3.79 \(\int \frac {1}{(d+e x^n)^2 (a+b x^n+c x^{2 n})^2} \, dx\)

Optimal. Leaf size=1129 \[ \frac {2 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right ) e^4}{d \left (c d^2-b e d+a e^2\right )^3}+\frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right ) e^4}{d^2 \left (c d^2-b e d+a e^2\right )^2}-\frac {2 c \left (3 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt {b^2-4 a c} d+a e\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}{\left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3}-\frac {2 c \left (3 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt {b^2-4 a c} d+a e\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}{\left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3}+\frac {c \left (e^2 (1-n) b^4-e \left (2 c d-\sqrt {b^2-4 a c} e\right ) (1-n) b^3-c \left (e \left (a e (5-7 n)+2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right ) b^2+c \left (c d \left (4 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-3 a \sqrt {b^2-4 a c} e^2 (1-n)\right ) b+4 a c^2 \left (e \left (a e (1-2 n)+\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 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 )}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n}+\frac {c \left (e^2 (1-n) b^4-e \left (2 c d+\sqrt {b^2-4 a c} e\right ) (1-n) b^3-c \left (e \left (a e (5-7 n)-2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right ) b^2+c \left (3 a \sqrt {b^2-4 a c} (1-n) e^2+c d \left (4 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right )\right ) b+4 a c^2 \left (e \left (a e (1-2 n)-\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 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 )}{a \left (b^2-4 a c\right ) \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n}-\frac {x \left (c \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right ) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n \left (b x^n+c x^{2 n}+a\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 3.34, antiderivative size = 1129, 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, 245, 1430, 1422} \[ \frac {2 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right ) e^4}{d \left (c d^2-b e d+a e^2\right )^3}+\frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right ) e^4}{d^2 \left (c d^2-b e d+a e^2\right )^2}-\frac {2 c \left (3 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt {b^2-4 a c} d+a e\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}{\left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3}-\frac {2 c \left (3 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt {b^2-4 a c} d+a e\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}{\left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3}+\frac {c \left (e^2 (1-n) b^4-e \left (2 c d-\sqrt {b^2-4 a c} e\right ) (1-n) b^3-c \left (e \left (a e (5-7 n)+2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right ) b^2+c \left (c d \left (4 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-3 a \sqrt {b^2-4 a c} e^2 (1-n)\right ) b+4 a c^2 \left (e \left (a e (1-2 n)+\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 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 )}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n}+\frac {c \left (e^2 (1-n) b^4-e \left (2 c d+\sqrt {b^2-4 a c} e\right ) (1-n) b^3-c \left (e \left (a e (5-7 n)-2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right ) b^2+c \left (3 a \sqrt {b^2-4 a c} (1-n) e^2+c d \left (4 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right )\right ) b+4 a c^2 \left (e \left (a e (1-2 n)-\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 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 )}{a \left (b^2-4 a c\right ) \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n}-\frac {x \left (c \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right ) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n \left (b x^n+c x^{2 n}+a\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^2),x]

[Out]

