3.67 \(\int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^2 \, dx\)
Optimal. Leaf size=132 \[ a^2 d x+\frac{x^{2 n+1} \left (2 a b e+2 a c d+b^2 d\right )}{2 n+1}+\frac{x^{3 n+1} \left (2 a c e+b^2 e+2 b c d\right )}{3 n+1}+\frac{a x^{n+1} (a e+2 b d)}{n+1}+\frac{c x^{4 n+1} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n+1}}{5 n+1} \]
[Out]
a^2*d*x + (a*(2*b*d + a*e)*x^(1 + n))/(1 + n) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^(
1 + 2*n))/(1 + 2*n) + ((2*b*c*d + b^2*e + 2*a*c*e)*x^(1 + 3*n))/(1 + 3*n) + (c*(
c*d + 2*b*e)*x^(1 + 4*n))/(1 + 4*n) + (c^2*e*x^(1 + 5*n))/(1 + 5*n)
_______________________________________________________________________________________
Rubi [A] time = 0.202595, antiderivative size = 132, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042
\[ a^2 d x+\frac{x^{2 n+1} \left (2 a b e+2 a c d+b^2 d\right )}{2 n+1}+\frac{x^{3 n+1} \left (2 a c e+b^2 e+2 b c d\right )}{3 n+1}+\frac{a x^{n+1} (a e+2 b d)}{n+1}+\frac{c x^{4 n+1} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n+1}}{5 n+1} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^2,x]
[Out]
a^2*d*x + (a*(2*b*d + a*e)*x^(1 + n))/(1 + n) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^(
1 + 2*n))/(1 + 2*n) + ((2*b*c*d + b^2*e + 2*a*c*e)*x^(1 + 3*n))/(1 + 3*n) + (c*(
c*d + 2*b*e)*x^(1 + 4*n))/(1 + 4*n) + (c^2*e*x^(1 + 5*n))/(1 + 5*n)
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a^{2} \int d\, dx + \frac{a x^{n + 1} \left (a e + 2 b d\right )}{n + 1} + \frac{c^{2} e x^{5 n + 1}}{5 n + 1} + \frac{c x^{4 n + 1} \left (2 b e + c d\right )}{4 n + 1} + \frac{x^{2 n + 1} \left (2 a \left (b e + c d\right ) + b^{2} d\right )}{2 n + 1} + \frac{x^{3 n + 1} \left (b^{2} e + 2 c \left (a e + b d\right )\right )}{3 n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d+e*x**n)*(a+b*x**n+c*x**(2*n))**2,x)
[Out]
a**2*Integral(d, x) + a*x**(n + 1)*(a*e + 2*b*d)/(n + 1) + c**2*e*x**(5*n + 1)/(
5*n + 1) + c*x**(4*n + 1)*(2*b*e + c*d)/(4*n + 1) + x**(2*n + 1)*(2*a*(b*e + c*d
) + b**2*d)/(2*n + 1) + x**(3*n + 1)*(b**2*e + 2*c*(a*e + b*d))/(3*n + 1)
_______________________________________________________________________________________
Mathematica [A] time = 0.363756, size = 123, normalized size = 0.93 \[ x \left (a^2 d+\frac{x^{2 n} \left (2 a b e+2 a c d+b^2 d\right )}{2 n+1}+\frac{x^{3 n} \left (2 a c e+b^2 e+2 b c d\right )}{3 n+1}+\frac{a x^n (a e+2 b d)}{n+1}+\frac{c x^{4 n} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n}}{5 n+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^2,x]
[Out]
x*(a^2*d + (a*(2*b*d + a*e)*x^n)/(1 + n) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^(2*n))
/(1 + 2*n) + ((2*b*c*d + b^2*e + 2*a*c*e)*x^(3*n))/(1 + 3*n) + (c*(c*d + 2*b*e)*
x^(4*n))/(1 + 4*n) + (c^2*e*x^(5*n))/(1 + 5*n))
_______________________________________________________________________________________
Maple [A] time = 0.022, size = 138, normalized size = 1.1 \[{a}^{2}dx+{\frac{ \left ( 2\,ace+{b}^{2}e+2\,bcd \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+{\frac{ \left ( 2\,abe+2\,acd+{b}^{2}d \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{a \left ( ae+2\,bd \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{c \left ( 2\,be+cd \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{1+4\,n}}+{\frac{{c}^{2}ex \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{1+5\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d+e*x^n)*(a+b*x^n+c*x^(2*n))^2,x)
[Out]
a^2*d*x+(2*a*c*e+b^2*e+2*b*c*d)/(1+3*n)*x*exp(n*ln(x))^3+(2*a*b*e+2*a*c*d+b^2*d)
