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

Optimal. Leaf size=1191 \[ -\frac {\left (-\left ((1-n) b^2\right )-\sqrt {b^2-4 a c} (1-n) b+4 a c (1-2 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 ) e^2}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) n}-\frac {\left (-\left ((1-n) b^2\right )+\sqrt {b^2-4 a c} (1-n) b+4 a c (1-2 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 ) e^2}{a \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 (b c x^n+b^2-2 a c\right ) e^2}{a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac {\left ((1-n) \left (-\left (\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3\right )+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )+\frac {-\left ((1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4\right )+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-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 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}-\frac {\left ((1-n) \left (-\left (\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3\right )+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )-\frac {-\left ((1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4\right )+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-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 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) n^2}+\frac {x \left (c \left (-\left (\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3\right )+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right ) x^n+2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c (1-2 n) d^2+2 a e^2 n\right )\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (\left (b c d^2-4 a c e d+a b e^2\right ) x^n+b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )}{2 a \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*d^2-2*a*b*d*e-2*a*(-a*e^2+c*d^2)+(a*b*e^2-4*a*c*d*e+b*c*d^2)*x^n)/a/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*
n))^2+e^2*x*(b^2-2*a*c+b*c*x^n)/a/c/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))+1/2*x*(2*a*b^3*c*d*e-a*b^2*c*(a*e^2*(1-
9*n)-5*c*d^2*(1-3*n))-4*a^2*c^2*(-a*e^2+c*d^2)*(1-4*n)-4*a^2*b*c^2*d*e*(2-3*n)-b^4*(c*d^2*(1-2*n)+2*a*e^2*n)+c
*(2*a*b^2*c*d*e-8*a^2*c^2*d*e*(1-3*n)+2*a*b*c*(c*d^2*(2-7*n)+a*e^2*n)-b^3*(c*d^2*(1-2*n)+2*a*e^2*n))*x^n)/a^2/
c/(-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)*(2*a*b^2*c*d*e-8*a^2*c^2*d*e*(1-3*n)+2*a*b*c*(c*d^2*(2-7*n)+a*e^2*n)-b^3*(c*d^2*(1-2*n)+2*a*e^2*n))+(2*a*b
^3*c*d*e*(1-n)-b^4*(1-n)*(c*d^2*(1-2*n)+2*a*e^2*n)-8*a^2*b*c^2*d*e*(-3*n^2-n+1)-8*a^2*c^2*(-a*e^2+c*d^2)*(8*n^
2-6*n+1)+2*a*b^2*c*(3*c*d^2*(3*n^2-4*n+1)-a*e^2*(15*n^2-10*n+1)))/(-4*a*c+b^2)^(1/2))/a^2/(-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)*(2*a*b^2*c*d*e-
8*a^2*c^2*d*e*(1-3*n)+2*a*b*c*(c*d^2*(2-7*n)+a*e^2*n)-b^3*(c*d^2*(1-2*n)+2*a*e^2*n))+(-2*a*b^3*c*d*e*(1-n)+b^4
*(1-n)*(c*d^2*(1-2*n)+2*a*e^2*n)+8*a^2*b*c^2*d*e*(-3*n^2-n+1)+8*a^2*c^2*(-a*e^2+c*d^2)*(8*n^2-6*n+1)-2*a*b^2*c
*(3*c*d^2*(3*n^2-4*n+1)-a*e^2*(15*n^2-10*n+1)))/(-4*a*c+b^2)^(1/2))/a^2/(-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)))*(4*a*c*(1-2*n)-b^2*(1-n)-b*(1-n)*(-4*a*c
+b^2)^(1/2))/a/(-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)))*(4*a*c*(1-2*n)-b^2*(1-n)+b*(1-n)*(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)/n/(b^2-4*a*c+b*(-4*a*c+
b^2)^(1/2))

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Rubi [A]  time = 4.01, antiderivative size = 1191, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1436, 1430, 1422, 245, 1345} \[ -\frac {\left (-(1-n) b^2-\sqrt {b^2-4 a c} (1-n) b+4 a c (1-2 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 ) e^2}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) n}-\frac {\left (-(1-n) b^2+\sqrt {b^2-4 a c} (1-n) b+4 a c (1-2 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 ) e^2}{a \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 (b c x^n+b^2-2 a c\right ) e^2}{a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac {\left ((1-n) \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )+\frac {-(1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-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 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}-\frac {\left ((1-n) \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )-\frac {-(1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-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 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) n^2}+\frac {x \left (c \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right ) x^n+2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c (1-2 n) d^2+2 a e^2 n\right )\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (\left (b c d^2-4 a c e d+a b e^2\right ) x^n+b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )}{2 a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a
