3.80 \(\int \frac {(d+e x^n)^3}{(a+b x^n+c x^{2 n})^3} \, dx\)

Optimal. Leaf size=1707 \[ \frac {\left (-e (1-n) b^3+\left (3 c d-\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-5 n)+3 \sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (6 c d (1-2 n)+\sqrt {b^2-4 a c} e (1-n)\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) e^2}{a c \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) n}+\frac {\left (-e (1-n) b^3+\left (3 c d+\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-5 n)-3 \sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (6 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) e^2}{a c \left (b^2-4 a c\right ) \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) n}+\frac {x \left (c \left (-e b^2+3 c d b-2 a c e\right ) x^n-6 a c^2 d+3 b^2 c d-b^3 e+a b c e\right ) e^2}{a c^2 \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}+\frac {\left ((1-n) \left (-2 a e^3 n b^4+c d \left (c (1-2 n) d^2+6 a e^2 n\right ) b^3-a c e \left (3 c d^2-a e^2 (2 n+1)\right ) b^2-2 a c^2 d \left (c (2-7 n) d^2+3 a e^2 n\right ) b+4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)\right )-\frac {2 a e^3 (1-n) n b^5-c d (1-n) \left (c (1-2 n) d^2+6 a e^2 n\right ) b^4+a c e \left (3 c (1-n) d^2+a e^2 \left (30 n^2-19 n+1\right )\right ) b^3+6 a c^2 d \left (c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-4 a^2 c^2 e \left (3 c \left (-3 n^2-n+1\right ) d^2+a e^2 \left (19 n^2-11 n+1\right )\right ) b-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 c \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}+\frac {\left ((1-n) \left (-2 a e^3 n b^4+c d \left (c (1-2 n) d^2+6 a e^2 n\right ) b^3-a c e \left (3 c d^2-a e^2 (2 n+1)\right ) b^2-2 a c^2 d \left (c (2-7 n) d^2+3 a e^2 n\right ) b+4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)\right )+\frac {2 a e^3 (1-n) n b^5-c d (1-n) \left (c (1-2 n) d^2+6 a e^2 n\right ) b^4+a c e \left (3 c (1-n) d^2+a e^2 \left (30 n^2-19 n+1\right )\right ) b^3+6 a c^2 d \left (c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-4 a^2 c^2 e \left (3 c \left (-3 n^2-n+1\right ) d^2+a e^2 \left (19 n^2-11 n+1\right )\right ) b-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 c \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) n^2}-\frac {x \left (c \left (-2 a e^3 n b^4+c d \left (c (1-2 n) d^2+6 a e^2 n\right ) b^3-a c e \left (3 c d^2-a e^2 (2 n+1)\right ) b^2-2 a c^2 d \left (c (2-7 n) d^2+3 a e^2 n\right ) b+4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)\right ) x^n+a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c (1-2 n) d^2+6 a e^2 n\right )\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (-\left (\left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )+b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )}{2 a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )^2} \]

[Out]

