\(\int (d+e x^n) (a+b x^n+c x^{2 n})^3 \, dx\) [62]

Optimal result

Integrand size = 24, antiderivative size = 218 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=a^3 d x+\frac {a^2 (3 b d+a e) x^{1+n}}{1+n}+\frac {3 a \left (b^2 d+a c d+a b e\right ) x^{1+2 n}}{1+2 n}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^{1+3 n}}{1+3 n}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^{1+4 n}}{1+4 n}+\frac {3 c \left (b c d+b^2 e+a c e\right ) x^{1+5 n}}{1+5 n}+\frac {c^2 (c d+3 b e) x^{1+6 n}}{1+6 n}+\frac {c^3 e x^{1+7 n}}{1+7 n} \] Output:

a^3*d*x+a^2*(a*e+3*b*d)*x^(1+n)/(1+n)+3*a*(a*b*e+a*c*d+b^2*d)*x^(1+2*n)/(1 
+2*n)+(3*a^2*c*e+3*a*b^2*e+6*a*b*c*d+b^3*d)*x^(1+3*n)/(1+3*n)+(6*a*b*c*e+3 
*a*c^2*d+b^3*e+3*b^2*c*d)*x^(1+4*n)/(1+4*n)+3*c*(a*c*e+b^2*e+b*c*d)*x^(1+5 
*n)/(1+5*n)+c^2*(3*b*e+c*d)*x^(1+6*n)/(1+6*n)+c^3*e*x^(1+7*n)/(1+7*n)
 

Mathematica [A] (verified)

Time = 4.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=x \left (a^3 d+\frac {a^2 (3 b d+a e) x^n}{1+n}+\frac {3 a \left (b^2 d+a c d+a b e\right ) x^{2 n}}{1+2 n}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^{3 n}}{1+3 n}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^{4 n}}{1+4 n}+\frac {3 c \left (b c d+b^2 e+a c e\right ) x^{5 n}}{1+5 n}+\frac {c^2 (c d+3 b e) x^{6 n}}{1+6 n}+\frac {c^3 e x^{7 n}}{1+7 n}\right ) \] Input:

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

Output:

x*(a^3*d + (a^2*(3*b*d + a*e)*x^n)/(1 + n) + (3*a*(b^2*d + a*c*d + a*b*e)* 
x^(2*n))/(1 + 2*n) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^(3*n)) 
/(1 + 3*n) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^(4*n))/(1 + 4* 
n) + (3*c*(b*c*d + b^2*e + a*c*e)*x^(5*n))/(1 + 5*n) + (c^2*(c*d + 3*b*e)* 
x^(6*n))/(1 + 6*n) + (c^3*e*x^(7*n))/(1 + 7*n))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1762, 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.

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

\(\Big \downarrow \) 1762

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

a^3*d*x + (a^2*(3*b*d + a*e)*x^(1 + n))/(1 + n) + (3*a*(b^2*d + a*c*d + a* 
b*e)*x^(1 + 2*n))/(1 + 2*n) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e) 
*x^(1 + 3*n))/(1 + 3*n) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^( 
1 + 4*n))/(1 + 4*n) + (3*c*(b*c*d + b^2*e + a*c*e)*x^(1 + 5*n))/(1 + 5*n) 
+ (c^2*(c*d + 3*b*e)*x^(1 + 6*n))/(1 + 6*n) + (c^3*e*x^(1 + 7*n))/(1 + 7*n 
)
 