/(1+2*n)*x*exp(n*ln(x))^2+a*(a*e+2*b*d)/(1+n)*x*exp(n*ln(x))+c*(2*b*e+c*d)/(1+4*
n)*x*exp(n*ln(x))^4+c^2*e/(1+5*n)*x*exp(n*ln(x))^5
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n + a)^2*(e*x^n + d),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.290458, size = 668, normalized size = 5.06 \[ \frac{{\left (24 \, c^{2} e n^{4} + 50 \, c^{2} e n^{3} + 35 \, c^{2} e n^{2} + 10 \, c^{2} e n + c^{2} e\right )} x x^{5 \, n} +{\left (30 \,{\left (c^{2} d + 2 \, b c e\right )} n^{4} + 61 \,{\left (c^{2} d + 2 \, b c e\right )} n^{3} + c^{2} d + 2 \, b c e + 41 \,{\left (c^{2} d + 2 \, b c e\right )} n^{2} + 11 \,{\left (c^{2} d + 2 \, b c e\right )} n\right )} x x^{4 \, n} +{\left (40 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} n^{4} + 78 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} n^{3} + 2 \, b c d + 49 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} n^{2} +{\left (b^{2} + 2 \, a c\right )} e + 12 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} n\right )} x x^{3 \, n} +{\left (60 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} n^{4} + 107 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} n^{3} + 2 \, a b e + 59 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} n^{2} +{\left (b^{2} + 2 \, a c\right )} d + 13 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} n\right )} x x^{2 \, n} +{\left (120 \,{\left (2 \, a b d + a^{2} e\right )} n^{4} + 154 \,{\left (2 \, a b d + a^{2} e\right )} n^{3} + 2 \, a b d + a^{2} e + 71 \,{\left (2 \, a b d + a^{2} e\right )} n^{2} + 14 \,{\left (2 \, a b d + a^{2} e\right )} n\right )} x x^{n} +{\left (120 \, a^{2} d n^{5} + 274 \, a^{2} d n^{4} + 225 \, a^{2} d n^{3} + 85 \, a^{2} d n^{2} + 15 \, a^{2} d n + a^{2} d\right )} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n + a)^2*(e*x^n + d),x, algorithm="fricas")
[Out]
((24*c^2*e*n^4 + 50*c^2*e*n^3 + 35*c^2*e*n^2 + 10*c^2*e*n + c^2*e)*x*x^(5*n) + (
30*(c^2*d + 2*b*c*e)*n^4 + 61*(c^2*d + 2*b*c*e)*n^3 + c^2*d + 2*b*c*e + 41*(c^2*
d + 2*b*c*e)*n^2 + 11*(c^2*d + 2*b*c*e)*n)*x*x^(4*n) + (40*(2*b*c*d + (b^2 + 2*a
*c)*e)*n^4 + 78*(2*b*c*d + (b^2 + 2*a*c)*e)*n^3 + 2*b*c*d + 49*(2*b*c*d + (b^2 +
2*a*c)*e)*n^2 + (b^2 + 2*a*c)*e + 12*(2*b*c*d + (b^2 + 2*a*c)*e)*n)*x*x^(3*n) +
(60*(2*a*b*e + (b^2 + 2*a*c)*d)*n^4 + 107*(2*a*b*e + (b^2 + 2*a*c)*d)*n^3 + 2*a
*b*e + 59*(2*a*b*e + (b^2 + 2*a*c)*d)*n^2 + (b^2 + 2*a*c)*d + 13*(2*a*b*e + (b^2
+ 2*a*c)*d)*n)*x*x^(2*n) + (120*(2*a*b*d + a^2*e)*n^4 + 154*(2*a*b*d + a^2*e)*n
^3 + 2*a*b*d + a^2*e + 71*(2*a*b*d + a^2*e)*n^2 + 14*(2*a*b*d + a^2*e)*n)*x*x^n
+ (120*a^2*d*n^5 + 274*a^2*d*n^4 + 225*a^2*d*n^3 + 85*a^2*d*n^2 + 15*a^2*d*n + a
^2*d)*x)/(120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)
_______________________________________________________________________________________
Sympy [A] time = 42.6775, size = 3128, normalized size = 23.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d+e*x**n)*(a+b*x**n+c*x**(2*n))**2,x)
[Out]
Piecewise((a**2*d*x + a**2*e*log(x) + 2*a*b*d*log(x) - 2*a*b*e/x - 2*a*c*d/x - a
*c*e/x**2 - b**2*d/x - b**2*e/(2*x**2) - b*c*d/x**2 - 2*b*c*e/(3*x**3) - c**2*d/
(3*x**3) - c**2*e/(4*x**4), Eq(n, -1)), (a**2*d*x + 2*a**2*e*sqrt(x) + 4*a*b*d*s
qrt(x) + 2*a*b*e*log(x) + 2*a*c*d*log(x) - 4*a*c*e/sqrt(x) + b**2*d*log(x) - 2*b
**2*e/sqrt(x) - 4*b*c*d/sqrt(x) - 2*b*c*e/x - c**2*d/x - 2*c**2*e/(3*x**(3/2)),
Eq(n, -1/2)), (a**2*d*x + 3*a**2*e*x**(2/3)/2 + 3*a*b*d*x**(2/3) + 6*a*b*e*x**(1
/3) + 6*a*c*d*x**(1/3) + 2*a*c*e*log(x) + 3*b**2*d*x**(1/3) + b**2*e*log(x) + 2*