1/2*x*(b^2*c*d^3-2*a*c*d*(-3*a*e^2+c*d^2)-a*b*e*(a*e^2+3*c*d^2)-(a*b^2*e^3+2*a*c*e*(-a*e^2+3*c*d^2)-b*c*d*(3*a
*e^2+c*d^2))*x^n)/a/c/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))^2+e^2*x*(3*b^2*c*d-6*a*c^2*d-b^3*e+a*b*c*e+c*(-2*a*c*
e-b^2*e+3*b*c*d)*x^n)/a/c^2/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))-1/2*x*(a*b^2*c^2*d*(3*a*e^2*(1-9*n)-5*c*d^2*(1-
3*n))+4*a^2*c^3*d*(-3*a*e^2+c*d^2)*(1-4*n)-2*a*b^5*e^3*n+2*a^2*b*c^2*e*(3*c*d^2*(2-3*n)-5*a*e^2*n)-3*a*b^3*c*e
*(-3*a*e^2*n+c*d^2)+b^4*c*d*(c*d^2*(1-2*n)+6*a*e^2*n)+c*(4*a^2*c^2*e*(-a*e^2+3*c*d^2)*(1-3*n)-2*a*b^4*e^3*n-2*
a*b*c^2*d*(c*d^2*(2-7*n)+3*a*e^2*n)+b^3*c*d*(c*d^2*(1-2*n)+6*a*e^2*n)-a*b^2*c*e*(3*c*d^2-a*e^2*(1+2*n)))*x^n)/
a^2/c^2/(-4*a*c+b^2)^2/n^2/(a+b*x^n+c*x^(2*n))+1/2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)
))*((1-n)*(4*a^2*c^2*e*(-a*e^2+3*c*d^2)*(1-3*n)-2*a*b^4*e^3*n-2*a*b*c^2*d*(c*d^2*(2-7*n)+3*a*e^2*n)+b^3*c*d*(c
*d^2*(1-2*n)+6*a*e^2*n)-a*b^2*c*e*(3*c*d^2-a*e^2*(1+2*n)))+(-2*a*b^5*e^3*(1-n)*n+b^4*c*d*(1-n)*(c*d^2*(1-2*n)+
6*a*e^2*n)+8*a^2*c^3*d*(-3*a*e^2+c*d^2)*(8*n^2-6*n+1)-6*a*b^2*c^2*d*(c*d^2*(3*n^2-4*n+1)-a*e^2*(15*n^2-10*n+1)
)+4*a^2*b*c^2*e*(3*c*d^2*(-3*n^2-n+1)+a*e^2*(19*n^2-11*n+1))-a*b^3*c*e*(3*c*d^2*(1-n)+a*e^2*(30*n^2-19*n+1)))/
(-4*a*c+b^2)^(1/2))/a^2/c/(-4*a*c+b^2)^2/n^2/(b-(-4*a*c+b^2)^(1/2))+1/2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/
(b+(-4*a*c+b^2)^(1/2)))*((1-n)*(4*a^2*c^2*e*(-a*e^2+3*c*d^2)*(1-3*n)-2*a*b^4*e^3*n-2*a*b*c^2*d*(c*d^2*(2-7*n)+
3*a*e^2*n)+b^3*c*d*(c*d^2*(1-2*n)+6*a*e^2*n)-a*b^2*c*e*(3*c*d^2-a*e^2*(1+2*n)))+(2*a*b^5*e^3*(1-n)*n-b^4*c*d*(
1-n)*(c*d^2*(1-2*n)+6*a*e^2*n)-8*a^2*c^3*d*(-3*a*e^2+c*d^2)*(8*n^2-6*n+1)+6*a*b^2*c^2*d*(c*d^2*(3*n^2-4*n+1)-a
*e^2*(15*n^2-10*n+1))-4*a^2*b*c^2*e*(3*c*d^2*(-3*n^2-n+1)+a*e^2*(19*n^2-11*n+1))+a*b^3*c*e*(3*c*d^2*(1-n)+a*e^
2*(30*n^2-19*n+1)))/(-4*a*c+b^2)^(1/2))/a^2/c/(-4*a*c+b^2)^2/n^2/(b+(-4*a*c+b^2)^(1/2))+e^2*x*hypergeom([1, 1/
n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))*(-b^3*e*(1-n)+b^2*(1-n)*(3*c*d+e*(-4*a*c+b^2)^(1/2))+b*c*(2*a*e*(2
-5*n)-3*d*(1-n)*(-4*a*c+b^2)^(1/2))-2*a*c*(6*c*d*(1-2*n)-e*(1-n)*(-4*a*c+b^2)^(1/2)))/a/c/(-4*a*c+b^2)/n/(b^2-
4*a*c+b*(-4*a*c+b^2)^(1/2))+e^2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*(-b^3*e*(1-n)+b^
2*(1-n)*(3*c*d-e*(-4*a*c+b^2)^(1/2))+b*c*(2*a*e*(2-5*n)+3*d*(1-n)*(-4*a*c+b^2)^(1/2))-2*a*c*(6*c*d*(1-2*n)+e*(
1-n)*(-4*a*c+b^2)^(1/2)))/a/c/(-4*a*c+b^2)/n/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 5.25, antiderivative size = 1707, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1436, 1430, 1422, 245} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) -
b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(2*a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(3*b^2*c*d - 6*a*c^
2*d - b^3*e + a*b*c*e + c*(3*b*c*d - b^2*e - 2*a*c*e)*x^n))/(a*c^2*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) -
(x*(a*b^2*c^2*d*(3*a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) + 4*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 4*n) - 2*a*b^5*e^
3*n + 2*a^2*b*c^2*e*(3*c*d^2*(2 - 3*n) - 5*a*e^2*n) - 3*a*b^3*c*e*(c*d^2 - 3*a*e^2*n) + b^4*c*d*(c*d^2*(1 - 2*
n) + 6*a*e^2*n) + c*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) +
3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n)))*x^n))/(2*a^2*c^2*(
b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n))) + (e^2*(b*c*(2*a*e*(2 - 5*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*
a*c*(6*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - n
))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4
*a*c - b*Sqrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^
2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*
n))) - (2*a*b^5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c*d^2 - 3*a*e^2)*
(1 - 6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)) - 4*a^2*b*c^2*e*(3*c*d
^2*(1 - n - 3*n^2) + a*e^2*(1 - 11*n + 19*n^2)) + a*b^3*c*e*(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqr
t[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*c*(b^2
- 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) + (e^2*(b*c*(2*a*e*(2 - 5*n) - 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*
(6*c*d*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x
*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4*a*c
+ b*Sqrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*
(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n)))
+ (2*a*b^5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 -
6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)) - 4*a^2*b*c^2*e*(3*c*d^2*(
1 - n - 3*n^2) + a*e^2*(1 - 11*n + 19*n^2)) + a*b^3*c*e*(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^
2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*c*(b^2 - 4*
a*c)^2*(b + Sqrt[b^2 - 4*a*c])*n^2)