Defintions of rubi rules used

rule 1762

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(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]
 

rule 2009

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

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.97

method	result	size
risch	\(a^{3} d x +\frac {\left (6 a b c e +3 a \,c^{2} d +b^{3} e +3 b^{2} c d \right ) x \,x^{4 n}}{1+4 n}+\frac {\left (3 a^{2} c e +3 a \,b^{2} e +6 a b c d +b^{3} d \right ) x \,x^{3 n}}{1+3 n}+\frac {a^{2} \left (a e +3 b d \right ) x \,x^{n}}{1+n}+\frac {c^{2} \left (3 e b +c d \right ) x \,x^{6 n}}{1+6 n}+\frac {e \,c^{3} x \,x^{7 n}}{7 n +1}+\frac {3 a \left (a b e +a c d +d \,b^{2}\right ) x \,x^{2 n}}{1+2 n}+\frac {3 c \left (a c e +b^{2} e +c b d \right ) x \,x^{5 n}}{1+5 n}\)	\(212\)
norman	\(a^{3} d x +\frac {\left (6 a b c e +3 a \,c^{2} d +b^{3} e +3 b^{2} c d \right ) x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}+\frac {\left (3 a^{2} c e +3 a \,b^{2} e +6 a b c d +b^{3} d \right ) x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {a^{2} \left (a e +3 b d \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {c^{2} \left (3 e b +c d \right ) x \,{\mathrm e}^{6 n \ln \left (x \right )}}{1+6 n}+\frac {e \,c^{3} x \,{\mathrm e}^{7 n \ln \left (x \right )}}{7 n +1}+\frac {3 a \left (a b e +a c d +d \,b^{2}\right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {3 c \left (a c e +b^{2} e +c b d \right ) x \,{\mathrm e}^{5 n \ln \left (x \right )}}{1+5 n}\)	\(226\)
parallelrisch	\(\text {Expression too large to display}\)	\(2404\)
orering	\(\text {Expression too large to display}\)	\(15487\)

Input:

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

Output:

a^3*d*x+(6*a*b*c*e+3*a*c^2*d+b^3*e+3*b^2*c*d)/(1+4*n)*x*(x^n)^4+(3*a^2*c*e 
+3*a*b^2*e+6*a*b*c*d+b^3*d)/(1+3*n)*x*(x^n)^3+a^2*(a*e+3*b*d)/(1+n)*x*x^n+ 
c^2*(3*b*e+c*d)/(1+6*n)*x*(x^n)^6+e*c^3/(7*n+1)*x*(x^n)^7+3*a*(a*b*e+a*c*d 
+b^2*d)/(1+2*n)*x*(x^n)^2+3*c*(a*c*e+b^2*e+b*c*d)/(1+5*n)*x*(x^n)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1209 vs. \(2 (218) = 436\).

Time = 0.09 (sec) , antiderivative size = 1209, normalized size of antiderivative = 5.55 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((720*c^3*e*n^6 + 1764*c^3*e*n^5 + 1624*c^3*e*n^4 + 735*c^3*e*n^3 + 175*c^ 
3*e*n^2 + 21*c^3*e*n + c^3*e)*x*x^(7*n) + (840*(c^3*d + 3*b*c^2*e)*n^6 + 2 
038*(c^3*d + 3*b*c^2*e)*n^5 + 1849*(c^3*d + 3*b*c^2*e)*n^4 + c^3*d + 3*b*c 
^2*e + 820*(c^3*d + 3*b*c^2*e)*n^3 + 190*(c^3*d + 3*b*c^2*e)*n^2 + 22*(c^3 
*d + 3*b*c^2*e)*n)*x*x^(6*n) + 3*(1008*(b*c^2*d + (b^2*c + a*c^2)*e)*n^6 + 
 2412*(b*c^2*d + (b^2*c + a*c^2)*e)*n^5 + 2144*(b*c^2*d + (b^2*c + a*c^2)* 
e)*n^4 + b*c^2*d + 925*(b*c^2*d + (b^2*c + a*c^2)*e)*n^3 + 207*(b*c^2*d + 
(b^2*c + a*c^2)*e)*n^2 + (b^2*c + a*c^2)*e + 23*(b*c^2*d + (b^2*c + a*c^2) 
*e)*n)*x*x^(5*n) + (1260*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*n^6 + 2 
952*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*n^5 + 2545*(3*(b^2*c + a*c^2 
)*d + (b^3 + 6*a*b*c)*e)*n^4 + 1056*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c) 
*e)*n^3 + 226*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*n^2 + 3*(b^2*c + a 
*c^2)*d + (b^3 + 6*a*b*c)*e + 24*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e) 
*n)*x*x^(4*n) + (1680*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*n^6 + 3796 
*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*n^5 + 3112*((b^3 + 6*a*b*c)*d + 
 3*(a*b^2 + a^2*c)*e)*n^4 + 1219*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e) 
*n^3 + 247*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*n^2 + (b^3 + 6*a*b*c) 
*d + 3*(a*b^2 + a^2*c)*e + 25*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*n) 
*x*x^(3*n) + 3*(2520*(a^2*b*e + (a*b^2 + a^2*c)*d)*n^6 + 5274*(a^2*b*e + ( 
a*b^2 + a^2*c)*d)*n^5 + 3929*(a^2*b*e + (a*b^2 + a^2*c)*d)*n^4 + a^2*b*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9190 vs. \(2 (212) = 424\).

