3.67 \(\int (d+e x^n) (a+b x^n+c x^{2 n})^2 \, dx\)
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)
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Rubi [A] time = 0.101675, 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, Rules used =
{1432} \[ a^2 d x+\frac{x^{2 n+1} \left (2 a b e+2 a c d+b^2 d\right )}{2 n+1}+\frac{x^{3 n+1} \left (2 a c e+b^2 e+2 b c d\right )}{3 n+1}+\frac{a x^{n+1} (a e+2 b d)}{n+1}+\frac{c x^{4 n+1} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n+1}}{5 n+1} \]
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)
Rule 1432
Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4
*a*c, 0]
Rubi steps
\begin{align*} \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^2 \, dx &=\int \left (a^2 d+a (2 b d+a e) x^n+\left (b^2 d+2 a c d+2 a b e\right ) x^{2 n}+\left (2 b c d+b^2 e+2 a c e\right ) x^{3 n}+c (c d+2 b e) x^{4 n}+c^2 e x^{5 n}\right ) \, dx\\ &=a^2 d x+\frac{a (2 b d+a e) x^{1+n}}{1+n}+\frac{\left (b^2 d+2 a c d+2 a b e\right ) x^{1+2 n}}{1+2 n}+\frac{\left (2 b c d+b^2 e+2 a c e\right ) x^{1+3 n}}{1+3 n}+\frac{c (c d+2 b e) x^{1+4 n}}{1+4 n}+\frac{c^2 e x^{1+5 n}}{1+5 n}\\ \end{align*}
Mathematica [A] time = 0.254792, 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))
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Maple [A] time = 0.013, size = 138, normalized size = 1.1 \begin{align*}{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}} \end{align*}
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
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.38937, size = 1161, normalized size = 8.8 \begin{align*} \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} \end{align*}
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, 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 + 1
07*(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)
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Sympy [A] time = 13.2015, size = 3128, normalized size = 23.7 \begin{align*} \text{result too large to display} \end{align*}
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*sqrt(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*l
og(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 + 85*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 + 27
4*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 + 15*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 + 2
74*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)/(1
20*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) + 122*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 + 1
5*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)/(1
20*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) + 10*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))