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 1430

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> -Simp[(x*(d*b^2 -
a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist
[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c*d*(2*n*p + 2*n + 1) + (2*n*p + 3*
n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[
n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]

Rule 1436

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandInt
egrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] &
& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ((IntegersQ[p, q] &&  !IntegerQ[n]) || IGtQ[p, 0] ||
(IGtQ[q, 0] &&  !IntegerQ[n]))

Rubi steps

\begin {align*} \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\int \left (\frac {c^2 d^3-3 a c d e^2+a b e^3+\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3\right ) x^n}{c^2 \left (a+b x^n+c x^{2 n}\right )^3}+\frac {e^2 \left (3 c d-b e+c e x^n\right )}{c^2 \left (a+b x^n+c x^{2 n}\right )^2}\right ) \, dx\\ &=\frac {\int \frac {c^2 d^3-3 a c d e^2+a b e^3+\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3\right ) x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{c^2}+\frac {e^2 \int \frac {3 c d-b e+c e x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{c^2}\\ &=\frac {x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac {\int \frac {-a b c e \left (3 c d^2+a e^2 (1-8 n)\right )-2 a c^2 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^3 e^3 n+b^2 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a c^2 \left (b^2-4 a c\right ) n}-\frac {e^2 \int \frac {-a b c e-2 a c (3 c d-b e) (1-2 n)+b^2 (3 c d-b e) (1-n)+c \left (3 b c d-b^2 e-2 a c e\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a c^2 \left (b^2-4 a c\right ) n}\\ &=\frac {x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac {x \left (a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )+c \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {\int \frac {-2 a b^5 e^3 (1-n) n+b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )+2 a^2 b c^2 e \left (3 c d^2 (2-5 n)-a e^2 (7-16 n) n\right )-a b^3 c e \left (3 c d^2 (1-n)-2 a e^2 (5-8 n) n\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )-a b^2 c^2 d \left (c d^2 \left (5-21 n+16 n^2\right )-3 a e^2 \left (1-11 n+16 n^2\right )\right )+c (1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2}-\frac {\left (e^2 \left (\frac {2 a b c e (2-5 n)-12 a c^2 d (1-2 n)+3 b^2 c d (1-n)-b^3 e (1-n)}{\sqrt {b^2-4 a c}}+\left (3 b c d-b^2 e-2 a c e\right ) (1-n)\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 a c \left (b^2-4 a c\right ) n}+\frac {\left (e^2 \left (b c \left (2 a e (2-5 n)-3 \sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (6 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )+b^2 \left (3 c d+\sqrt {b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 a c \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac {x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac {x \left (a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )+c \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {e^2 \left (\frac {2 a b c e (2-5 n)-12 a c^2 d (1-2 n)+3 b^2 c d (1-n)-b^3 e (1-n)}{\sqrt {b^2-4 a c}}+\left (3 b c d-b^2 e-2 a c e\right ) (1-n)\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right ) n}+\frac {e^2 \left (b c \left (2 a e (2-5 n)-3 \sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (6 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )+b^2 \left (3 c d+\sqrt {b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) n}+\frac {\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )-\frac {2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{4 a^2 c \left (b^2-4 a c\right )^2 n^2}+\frac {\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )+\frac {2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{4 a^2 c \left (b^2-4 a c\right )^2 n^2}\\ &=\frac {x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac {x \left (a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )+c \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {e^2 \left (\frac {2 a b c e (2-5 n)-12 a c^2 d (1-2 n)+3 b^2 c d (1-n)-b^3 e (1-n)}{\sqrt {b^2-4 a c}}+\left (3 b c d-b^2 e-2 a c e\right ) (1-n)\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right ) n}+\frac {\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )-\frac {2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 c \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}+\frac {e^2 \left (b c \left (2 a e (2-5 n)-3 \sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (6 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )+b^2 \left (3 c d+\sqrt {b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) n}+\frac {\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )+\frac {2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 c \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) n^2}\\ \end {align*}

