + a^2*c^2)*e^4)*sqrt(-b^2 + 4*a*c)*n*arc 
tan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 60*(10*c^5*d^4 - 1...
 

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.53 \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=-\frac {1}{25} \, {\left (2 \, e^{4} n - 5 \, e^{4} \log \left (d\right )\right )} x^{5} - \frac {{\left (10 \, c d e^{3} n - b e^{4} n - 20 \, c d e^{3} \log \left (d\right )\right )} x^{4}}{20 \, c} - \frac {{\left (20 \, c^{2} d^{2} e^{2} n - 5 \, b c d e^{3} n + b^{2} e^{4} n - 2 \, a c e^{4} n - 30 \, c^{2} d^{2} e^{2} \log \left (d\right )\right )} x^{3}}{15 \, c^{2}} + \frac {1}{5} \, {\left (e^{4} n x^{5} + 5 \, d e^{3} n x^{4} + 10 \, d^{2} e^{2} n x^{3} + 10 \, d^{3} e n x^{2} + 5 \, d^{4} n x\right )} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (20 \, c^{3} d^{3} e n - 10 \, b c^{2} d^{2} e^{2} n + 5 \, b^{2} c d e^{3} n - 10 \, a c^{2} d e^{3} n - b^{3} e^{4} n + 3 \, a b c e^{4} n - 20 \, c^{3} d^{3} e \log \left (d\right )\right )} x^{2}}{10 \, c^{3}} - \frac {{\left (10 \, c^{4} d^{4} n - 10 \, b c^{3} d^{3} e n + 10 \, b^{2} c^{2} d^{2} e^{2} n - 20 \, a c^{3} d^{2} e^{2} n - 5 \, b^{3} c d e^{3} n + 15 \, a b c^{2} d e^{3} n + b^{4} e^{4} n - 4 \, a b^{2} c e^{4} n + 2 \, a^{2} c^{2} e^{4} n - 5 \, c^{4} d^{4} \log \left (d\right )\right )} x}{5 \, c^{4}} + \frac {{\left (5 \, b c^{4} d^{4} n - 10 \, b^{2} c^{3} d^{3} e n + 20 \, a c^{4} d^{3} e n + 10 \, b^{3} c^{2} d^{2} e^{2} n - 30 \, a b c^{3} d^{2} e^{2} n - 5 \, b^{4} c d e^{3} n + 20 \, a b^{2} c^{2} d e^{3} n - 10 \, a^{2} c^{3} d e^{3} n + b^{5} e^{4} n - 5 \, a b^{3} c e^{4} n + 5 \, a^{2} b c^{2} e^{4} n\right )} \log \left (c x^{2} + b x + a\right )}{10 \, c^{5}} - \frac {{\left (5 \, b^{2} c^{4} d^{4} n - 20 \, a c^{5} d^{4} n - 10 \, b^{3} c^{3} d^{3} e n + 40 \, a b c^{4} d^{3} e n + 10 \, b^{4} c^{2} d^{2} e^{2} n - 50 \, a b^{2} c^{3} d^{2} e^{2} n + 40 \, a^{2} c^{4} d^{2} e^{2} n - 5 \, b^{5} c d e^{3} n + 30 \, a b^{3} c^{2} d e^{3} n - 40 \, a^{2} b c^{3} d e^{3} n + b^{6} e^{4} n - 7 \, a b^{4} c e^{4} n + 13 \, a^{2} b^{2} c^{2} e^{4} n - 4 \, a^{3} c^{3} e^{4} n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{5 \, \sqrt {-b^{2} + 4 \, a c} c^{5}} \] Input:

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

Output:

-1/25*(2*e^4*n - 5*e^4*log(d))*x^5 - 1/20*(10*c*d*e^3*n - b*e^4*n - 20*c*d 
*e^3*log(d))*x^4/c - 1/15*(20*c^2*d^2*e^2*n - 5*b*c*d*e^3*n + b^2*e^4*n - 
2*a*c*e^4*n - 30*c^2*d^2*e^2*log(d))*x^3/c^2 + 1/5*(e^4*n*x^5 + 5*d*e^3*n* 
x^4 + 10*d^2*e^2*n*x^3 + 10*d^3*e*n*x^2 + 5*d^4*n*x)*log(c*x^2 + b*x + a) 
- 1/10*(20*c^3*d^3*e*n - 10*b*c^2*d^2*e^2*n + 5*b^2*c*d*e^3*n - 10*a*c^2*d 
*e^3*n - b^3*e^4*n + 3*a*b*c*e^4*n - 20*c^3*d^3*e*log(d))*x^2/c^3 - 1/5*(1 
0*c^4*d^4*n - 10*b*c^3*d^3*e*n + 10*b^2*c^2*d^2*e^2*n - 20*a*c^3*d^2*e^2*n 
 - 5*b^3*c*d*e^3*n + 15*a*b*c^2*d*e^3*n + b^4*e^4*n - 4*a*b^2*c*e^4*n + 2* 
a^2*c^2*e^4*n - 5*c^4*d^4*log(d))*x/c^4 + 1/10*(5*b*c^4*d^4*n - 10*b^2*c^3 
*d^3*e*n + 20*a*c^4*d^3*e*n + 10*b^3*c^2*d^2*e^2*n - 30*a*b*c^3*d^2*e^2*n 
- 5*b^4*c*d*e^3*n + 20*a*b^2*c^2*d*e^3*n - 10*a^2*c^3*d*e^3*n + b^5*e^4*n 
- 5*a*b^3*c*e^4*n + 5*a^2*b*c^2*e^4*n)*log(c*x^2 + b*x + a)/c^5 - 1/5*(5*b 
^2*c^4*d^4*n - 20*a*c^5*d^4*n - 10*b^3*c^3*d^3*e*n + 40*a*b*c^4*d^3*e*n + 
10*b^4*c^2*d^2*e^2*n - 50*a*b^2*c^3*d^2*e^2*n + 40*a^2*c^4*d^2*e^2*n - 5*b 
^5*c*d*e^3*n + 30*a*b^3*c^2*d*e^3*n - 40*a^2*b*c^3*d*e^3*n + b^6*e^4*n - 7 
*a*b^4*c*e^4*n + 13*a^2*b^2*c^2*e^4*n - 4*a^3*c^3*e^4*n)*arctan((2*c*x + b 
)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)
 