+ b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2 - 2*a*c + b*c*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (x*(
2*a*b^3*c*d*e - a*b^2*c*(a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) - 4*a^2*c^2*(c*d^2 - a*e^2)*(1 - 4*n) - 4*a^2*b*
c^2*d*e*(2 - 3*n) - b^4*(c*d^2*(1 - 2*n) + 2*a*e^2*n) + c*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(
c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n))*x^n))/(2*a^2*c*(b^2 - 4*a*c)^2*n^2*(a + b*x^n
+ c*x^(2*n))) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-
1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) -
(((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*
n) + 2*a*e^2*n)) + (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e^2*n) - 8*a^2*b*c^2*d*e*(1 - n
- 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10
*n + 15*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*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4
*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c
)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2
*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n)) - (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 -
2*n) + 2*a*e^2*n) - 8*a^2*b*c^2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*
(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*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*(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 1345

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

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 )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\int \left (\frac {c d^2-a e^2+\left (2 c d e-b e^2\right ) x^n}{c \left (a+b x^n+c x^{2 n}\right )^3}+\frac {e^2}{c \left (a+b x^n+c x^{2 n}\right )^2}\right ) \, dx\\ &=\frac {\int \frac {c d^2-a e^2+\left (2 c d e-b e^2\right ) x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{c}+\frac {e^2 \int \frac {1}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{c}\\ &=\frac {x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac {\int \frac {-2 a b c d e-2 a c \left (c d^2-a e^2\right ) (1-4 n)+b^2 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (b c d^2-4 a c d e+a b e^2\right ) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a c \left (b^2-4 a c\right ) n}-\frac {e^2 \int \frac {b^2-2 a c-\left (b^2-4 a c\right ) n+b c (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a c \left (b^2-4 a c\right ) n}\\ &=\frac {x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {\int \frac {4 a^2 b c^2 d e (2-5 n)-2 a b^3 c d e (1-n)+b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )+4 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )-a b^2 c \left (c d^2 \left (5-21 n+16 n^2\right )-a e^2 \left (1-11 n+16 n^2\right )\right )-c (1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 c \left (b^2-4 a c\right )^2 n^2}+\frac {\left (e^2 \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt {b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} n}-\frac {\left (e^2 \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt {b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac {x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {e^2 \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt {b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right ) n}-\frac {e^2 \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt {b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) n}-\frac {\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )-\frac {2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 n^2}-\frac {\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )+\frac {2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 n^2}\\ &=\frac {x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {e^2 \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt {b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right ) n}-\frac {\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )+\frac {2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}-\frac {e^2 \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt {b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) n}-\frac {\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )-\frac {2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \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 = 6.87, size = 10910, normalized size = 9.16 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}{c^{3} x^{6 \, n} + b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3} + 3 \, {\left (b