b*c*d*log(x) - 6*b*c*e/x**(1/3) - 3*c**2*d/x**(1/3) - 3*c**2*e/(2*x**(2/3)), Eq(
n, -1/3)), (a**2*d*x + 4*a**2*e*x**(3/4)/3 + 8*a*b*d*x**(3/4)/3 + 4*a*b*e*sqrt(x
) + 4*a*c*d*sqrt(x) + 8*a*c*e*x**(1/4) + 2*b**2*d*sqrt(x) + 4*b**2*e*x**(1/4) +
8*b*c*d*x**(1/4) + 2*b*c*e*log(x) + c**2*d*log(x) - 4*c**2*e/x**(1/4), Eq(n, -1/
4)), (a**2*d*x + 5*a**2*e*x**(4/5)/4 + 5*a*b*d*x**(4/5)/2 + 10*a*b*e*x**(3/5)/3
+ 10*a*c*d*x**(3/5)/3 + 5*a*c*e*x**(2/5) + 5*b**2*d*x**(3/5)/3 + 5*b**2*e*x**(2/
5)/2 + 5*b*c*d*x**(2/5) + 10*b*c*e*x**(1/5) + 5*c**2*d*x**(1/5) + c**2*e*log(x),
Eq(n, -1/5)), (120*a**2*d*n**5*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15
*n + 1) + 274*a**2*d*n**4*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1
) + 225*a**2*d*n**3*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 85
*a**2*d*n**2*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 15*a**2*d
*n*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + a**2*d*x/(120*n**5
+ 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 120*a**2*e*n**4*x*x**n/(120*n**5 +
274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 154*a**2*e*n**3*x*x**n/(120*n**5 +
274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 71*a**2*e*n**2*x*x**n/(120*n**5 + 27
4*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 14*a**2*e*n*x*x**n/(120*n**5 + 274*n**
4 + 225*n**3 + 85*n**2 + 15*n + 1) + a**2*e*x*x**n/(120*n**5 + 274*n**4 + 225*n*
*3 + 85*n**2 + 15*n + 1) + 240*a*b*d*n**4*x*x**n/(120*n**5 + 274*n**4 + 225*n**3
+ 85*n**2 + 15*n + 1) + 308*a*b*d*n**3*x*x**n/(120*n**5 + 274*n**4 + 225*n**3 +
85*n**2 + 15*n + 1) + 142*a*b*d*n**2*x*x**n/(120*n**5 + 274*n**4 + 225*n**3 + 8
5*n**2 + 15*n + 1) + 28*a*b*d*n*x*x**n/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2
+ 15*n + 1) + 2*a*b*d*x*x**n/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + 120*a*b*e*n**4*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + 214*a*b*e*n**3*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15
*n + 1) + 118*a*b*e*n**2*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 +
15*n + 1) + 26*a*b*e*n*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15
*n + 1) + 2*a*b*e*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + 120*a*c*d*n**4*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + 214*a*c*d*n**3*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*
n + 1) + 118*a*c*d*n**2*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 1
5*n + 1) + 26*a*c*d*n*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*
n + 1) + 2*a*c*d*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1
) + 80*a*c*e*n**4*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + 156*a*c*e*n**3*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + 98*a*c*e*n**2*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + 24*a*c*e*n*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + 2*a*c*e*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
60*b**2*d*n**4*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1)
+ 107*b**2*d*n**3*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + 59*b**2*d*n**2*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + 13*b**2*d*n*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + b**2*d*x*x**(2*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
40*b**2*e*n**4*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1)
+ 78*b**2*e*n**3*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + 49*b**2*e*n**2*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n