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Mathematica [B]  time = 7.79, size = 13018, normalized size = 7.63 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c^{3} x^{6 \, n} + b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3} + 3 \, {\left (b c^{2} x^{n} + a c^{2}\right )} x^{4 \, n} + 3 \, {\left (b^{2} c x^{2 \, n} + 2 \, a b c x^{n} + a^{2} c\right )} x^{2 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3)/(c^3*x^(6*n) + b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*
a^2*b*x^n + a^3 + 3*(b*c^2*x^n + a*c^2)*x^(4*n) + 3*(b^2*c*x^(2*n) + 2*a*b*c*x^n + a^2*c)*x^(2*n)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+d \right )^{3}}{\left (b \,x^{n}+c \,x^{2 n}+a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b^3*c^2*d^3*(2*n - 1) + 4*a^3*c^2*e^3*(n + 1) + (12*c^3*d^2*e*(3*n - 1) + b^2*c*e^3*(2*n - 1) - 18*b*c^2
*d*e^2*n)*a^2 - (2*b*c^3*d^3*(7*n - 2) - 3*b^2*c^2*d^2*e)*a)*x*x^(3*n) + (2*b^4*c*d^3*(2*n - 1) + 2*(b*c*e^3*(
3*n + 2) + 6*c^2*d*e^2)*a^3 - (3*b^2*c*d*e^2*(9*n + 1) - 6*b*c^2*d^2*e*(9*n - 4) - 4*c^3*d^3*(4*n - 1) - b^3*e
^3*(3*n - 1))*a^2 - (b^2*c^2*d^3*(29*n - 9) - 6*b^3*c*d^2*e)*a)*x*x^(2*n) + (b^5*d^3*(2*n - 1) - 4*a^4*c*e^3*(
n - 1) + (b^2*e^3*(10*n - 1) + 12*c^2*d^2*e*(5*n - 1) - 6*b*c*d*e^2*(5*n - 2))*a^3 + (3*b^2*c*d^2*e*(4*n - 3)
- 3*b^3*d*e^2*(2*n + 1) - 2*b*c^2*d^3*n)*a^2 - (4*b^3*c*d^3*(3*n - 1) - 3*b^4*d^2*e)*a)*x*x^n + (a*b^4*d^3*(3*
n - 1) - 6*(2*c*d*e^2*(2*n - 1) - b*e^3*n)*a^4 + (4*c^2*d^3*(6*n - 1) + 6*b*c*d^2*e*(5*n - 2) - 3*b^2*d*e^2*(n
+ 1))*a^3 - (b^2*c*d^3*(21*n - 5) + 3*b^3*d^2*e*(n - 1))*a^2)*x)/(a^4*b^4*n^2 - 8*a^5*b^2*c*n^2 + 16*a^6*c^2*
n^2 + (a^2*b^4*c^2*n^2 - 8*a^3*b^2*c^3*n^2 + 16*a^4*c^4*n^2)*x^(4*n) + 2*(a^2*b^5*c*n^2 - 8*a^3*b^3*c^2*n^2 +
16*a^4*b*c^3*n^2)*x^(3*n) + (a^2*b^6*n^2 - 6*a^3*b^4*c*n^2 + 32*a^5*c^3*n^2)*x^(2*n) + 2*(a^3*b^5*n^2 - 8*a^4*
b^3*c*n^2 + 16*a^5*b*c^2*n^2)*x^n) + integrate(1/2*((2*n^2 - 3*n + 1)*b^4*d^3 + 6*(2*c*d*e^2*(2*n - 1) - b*e^3
*n)*a^3 + (4*(8*n^2 - 6*n + 1)*c^2*d^3 - 6*b*c*d^2*e*(5*n - 2) + 3*b^2*d*e^2*(n + 1))*a^2 - ((16*n^2 - 21*n +
5)*b^2*c*d^3 - 3*b^3*d^2*e*(n - 1))*a + ((2*n^2 - 3*n + 1)*b^3*c*d^3 + 4*(n^2 - 1)*a^3*c*e^3 + (12*(3*n^2 - 4*
n + 1)*c^2*d^2*e - 18*(n^2 - n)*b*c*d*e^2 + (2*n^2 - 3*n + 1)*b^2*e^3)*a^2 - (2*(7*n^2 - 9*n + 2)*b*c^2*d^3 -
3*b^2*c*d^2*e*(n - 1))*a)*x^n)/(a^3*b^4*n^2 - 8*a^4*b^2*c*n^2 + 16*a^5*c^2*n^2 + (a^2*b^4*c*n^2 - 8*a^3*b^2*c^
2*n^2 + 16*a^4*c^3*n^2)*x^(2*n) + (a^2*b^5*n^2 - 8*a^3*b^3*c*n^2 + 16*a^4*b*c^2*n^2)*x^n), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x^n\right )}^3}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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