Time = 6.06 (sec) , antiderivative size = 9190, normalized size of antiderivative = 42.16 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**3*d*x + a**3*e*log(x) + 3*a**2*b*d*log(x) - 3*a**2*b*e/x - 3 
*a**2*c*d/x - 3*a**2*c*e/(2*x**2) - 3*a*b**2*d/x - 3*a*b**2*e/(2*x**2) - 3 
*a*b*c*d/x**2 - 2*a*b*c*e/x**3 - a*c**2*d/x**3 - 3*a*c**2*e/(4*x**4) - b** 
3*d/(2*x**2) - b**3*e/(3*x**3) - b**2*c*d/x**3 - 3*b**2*c*e/(4*x**4) - 3*b 
*c**2*d/(4*x**4) - 3*b*c**2*e/(5*x**5) - c**3*d/(5*x**5) - c**3*e/(6*x**6) 
, Eq(n, -1)), (a**3*d*x + 2*a**3*e*sqrt(x) + 6*a**2*b*d*sqrt(x) + 3*a**2*b 
*e*log(x) + 3*a**2*c*d*log(x) - 6*a**2*c*e/sqrt(x) + 3*a*b**2*d*log(x) - 6 
*a*b**2*e/sqrt(x) - 12*a*b*c*d/sqrt(x) - 6*a*b*c*e/x - 3*a*c**2*d/x - 2*a* 
c**2*e/x**(3/2) - 2*b**3*d/sqrt(x) - b**3*e/x - 3*b**2*c*d/x - 2*b**2*c*e/ 
x**(3/2) - 2*b*c**2*d/x**(3/2) - 3*b*c**2*e/(2*x**2) - c**3*d/(2*x**2) - 2 
*c**3*e/(5*x**(5/2)), Eq(n, -1/2)), (a**3*d*x + 3*a**3*e*x**(2/3)/2 + 9*a* 
*2*b*d*x**(2/3)/2 + 9*a**2*b*e*x**(1/3) + 9*a**2*c*d*x**(1/3) + 3*a**2*c*e 
*log(x) + 9*a*b**2*d*x**(1/3) + 3*a*b**2*e*log(x) + 6*a*b*c*d*log(x) - 18* 
a*b*c*e/x**(1/3) - 9*a*c**2*d/x**(1/3) - 9*a*c**2*e/(2*x**(2/3)) + b**3*d* 
log(x) - 3*b**3*e/x**(1/3) - 9*b**2*c*d/x**(1/3) - 9*b**2*c*e/(2*x**(2/3)) 
 - 9*b*c**2*d/(2*x**(2/3)) - 3*b*c**2*e/x - c**3*d/x - 3*c**3*e/(4*x**(4/3 
)), Eq(n, -1/3)), (a**3*d*x + 4*a**3*e*x**(3/4)/3 + 4*a**2*b*d*x**(3/4) + 
6*a**2*b*e*sqrt(x) + 6*a**2*c*d*sqrt(x) + 12*a**2*c*e*x**(1/4) + 6*a*b**2* 
d*sqrt(x) + 12*a*b**2*e*x**(1/4) + 24*a*b*c*d*x**(1/4) + 6*a*b*c*e*log(x) 
+ 3*a*c**2*d*log(x) - 12*a*c**2*e/x**(1/4) + 4*b**3*d*x**(1/4) + b**3*e...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.77 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=a^{3} d x + \frac {c^{3} e x^{7 \, n + 1}}{7 \, n + 1} + \frac {c^{3} d x^{6 \, n + 1}}{6 \, n + 1} + \frac {3 \, b c^{2} e x^{6 \, n + 1}}{6 \, n + 1} + \frac {3 \, b c^{2} d x^{5 \, n + 1}}{5 \, n + 1} + \frac {3 \, b^{2} c e x^{5 \, n + 1}}{5 \, n + 1} + \frac {3 \, a c^{2} e x^{5 \, n + 1}}{5 \, n + 1} + \frac {3 \, b^{2} c d x^{4 \, n + 1}}{4 \, n + 1} + \frac {3 \, a c^{2} d x^{4 \, n + 1}}{4 \, n + 1} + \frac {b^{3} e x^{4 \, n + 1}}{4 \, n + 1} + \frac {6 \, a b c e x^{4 \, n + 1}}{4 \, n + 1} + \frac {b^{3} d x^{3 \, n + 1}}{3 \, n + 1} + \frac {6 \, a b c d x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, a b^{2} e x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, a^{2} c e x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, a b^{2} d x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} c d x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} b e x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} b d x^{n + 1}}{n + 1} + \frac {a^{3} e x^{n + 1}}{n + 1} \] Input:

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

Output:

a^3*d*x + c^3*e*x^(7*n + 1)/(7*n + 1) + c^3*d*x^(6*n + 1)/(6*n + 1) + 3*b* 
c^2*e*x^(6*n + 1)/(6*n + 1) + 3*b*c^2*d*x^(5*n + 1)/(5*n + 1) + 3*b^2*c*e* 
x^(5*n + 1)/(5*n + 1) + 3*a*c^2*e*x^(5*n + 1)/(5*n + 1) + 3*b^2*c*d*x^(4*n 
 + 1)/(4*n + 1) + 3*a*c^2*d*x^(4*n + 1)/(4*n + 1) + b^3*e*x^(4*n + 1)/(4*n 
 + 1) + 6*a*b*c*e*x^(4*n + 1)/(4*n + 1) + b^3*d*x^(3*n + 1)/(3*n + 1) + 6* 
a*b*c*d*x^(3*n + 1)/(3*n + 1) + 3*a*b^2*e*x^(3*n + 1)/(3*n + 1) + 3*a^2*c* 
e*x^(3*n + 1)/(3*n + 1) + 3*a*b^2*d*x^(2*n + 1)/(2*n + 1) + 3*a^2*c*d*x^(2 
*n + 1)/(2*n + 1) + 3*a^2*b*e*x^(2*n + 1)/(2*n + 1) + 3*a^2*b*d*x^(n + 1)/ 
(n + 1) + a^3*e*x^(n + 1)/(n + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2064 vs. \(2 (218) = 436\).