-((x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d
*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*n*(a + b*
x^n + c*x^(2*n)))) - (2*c*e^2*(3*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d
+ a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt
[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c*(4*a*c^2*(e*(a*e*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d
^2*(1 - 2*n)) - b^2*c*(e*(a*e*(5 - 7*n) + 2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(2
- 3*n) + Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + b^4*e^2*(1 - n) - b^3*e*(2*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*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) - (2*c*e^2*(3*c^2*d^2 + b*(
b - Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d - 2*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(
-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c
*(4*a*c^2*(e*(a*e*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(5 - 7*n) - 2*Sq
rt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(2 - 3*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) + 3*a*S
qrt[b^2 - 4*a*c]*e^2*(1 - n)) + b^4*e^2*(1 - n) - b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometri
c2F1[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])*(c*d^2 - b*d*e + a*e^2)^2*n) + (2*e^4*(2*c*d - b*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^
n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c
*d^2 - b*d*e + a*e^2)^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 {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^2} \, dx &=\int \left (\frac {e^4}{\left (c d^2-b d e+a e^2\right )^2 \left (d+e x^n\right )^2}-\frac {2 e^4 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^3 \left (d+e x^n\right )}+\frac {c^2 d^2-2 b c d e+b^2 e^2-a c e^2-\left (2 c^2 d e-b c e^2\right ) x^n}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 \left (3 c^2 d^2-5 b c d e+2 b^2 e^2-a c e^2+\left (-4 c^2 d e+2 b c e^2\right ) x^n\right )}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x^n+c x^{2 n}\right )}\right ) \, dx\\ &=\frac {e^2 \int \frac {3 c^2 d^2-5 b c d e+2 b^2 e^2-a c e^2+\left (-4 c^2 d e+2 b c e^2\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\left (2 e^4 (2 c d-b e)\right ) \int \frac {1}{d+e x^n} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {c^2 d^2-2 b c d e+b^2 e^2-a c e^2-\left (2 c^2 d e-b c e^2\right ) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac {e^4 \int \frac {1}{\left (d+e x^n\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {2 e^4 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (c e^2 \left (3 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt {b^2-4 a c} d+a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (c e^2 \left (3 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt {b^2-4 a c} d+a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}-\frac {\int \frac {-b^2 c \left (a e^2 (4-5 n)-c d^2 (1-n)\right )+2 a b c^2 d e (3-4 n)-2 a c^2 \left (c d^2-a e^2\right ) (1-2 n)-2 b^3 c d e (1-n)+b^4 e^2 (1-n)-c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n}\\ &=-\frac {x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {2 c e^2 \left (3 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt {b^2-4 a c} d+a e\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 )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {2 c e^2 \left (3 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt {b^2-4 a c} d+a e\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 )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {2 e^4 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (c \left (4 a c^2 \left (e \left (a e (1-2 n)+\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (5-7 n)+2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right )+b c \left (c d \left (4 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-3 a \sqrt {b^2-4 a c} e^2 (1-n)\right )+b^4 e^2 (1-n)-b^3 e \left (2 c d-\sqrt {b^2-4 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 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 n}+\frac {\left (c \left (4 a c^2 \left (e \left (a e (1-2 n)-\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (5-7 n)-2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right )+b c \left (c d \left (4 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right )+3 a \sqrt {b^2-4 a c} e^2 (1-n)\right )+b^4 e^2 (1-n)-b^3 e \left (2 c d+\sqrt {b^2-4 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 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 n}\\ &=-\frac {x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {2 c e^2 \left (3 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt {b^2-4 a c} d+a e\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 )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c \left (4 a c^2 \left (e \left (a e (1-2 n)+\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (5-7 n)+2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right )+b c \left (c d \left (4 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-3 a \sqrt {b^2-4 a c} e^2 (1-n)\right )+b^4 e^2 (1-n)-b^3 e \left (2 c d-\sqrt {b^2-4 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 \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\frac {2 c e^2 \left (3 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt {b^2-4 a c} d+a e\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 )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {c \left (4 a c^2 \left (e \left (a e (1-2 n)-\sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (5-7 n)-2 \sqrt {b^2-4 a c} d (1-n)\right )-c d^2 (1-n)\right )+b c \left (c d \left (4 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right )+3 a \sqrt {b^2-4 a c} e^2 (1-n)\right )+b^4 e^2 (1-n)-b^3 e \left (2 c d+\sqrt {b^2-4 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 \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {2 e^4 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^2}\\ \end {align*}

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Mathematica [B]  time = 8.02, size = 16855, normalized size = 14.93 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^2),x]

[Out]