Mupad [B] (verification not implemented)

Time = 26.18 (sec) , antiderivative size = 1240, normalized size of antiderivative = 2.56 \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

x^3*((b*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/(3*c) + (2*a*e 
^4*n)/(15*c) - (d*e^2*n*(b*e + 4*c*d))/(3*c)) - x*((a*((b*((e^3*n*(b*e + 1 
0*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/c + (2*a*e^4*n)/(5*c) - (d*e^2*n*(b*e 
+ 4*c*d))/c))/c - (b*((b*((b*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/( 
5*c)))/c + (2*a*e^4*n)/(5*c) - (d*e^2*n*(b*e + 4*c*d))/c))/c - (a*((e^3*n* 
(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/c + (2*d^2*e*n*(b*e + 2*c*d))/ 
c))/c + (2*d^3*n*(b*e + c*d))/c) - x^2*((b*((b*((e^3*n*(b*e + 10*c*d))/(5* 
c) - (2*b*e^4*n)/(5*c)))/c + (2*a*e^4*n)/(5*c) - (d*e^2*n*(b*e + 4*c*d))/c 
))/(2*c) - (a*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/(2*c) + 
(d^2*e*n*(b*e + 2*c*d))/c) - x^4*((e^3*n*(b*e + 10*c*d))/(20*c) - (b*e^4*n 
)/(10*c)) + log(d*(a + b*x + c*x^2)^n)*(d^4*x + (e^4*x^5)/5 + 2*d^3*e*x^2 
+ d*e^3*x^4 + 2*d^2*e^2*x^3) + (log(b*(b^2 - 4*a*c)^(1/2) - 4*a*c + b^2 + 
2*c*x*(b^2 - 4*a*c)^(1/2))*(b^5*e^4*n + 5*b*c^4*d^4*n + b^4*e^4*n*(b^2 - 4 
*a*c)^(1/2) + 5*c^4*d^4*n*(b^2 - 4*a*c)^(1/2) - 5*a*b^3*c*e^4*n + 20*a*c^4 
*d^3*e*n - 5*b^4*c*d*e^3*n + 5*a^2*b*c^2*e^4*n - 10*a^2*c^3*d*e^3*n - 10*b 
^2*c^3*d^3*e*n + a^2*c^2*e^4*n*(b^2 - 4*a*c)^(1/2) + 10*b^3*c^2*d^2*e^2*n 
- 10*a*c^3*d^2*e^2*n*(b^2 - 4*a*c)^(1/2) + 10*b^2*c^2*d^2*e^2*n*(b^2 - 4*a 
*c)^(1/2) - 3*a*b^2*c*e^4*n*(b^2 - 4*a*c)^(1/2) - 10*b*c^3*d^3*e*n*(b^2 - 
4*a*c)^(1/2) - 5*b^3*c*d*e^3*n*(b^2 - 4*a*c)^(1/2) - 30*a*b*c^3*d^2*e^2*n 
+ 20*a*b^2*c^2*d*e^3*n + 10*a*b*c^2*d*e^3*n*(b^2 - 4*a*c)^(1/2)))/(10*c...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1118, normalized size of antiderivative = 2.31 \[ \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c**2*e**4 
*n - 180*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c* 
e**4*n + 600*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c 
**2*d*e**3*n - 600*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a*c**3*d**2*e**2*n + 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
b**2))*b**4*e**4*n - 300*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
b**2))*b**3*c*d*e**3*n + 600*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* 
c - b**2))*b**2*c**2*d**2*e**2*n - 600*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*b*c**3*d**3*e*n + 300*sqrt(4*a*c - b**2)*atan((b + 2* 
c*x)/sqrt(4*a*c - b**2))*c**4*d**4*n + 150*log((a + b*x + c*x**2)**n*d)*a* 
*2*b*c**2*e**4 - 300*log((a + b*x + c*x**2)**n*d)*a**2*c**3*d*e**3 - 150*l 
og((a + b*x + c*x**2)**n*d)*a*b**3*c*e**4 + 600*log((a + b*x + c*x**2)**n* 
d)*a*b**2*c**2*d*e**3 - 900*log((a + b*x + c*x**2)**n*d)*a*b*c**3*d**2*e** 
2 + 600*log((a + b*x + c*x**2)**n*d)*a*c**4*d**3*e + 30*log((a + b*x + c*x 
**2)**n*d)*b**5*e**4 - 150*log((a + b*x + c*x**2)**n*d)*b**4*c*d*e**3 + 30 
0*log((a + b*x + c*x**2)**n*d)*b**3*c**2*d**2*e**2 - 300*log((a + b*x + c* 
x**2)**n*d)*b**2*c**3*d**3*e + 150*log((a + b*x + c*x**2)**n*d)*b*c**4*d** 
4 + 300*log((a + b*x + c*x**2)**n*d)*c**5*d**4*x + 600*log((a + b*x + c*x* 
*2)**n*d)*c**5*d**3*e*x**2 + 600*log((a + b*x + c*x**2)**n*d)*c**5*d**2*e* 
*2*x**3 + 300*log((a + b*x + c*x**2)**n*d)*c**5*d*e**3*x**4 + 60*log((a...