Time = 0.17 (sec) , antiderivative size = 2064, normalized size of antiderivative = 9.47 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(5040*a^3*d*n^7*x + 720*c^3*e*n^6*x*x^(7*n) + 840*c^3*d*n^6*x*x^(6*n) + 25 
20*b*c^2*e*n^6*x*x^(6*n) + 3024*b*c^2*d*n^6*x*x^(5*n) + 3024*b^2*c*e*n^6*x 
*x^(5*n) + 3024*a*c^2*e*n^6*x*x^(5*n) + 3780*b^2*c*d*n^6*x*x^(4*n) + 3780* 
a*c^2*d*n^6*x*x^(4*n) + 1260*b^3*e*n^6*x*x^(4*n) + 7560*a*b*c*e*n^6*x*x^(4 
*n) + 1680*b^3*d*n^6*x*x^(3*n) + 10080*a*b*c*d*n^6*x*x^(3*n) + 5040*a*b^2* 
e*n^6*x*x^(3*n) + 5040*a^2*c*e*n^6*x*x^(3*n) + 7560*a*b^2*d*n^6*x*x^(2*n) 
+ 7560*a^2*c*d*n^6*x*x^(2*n) + 7560*a^2*b*e*n^6*x*x^(2*n) + 15120*a^2*b*d* 
n^6*x*x^n + 5040*a^3*e*n^6*x*x^n + 13068*a^3*d*n^6*x + 1764*c^3*e*n^5*x*x^ 
(7*n) + 2038*c^3*d*n^5*x*x^(6*n) + 6114*b*c^2*e*n^5*x*x^(6*n) + 7236*b*c^2 
*d*n^5*x*x^(5*n) + 7236*b^2*c*e*n^5*x*x^(5*n) + 7236*a*c^2*e*n^5*x*x^(5*n) 
 + 8856*b^2*c*d*n^5*x*x^(4*n) + 8856*a*c^2*d*n^5*x*x^(4*n) + 2952*b^3*e*n^ 
5*x*x^(4*n) + 17712*a*b*c*e*n^5*x*x^(4*n) + 3796*b^3*d*n^5*x*x^(3*n) + 227 
76*a*b*c*d*n^5*x*x^(3*n) + 11388*a*b^2*e*n^5*x*x^(3*n) + 11388*a^2*c*e*n^5 
*x*x^(3*n) + 15822*a*b^2*d*n^5*x*x^(2*n) + 15822*a^2*c*d*n^5*x*x^(2*n) + 1 
5822*a^2*b*e*n^5*x*x^(2*n) + 24084*a^2*b*d*n^5*x*x^n + 8028*a^3*e*n^5*x*x^ 
n + 13132*a^3*d*n^5*x + 1624*c^3*e*n^4*x*x^(7*n) + 1849*c^3*d*n^4*x*x^(6*n 
) + 5547*b*c^2*e*n^4*x*x^(6*n) + 6432*b*c^2*d*n^4*x*x^(5*n) + 6432*b^2*c*e 
*n^4*x*x^(5*n) + 6432*a*c^2*e*n^4*x*x^(5*n) + 7635*b^2*c*d*n^4*x*x^(4*n) + 
 7635*a*c^2*d*n^4*x*x^(4*n) + 2545*b^3*e*n^4*x*x^(4*n) + 15270*a*b*c*e*n^4 
*x*x^(4*n) + 3112*b^3*d*n^4*x*x^(3*n) + 18672*a*b*c*d*n^4*x*x^(3*n) + 9...
 

Mupad [B] (verification not implemented)

Time = 20.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx=a^3\,d\,x+\frac {x\,x^n\,\left (e\,a^3+3\,b\,d\,a^2\right )}{n+1}+\frac {x\,x^{2\,n}\,\left (3\,e\,a^2\,b+3\,c\,d\,a^2+3\,d\,a\,b^2\right )}{2\,n+1}+\frac {x\,x^{5\,n}\,\left (3\,e\,b^2\,c+3\,d\,b\,c^2+3\,a\,e\,c^2\right )}{5\,n+1}+\frac {x\,x^{3\,n}\,\left (3\,c\,e\,a^2+3\,e\,a\,b^2+6\,c\,d\,a\,b+d\,b^3\right )}{3\,n+1}+\frac {x\,x^{4\,n}\,\left (e\,b^3+3\,d\,b^2\,c+6\,a\,e\,b\,c+3\,a\,d\,c^2\right )}{4\,n+1}+\frac {x\,x^{6\,n}\,\left (d\,c^3+3\,b\,e\,c^2\right )}{6\,n+1}+\frac {c^3\,e\,x\,x^{7\,n}}{7\,n+1} \] Input:

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

Output:

a^3*d*x + (x*x^n*(a^3*e + 3*a^2*b*d))/(n + 1) + (x*x^(2*n)*(3*a*b^2*d + 3* 
a^2*b*e + 3*a^2*c*d))/(2*n + 1) + (x*x^(5*n)*(3*a*c^2*e + 3*b*c^2*d + 3*b^ 
2*c*e))/(5*n + 1) + (x*x^(3*n)*(b^3*d + 3*a*b^2*e + 3*a^2*c*e + 6*a*b*c*d) 
)/(3*n + 1) + (x*x^(4*n)*(b^3*e + 3*a*c^2*d + 3*b^2*c*d + 6*a*b*c*e))/(4*n 
 + 1) + (x*x^(6*n)*(c^3*d + 3*b*c^2*e))/(6*n + 1) + (c^3*e*x*x^(7*n))/(7*n 
 + 1)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1924, normalized size of antiderivative = 8.83 \[ \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(720*x**(7*n)*c**3*e*n**6 + 1764*x**(7*n)*c**3*e*n**5 + 1624*x**(7*n)*c 
**3*e*n**4 + 735*x**(7*n)*c**3*e*n**3 + 175*x**(7*n)*c**3*e*n**2 + 21*x**( 
7*n)*c**3*e*n + x**(7*n)*c**3*e + 2520*x**(6*n)*b*c**2*e*n**6 + 6114*x**(6 
*n)*b*c**2*e*n**5 + 5547*x**(6*n)*b*c**2*e*n**4 + 2460*x**(6*n)*b*c**2*e*n 
**3 + 570*x**(6*n)*b*c**2*e*n**2 + 66*x**(6*n)*b*c**2*e*n + 3*x**(6*n)*b*c 
**2*e + 840*x**(6*n)*c**3*d*n**6 + 2038*x**(6*n)*c**3*d*n**5 + 1849*x**(6* 
n)*c**3*d*n**4 + 820*x**(6*n)*c**3*d*n**3 + 190*x**(6*n)*c**3*d*n**2 + 22* 
x**(6*n)*c**3*d*n + x**(6*n)*c**3*d + 3024*x**(5*n)*a*c**2*e*n**6 + 7236*x 
**(5*n)*a*c**2*e*n**5 + 6432*x**(5*n)*a*c**2*e*n**4 + 2775*x**(5*n)*a*c**2 
*e*n**3 + 621*x**(5*n)*a*c**2*e*n**2 + 69*x**(5*n)*a*c**2*e*n + 3*x**(5*n) 
*a*c**2*e + 3024*x**(5*n)*b**2*c*e*n**6 + 7236*x**(5*n)*b**2*c*e*n**5 + 64 
32*x**(5*n)*b**2*c*e*n**4 + 2775*x**(5*n)*b**2*c*e*n**3 + 621*x**(5*n)*b** 
2*c*e*n**2 + 69*x**(5*n)*b**2*c*e*n + 3*x**(5*n)*b**2*c*e + 3024*x**(5*n)* 
b*c**2*d*n**6 + 7236*x**(5*n)*b*c**2*d*n**5 + 6432*x**(5*n)*b*c**2*d*n**4 
+ 2775*x**(5*n)*b*c**2*d*n**3 + 621*x**(5*n)*b*c**2*d*n**2 + 69*x**(5*n)*b 
*c**2*d*n + 3*x**(5*n)*b*c**2*d + 7560*x**(4*n)*a*b*c*e*n**6 + 17712*x**(4 
*n)*a*b*c*e*n**5 + 15270*x**(4*n)*a*b*c*e*n**4 + 6336*x**(4*n)*a*b*c*e*n** 
3 + 1356*x**(4*n)*a*b*c*e*n**2 + 144*x**(4*n)*a*b*c*e*n + 6*x**(4*n)*a*b*c 
*e + 3780*x**(4*n)*a*c**2*d*n**6 + 8856*x**(4*n)*a*c**2*d*n**5 + 7635*x**( 
4*n)*a*c**2*d*n**4 + 3168*x**(4*n)*a*c**2*d*n**3 + 678*x**(4*n)*a*c**2*...