Result too large to show

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fricas [F]  time = 6.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} e^{2} x^{4 \, n} + a^{2} d^{2} + {\left (c^{2} e^{2} x^{2 \, n} + 2 \, c^{2} d e x^{n} + c^{2} d^{2}\right )} x^{4 \, n} + 2 \, {\left (b^{2} d e + a b e^{2}\right )} x^{3 \, n} + 2 \, {\left (b c e^{2} x^{3 \, n} + a c d^{2} + {\left (2 \, b c d e + a c e^{2}\right )} x^{2 \, n} + {\left (b c d^{2} + 2 \, a c d e\right )} x^{n}\right )} x^{2 \, n} + {\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} x^{2 \, n} + 2 \, {\left (a b d^{2} + a^{2} d e\right )} x^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*e^2*x^(4*n) + a^2*d^2 + (c^2*e^2*x^(2*n) + 2*c^2*d*e*x^n + c^2*d^2)*x^(4*n) + 2*(b^2*d*e + a*b
*e^2)*x^(3*n) + 2*(b*c*e^2*x^(3*n) + a*c*d^2 + (2*b*c*d*e + a*c*e^2)*x^(2*n) + (b*c*d^2 + 2*a*c*d*e)*x^n)*x^(2
*n) + (b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^(2*n) + 2*(a*b*d^2 + a^2*d*e)*x^n), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2} {\left (e x^{n} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^2,x, algorithm="giac")

[Out]

integrate(1/((c*x^(2*n) + b*x^n + a)^2*(e*x^n + d)^2), x)

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{n}+d \right )^{2} \left (b \,x^{n}+c \,x^{2 n}+a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^n+d)^2/(b*x^n+c*x^(2*n)+a)^2,x)

[Out]

int(1/(e*x^n+d)^2/(b*x^n+c*x^(2*n)+a)^2,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(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^2,x, algorithm="maxima")

[Out]