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Giac [B] time = 1.13853, size = 1118, normalized size = 8.47 \begin{align*} \frac{120 \, a^{2} d n^{5} x + 30 \, c^{2} d n^{4} x x^{4 \, n} + 80 \, b c d n^{4} x x^{3 \, n} + 60 \, b^{2} d n^{4} x x^{2 \, n} + 120 \, a c d n^{4} x x^{2 \, n} + 240 \, a b d n^{4} x x^{n} + 24 \, c^{2} n^{4} x x^{5 \, n} e + 60 \, b c n^{4} x x^{4 \, n} e + 40 \, b^{2} n^{4} x x^{3 \, n} e + 80 \, a c n^{4} x x^{3 \, n} e + 120 \, a b n^{4} x x^{2 \, n} e + 120 \, a^{2} n^{4} x x^{n} e + 274 \, a^{2} d n^{4} x + 61 \, c^{2} d n^{3} x x^{4 \, n} + 156 \, b c d n^{3} x x^{3 \, n} + 107 \, b^{2} d n^{3} x x^{2 \, n} + 214 \, a c d n^{3} x x^{2 \, n} + 308 \, a b d n^{3} x x^{n} + 50 \, c^{2} n^{3} x x^{5 \, n} e + 122 \, b c n^{3} x x^{4 \, n} e + 78 \, b^{2} n^{3} x x^{3 \, n} e + 156 \, a c n^{3} x x^{3 \, n} e + 214 \, a b n^{3} x x^{2 \, n} e + 154 \, a^{2} n^{3} x x^{n} e + 225 \, a^{2} d n^{3} x + 41 \, c^{2} d n^{2} x x^{4 \, n} + 98 \, b c d n^{2} x x^{3 \, n} + 59 \, b^{2} d n^{2} x x^{2 \, n} + 118 \, a c d n^{2} x x^{2 \, n} + 142 \, a b d n^{2} x x^{n} + 35 \, c^{2} n^{2} x x^{5 \, n} e + 82 \, b c n^{2} x x^{4 \, n} e + 49 \, b^{2} n^{2} x x^{3 \, n} e + 98 \, a c n^{2} x x^{3 \, n} e + 118 \, a b n^{2} x x^{2 \, n} e + 71 \, a^{2} n^{2} x x^{n} e + 85 \, a^{2} d n^{2} x + 11 \, c^{2} d n x x^{4 \, n} + 24 \, b c d n x x^{3 \, n} + 13 \, b^{2} d n x x^{2 \, n} + 26 \, a c d n x x^{2 \, n} + 28 \, a b d n x x^{n} + 10 \, c^{2} n x x^{5 \, n} e + 22 \, b c n x x^{4 \, n} e + 12 \, b^{2} n x x^{3 \, n} e + 24 \, a c n x x^{3 \, n} e + 26 \, a b n x x^{2 \, n} e + 14 \, a^{2} n x x^{n} e + 15 \, a^{2} d n x + c^{2} d x x^{4 \, n} + 2 \, b c d x x^{3 \, n} + b^{2} d x x^{2 \, n} + 2 \, a c d x x^{2 \, n} + 2 \, a b d x x^{n} + c^{2} x x^{5 \, n} e + 2 \, b c x x^{4 \, n} e + b^{2} x x^{3 \, n} e + 2 \, a c x x^{3 \, n} e + 2 \, a b x x^{2 \, n} e + a^{2} x x^{n} e + a^{2} d x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \end{align*}
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, algorithm="giac")
[Out]
(120*a^2*d*n^5*x + 30*c^2*d*n^4*x*x^(4*n) + 80*b*c*d*n^4*x*x^(3*n) + 60*b^2*d*n^4*x*x^(2*n) + 120*a*c*d*n^4*x*
x^(2*n) + 240*a*b*d*n^4*x*x^n + 24*c^2*n^4*x*x^(5*n)*e + 60*b*c*n^4*x*x^(4*n)*e + 40*b^2*n^4*x*x^(3*n)*e + 80*
a*c*n^4*x*x^(3*n)*e + 120*a*b*n^4*x*x^(2*n)*e + 120*a^2*n^4*x*x^n*e + 274*a^2*d*n^4*x + 61*c^2*d*n^3*x*x^(4*n)
+ 156*b*c*d*n^3*x*x^(3*n) + 107*b^2*d*n^3*x*x^(2*n) + 214*a*c*d*n^3*x*x^(2*n) + 308*a*b*d*n^3*x*x^n + 50*c^2*
n^3*x*x^(5*n)*e + 122*b*c*n^3*x*x^(4*n)*e + 78*b^2*n^3*x*x^(3*n)*e + 156*a*c*n^3*x*x^(3*n)*e + 214*a*b*n^3*x*x
^(2*n)*e + 154*a^2*n^3*x*x^n*e + 225*a^2*d*n^3*x + 41*c^2*d*n^2*x*x^(4*n) + 98*b*c*d*n^2*x*x^(3*n) + 59*b^2*d*
n^2*x*x^(2*n) + 118*a*c*d*n^2*x*x^(2*n) + 142*a*b*d*n^2*x*x^n + 35*c^2*n^2*x*x^(5*n)*e + 82*b*c*n^2*x*x^(4*n)*
e + 49*b^2*n^2*x*x^(3*n)*e + 98*a*c*n^2*x*x^(3*n)*e + 118*a*b*n^2*x*x^(2*n)*e + 71*a^2*n^2*x*x^n*e + 85*a^2*d*
n^2*x + 11*c^2*d*n*x*x^(4*n) + 24*b*c*d*n*x*x^(3*n) + 13*b^2*d*n*x*x^(2*n) + 26*a*c*d*n*x*x^(2*n) + 28*a*b*d*n
*x*x^n + 10*c^2*n*x*x^(5*n)*e + 22*b*c*n*x*x^(4*n)*e + 12*b^2*n*x*x^(3*n)*e + 24*a*c*n*x*x^(3*n)*e + 26*a*b*n*
x*x^(2*n)*e + 14*a^2*n*x*x^n*e + 15*a^2*d*n*x + c^2*d*x*x^(4*n) + 2*b*c*d*x*x^(3*n) + b^2*d*x*x^(2*n) + 2*a*c*
d*x*x^(2*n) + 2*a*b*d*x*x^n + c^2*x*x^(5*n)*e + 2*b*c*x*x^(4*n)*e + b^2*x*x^(3*n)*e + 2*a*c*x*x^(3*n)*e + 2*a*
b*x*x^(2*n)*e + a^2*x*x^n*e + a^2*d*x)/(120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)