+ 1) + 12*b**2*e*n*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n +
1) + b**2*e*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
80*b*c*d*n**4*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
156*b*c*d*n**3*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1)
+ 98*b*c*d*n**2*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1
) + 24*b*c*d*n*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1)
+ 2*b*c*d*x*x**(3*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 60*
b*c*e*n**4*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 12
2*b*c*e*n**3*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
82*b*c*e*n**2*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
22*b*c*e*n*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 2
*b*c*e*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 30*c**
2*d*n**4*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 61*c
**2*d*n**3*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 41
*c**2*d*n**2*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) +
11*c**2*d*n*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + c
**2*d*x*x**(4*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 24*c**2
*e*n**4*x*x**(5*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 50*c*
*2*e*n**3*x*x**(5*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 35*
c**2*e*n**2*x*x**(5*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 1
0*c**2*e*n*x*x**(5*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + c*
*2*e*x*x**(5*n)/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.289965, size = 1206, normalized size = 9.14 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^(2*n) + b*x^n + a)^2*(e*x^n + d),x, algorithm="giac")
[Out]
(120*a^2*d*n^5*x + 30*c^2*d*n^4*x*e^(4*n*ln(x)) + 80*b*c*d*n^4*x*e^(3*n*ln(x)) +
60*b^2*d*n^4*x*e^(2*n*ln(x)) + 120*a*c*d*n^4*x*e^(2*n*ln(x)) + 240*a*b*d*n^4*x*
e^(n*ln(x)) + 274*a^2*d*n^4*x + 61*c^2*d*n^3*x*e^(4*n*ln(x)) + 156*b*c*d*n^3*x*e
^(3*n*ln(x)) + 107*b^2*d*n^3*x*e^(2*n*ln(x)) + 214*a*c*d*n^3*x*e^(2*n*ln(x)) + 2
4*c^2*n^4*x*e^(5*n*ln(x) + 1) + 60*b*c*n^4*x*e^(4*n*ln(x) + 1) + 40*b^2*n^4*x*e^
(3*n*ln(x) + 1) + 80*a*c*n^4*x*e^(3*n*ln(x) + 1) + 120*a*b*n^4*x*e^(2*n*ln(x) +
1) + 120*a^2*n^4*x*e^(n*ln(x) + 1) + 308*a*b*d*n^3*x*e^(n*ln(x)) + 225*a^2*d*n^3
*x + 41*c^2*d*n^2*x*e^(4*n*ln(x)) + 98*b*c*d*n^2*x*e^(3*n*ln(x)) + 59*b^2*d*n^2*
x*e^(2*n*ln(x)) + 118*a*c*d*n^2*x*e^(2*n*ln(x)) + 50*c^2*n^3*x*e^(5*n*ln(x) + 1)
+ 122*b*c*n^3*x*e^(4*n*ln(x) + 1) + 78*b^2*n^3*x*e^(3*n*ln(x) + 1) + 156*a*c*n^
3*x*e^(3*n*ln(x) + 1) + 214*a*b*n^3*x*e^(2*n*ln(x) + 1) + 154*a^2*n^3*x*e^(n*ln(
x) + 1) + 142*a*b*d*n^2*x*e^(n*ln(x)) + 85*a^2*d*n^2*x + 11*c^2*d*n*x*e^(4*n*ln(
x)) + 24*b*c*d*n*x*e^(3*n*ln(x)) + 13*b^2*d*n*x*e^(2*n*ln(x)) + 26*a*c*d*n*x*e^(
2*n*ln(x)) + 35*c^2*n^2*x*e^(5*n*ln(x) + 1) + 82*b*c*n^2*x*e^(4*n*ln(x) + 1) + 4
9*b^2*n^2*x*e^(3*n*ln(x) + 1) + 98*a*c*n^2*x*e^(3*n*ln(x) + 1) + 118*a*b*n^2*x*e
^(2*n*ln(x) + 1) + 71*a^2*n^2*x*e^(n*ln(x) + 1) + 28*a*b*d*n*x*e^(n*ln(x)) + 15*
a^2*d*n*x + c^2*d*x*e^(4*n*ln(x)) + 2*b*c*d*x*e^(3*n*ln(x)) + b^2*d*x*e^(2*n*ln(
x)) + 2*a*c*d*x*e^(2*n*ln(x)) + 10*c^2*n*x*e^(5*n*ln(x) + 1) + 22*b*c*n*x*e^(4*n
*ln(x) + 1) + 12*b^2*n*x*e^(3*n*ln(x) + 1) + 24*a*c*n*x*e^(3*n*ln(x) + 1) + 26*a
*b*n*x*e^(2*n*ln(x) + 1) + 14*a^2*n*x*e^(n*ln(x) + 1) + 2*a*b*d*x*e^(n*ln(x)) +
a^2*d*x + c^2*x*e^(5*n*ln(x) + 1) + 2*b*c*x*e^(4*n*ln(x) + 1) + b^2*x*e^(3*n*ln(
x) + 1) + 2*a*c*x*e^(3*n*ln(x) + 1) + 2*a*b*x*e^(2*n*ln(x) + 1) + a^2*x*e^(n*ln(
x) + 1))/(120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)