c^{2} x^{n} + a c^{2}\right )} x^{4 \, n} + 3 \, {\left (b^{2} c x^{2 \, n} + 2 \, a b c x^{n} + a^{2} c\right )} x^{2 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e^2*x^(2*n) + 2*d*e*x^n + d^2)/(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 \[ \int \frac {{\left (e x^{n} + d\right )}^{2}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+d \right )^{2}}{\left (b \,x^{n}+c \,x^{2 n}+a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x^n+d)^2/(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 \[ \frac {{\left (b^{3} c^{2} d^{2} {\left (2 \, n - 1\right )} + 2 \, {\left (4 \, c^{3} d e {\left (3 \, n - 1\right )} - 3 \, b c^{2} e^{2} n\right )} a^{2} - 2 \, {\left (b c^{3} d^{2} {\left (7 \, n - 2\right )} - b^{2} c^{2} d e\right )} a\right )} x x^{3 \, n} + {\left (2 \, b^{4} c d^{2} {\left (2 \, n - 1\right )} + 4 \, a^{3} c^{2} e^{2} - {\left (b^{2} c e^{2} {\left (9 \, n + 1\right )} - 4 \, b c^{2} d e {\left (9 \, n - 4\right )} - 4 \, c^{3} d^{2} {\left (4 \, n - 1\right )}\right )} a^{2} - {\left (b^{2} c^{2} d^{2} {\left (29 \, n - 9\right )} - 4 \, b^{3} c d e\right )} a\right )} x x^{2 \, n} + {\left (b^{5} d^{2} {\left (2 \, n - 1\right )} + 2 \, {\left (4 \, c^{2} d e {\left (5 \, n - 1\right )} - b c e^{2} {\left (5 \, n - 2\right )}\right )} a^{3} + {\left (2 \, b^{2} c d e {\left (4 \, n - 3\right )} - b^{3} e^{2} {\left (2 \, n + 1\right )} - 2 \, b c^{2} d^{2} n\right )} a^{2} - 2 \, {\left (2 \, b^{3} c d^{2} {\left (3 \, n - 1\right )} - b^{4} d e\right )} a\right )} x x^{n} + {\left (a b^{4} d^{2} {\left (3 \, n - 1\right )} - 4 \, a^{4} c e^{2} {\left (2 \, n - 1\right )} + {\left (4 \, c^{2} d^{2} {\left (6 \, n - 1\right )} + 4 \, b c d e {\left (5 \, n - 2\right )} - b^{2} e^{2} {\left (n + 1\right )}\right )} a^{3} - {\left (b^{2} c d^{2} {\left (21 \, n - 5\right )} + 2 \, b^{3} d e {\left (n - 1\right )}\right )} a^{2}\right )} x}{2 \, {\left (a^{4} b^{4} n^{2} - 8 \, a^{5} b^{2} c n^{2} + 16 \, a^{6} c^{2} n^{2} + {\left (a^{2} b^{4} c^{2} n^{2} - 8 \, a^{3} b^{2} c^{3} n^{2} + 16 \, a^{4} c^{4} n^{2}\right )} x^{4 \, n} + 2 \, {\left (a^{2} b^{5} c n^{2} - 8 \, a^{3} b^{3} c^{2} n^{2} + 16 \, a^{4} b c^{3} n^{2}\right )} x^{3 \, n} + {\left (a^{2} b^{6} n^{2} - 6 \, a^{3} b^{4} c n^{2} + 32 \, a^{5} c^{3} n^{2}\right )} x^{2 \, n} + 2 \, {\left (a^{3} b^{5} n^{2} - 8 \, a^{4} b^{3} c n^{2} + 16 \, a^{5} b c^{2} n^{2}\right )} x^{n}\right )}} - \int -\frac {{\left (2 \, n^{2} - 3 \, n + 1\right )} b^{4} d^{2} + 4 \, a^{3} c e^{2} {\left (2 \, n - 1\right )} + {\left (4 \, {\left (8 \, n^{2} - 6 \, n + 1\right )} c^{2} d^{2} - 4 \, b c d e {\left (5 \, n - 2\right )} + b^{2} e^{2} {\left (n + 1\right )}\right )} a^{2} - {\left ({\left (16 \, n^{2} - 21 \, n + 5\right )} b^{2} c d^{2} - 2 \, b^{3} d e {\left (n - 1\right )}\right )} a + {\left ({\left (2 \, n^{2} - 3 \, n + 1\right )} b^{3} c d^{2} + 2 \, {\left (4 \, {\left (3 \, n^{2} - 4 \, n + 1\right )} c^{2} d e - 3 \, {\left (n^{2} - n\right )} b c e^{2}\right )} a^{2} - 2 \, {\left ({\left (7 \, n^{2} - 9 \, n + 2\right )} b c^{2} d^{2} - b^{2} c d e {\left (n - 1\right )}\right )} a\right )} x^{n}}{2 \, {\left (a^{3} b^{4} n^{2} - 8 \, a^{4} b^{2} c n^{2} + 16 \, a^{5} c^{2} n^{2} + {\left (a^{2} b^{4} c n^{2} - 8 \, a^{3} b^{2} c^{2} n^{2} + 16 \, a^{4} c^{3} n^{2}\right )} x^{2 \, n} + {\left (a^{2} b^{5} n^{2} - 8 \, a^{3} b^{3} c n^{2} + 16 \, a^{4} b c^{2} n^{2}\right )} x^{n}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b^3*c^2*d^2*(2*n - 1) + 2*(4*c^3*d*e*(3*n - 1) - 3*b*c^2*e^2*n)*a^2 - 2*(b*c^3*d^2*(7*n - 2) - b^2*c^2*d
*e)*a)*x*x^(3*n) + (2*b^4*c*d^2*(2*n - 1) + 4*a^3*c^2*e^2 - (b^2*c*e^2*(9*n + 1) - 4*b*c^2*d*e*(9*n - 4) - 4*c
^3*d^2*(4*n - 1))*a^2 - (b^2*c^2*d^2*(29*n - 9) - 4*b^3*c*d*e)*a)*x*x^(2*n) + (b^5*d^2*(2*n - 1) + 2*(4*c^2*d*
e*(5*n - 1) - b*c*e^2*(5*n - 2))*a^3 + (2*b^2*c*d*e*(4*n - 3) - b^3*e^2*(2*n + 1) - 2*b*c^2*d^2*n)*a^2 - 2*(2*
b^3*c*d^2*(3*n - 1) - b^4*d*e)*a)*x*x^n + (a*b^4*d^2*(3*n - 1) - 4*a^4*c*e^2*(2*n - 1) + (4*c^2*d^2*(6*n - 1)
+ 4*b*c*d*e*(5*n - 2) - b^2*e^2*(n + 1))*a^3 - (b^2*c*d^2*(21*n - 5) + 2*b^3*d*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^2
+ 4*a^3*c*e^2*(2*n - 1) + (4*(8*n^2 - 6*n + 1)*c^2*d^2 - 4*b*c*d*e*(5*n - 2) + b^2*e^2*(n + 1))*a^2 - ((16*n^
2 - 21*n + 5)*b^2*c*d^2 - 2*b^3*d*e*(n - 1))*a + ((2*n^2 - 3*n + 1)*b^3*c*d^2 + 2*(4*(3*n^2 - 4*n + 1)*c^2*d*e
- 3*(n^2 - n)*b*c*e^2)*a^2 - 2*((7*n^2 - 9*n + 2)*b*c^2*d^2 - b^2*c*d*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 \[ \int \frac {{\left (d+e\,x^n\right )}^2}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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