(c*d^2*e^4*(5*n - 1) - b*d*e^5*(3*n - 1) + a*e^6*(n - 1))*integrate(1/(c^3*d^8*n - 3*b*c^2*d^7*e*n + 3*b^2*c*d
^6*e^2*n - b^3*d^5*e^3*n + a^3*d^2*e^6*n + 3*(c*d^4*e^4*n - b*d^3*e^5*n)*a^2 + 3*(c^2*d^6*e^2*n - 2*b*c*d^5*e^
3*n + b^2*d^4*e^4*n)*a + (c^3*d^7*e*n - 3*b*c^2*d^6*e^2*n + 3*b^2*c*d^5*e^3*n - b^3*d^4*e^4*n + a^3*d*e^7*n +
3*(c*d^3*e^5*n - b*d^2*e^6*n)*a^2 + 3*(c^2*d^5*e^3*n - 2*b*c*d^4*e^4*n + b^2*d^3*e^5*n)*a)*x^n), x) - ((b*c^3*
d^3*e - 2*b^2*c^2*d^2*e^2 + b^3*c*d*e^3 - 4*a^2*c^2*e^4 + (4*c^3*d^2*e^2 - 3*b*c^2*d*e^3 + b^2*c*e^4)*a)*x*x^(
2*n) + (b*c^3*d^4 - b^2*c^2*d^3*e - b^3*c*d^2*e^2 + b^4*d*e^3 + 2*(c^2*d*e^3 - 2*b*c*e^4)*a^2 + (2*c^3*d^3*e +
3*b*c^2*d^2*e^2 - 4*b^2*c*d*e^3 + b^3*e^4)*a)*x*x^n + (b^2*c^2*d^4 - 2*b^3*c*d^3*e + b^4*d^2*e^2 - 4*a^3*c*e^
4 + (2*c^2*d^2*e^2 + b^2*e^4)*a^2 - 2*(c^3*d^4 - 3*b*c^2*d^3*e + 2*b^2*c*d^2*e^2)*a)*x)/(4*a^5*c*d^2*e^4*n + (
8*c^2*d^4*e^2*n - 8*b*c*d^3*e^3*n - b^2*d^2*e^4*n)*a^4 + 2*(2*c^3*d^6*n - 4*b*c^2*d^5*e*n + b^2*c*d^4*e^2*n +
b^3*d^3*e^3*n)*a^3 - (b^2*c^2*d^6*n - 2*b^3*c*d^5*e*n + b^4*d^4*e^2*n)*a^2 + (4*a^4*c^2*d*e^5*n + (8*c^3*d^3*e
^3*n - 8*b*c^2*d^2*e^4*n - b^2*c*d*e^5*n)*a^3 + 2*(2*c^4*d^5*e*n - 4*b*c^3*d^4*e^2*n + b^2*c^2*d^3*e^3*n + b^3
*c*d^2*e^4*n)*a^2 - (b^2*c^3*d^5*e*n - 2*b^3*c^2*d^4*e^2*n + b^4*c*d^3*e^3*n)*a)*x^(3*n) + (4*(c^2*d^2*e^4*n +
b*c*d*e^5*n)*a^4 + (8*c^3*d^4*e^2*n - 9*b^2*c*d^2*e^4*n - b^3*d*e^5*n)*a^3 + 2*(2*c^4*d^6*n - 2*b*c^3*d^5*e*n
- 3*b^2*c^2*d^4*e^2*n + 2*b^3*c*d^3*e^3*n + b^4*d^2*e^4*n)*a^2 - (b^2*c^3*d^6*n - b^3*c^2*d^5*e*n - b^4*c*d^4
*e^2*n + b^5*d^3*e^3*n)*a)*x^(2*n) + (4*a^5*c*d*e^5*n + (8*c^2*d^3*e^3*n - 4*b*c*d^2*e^4*n - b^2*d*e^5*n)*a^4
+ (4*c^3*d^5*e*n - 6*b^2*c*d^3*e^3*n + b^3*d^2*e^4*n)*a^3 + (4*b*c^3*d^6*n - 9*b^2*c^2*d^5*e*n + 4*b^3*c*d^4*e
^2*n + b^4*d^3*e^3*n)*a^2 - (b^3*c^2*d^6*n - 2*b^4*c*d^5*e*n + b^5*d^4*e^2*n)*a)*x^n) + integrate(-(2*a^3*c^2*
e^4*(4*n - 1) + b^2*c^3*d^4*(n - 1) - 3*b^3*c^2*d^3*e*(n - 1) + 3*b^4*c*d^2*e^2*(n - 1) - b^5*d*e^3*(n - 1) -
2*(b^2*c*e^4*(7*n - 2) - 2*b*c^2*d*e^3*(6*n - 1) + 6*c^3*d^2*e^2*n)*a^2 + (b^4*e^4*(3*n - 1) + 4*b*c^3*d^3*e*(
3*n - 2) - 2*c^4*d^4*(2*n - 1) - 2*b^3*c*d*e^3*(n + 1) - 9*b^2*c^2*d^2*e^2*(n - 1))*a + (b*c^4*d^4*(n - 1) - 3
*b^2*c^3*d^3*e*(n - 1) + 3*b^3*c^2*d^2*e^2*(n - 1) - b^4*c*d*e^3*(n - 1) - (b*c^2*e^4*(11*n - 3) - 4*c^3*d*e^3
*(5*n - 1))*a^2 - (b^2*c^2*d*e^3*(3*n + 1) - b^3*c*e^4*(3*n - 1) - 4*c^4*d^3*e*(n - 1) + 6*b*c^3*d^2*e^2*(n -
1))*a)*x^n)/(4*a^6*c*e^6*n + (12*c^2*d^2*e^4*n - 12*b*c*d*e^5*n - b^2*e^6*n)*a^5 + 3*(4*c^3*d^4*e^2*n - 8*b*c^
2*d^3*e^3*n + 3*b^2*c*d^2*e^4*n + b^3*d*e^5*n)*a^4 + (4*c^4*d^6*n - 12*b*c^3*d^5*e*n + 9*b^2*c^2*d^4*e^2*n + 2
*b^3*c*d^3*e^3*n - 3*b^4*d^2*e^4*n)*a^3 - (b^2*c^3*d^6*n - 3*b^3*c^2*d^5*e*n + 3*b^4*c*d^4*e^2*n - b^5*d^3*e^3
*n)*a^2 + (4*a^5*c^2*e^6*n + (12*c^3*d^2*e^4*n - 12*b*c^2*d*e^5*n - b^2*c*e^6*n)*a^4 + 3*(4*c^4*d^4*e^2*n - 8*
b*c^3*d^3*e^3*n + 3*b^2*c^2*d^2*e^4*n + b^3*c*d*e^5*n)*a^3 + (4*c^5*d^6*n - 12*b*c^4*d^5*e*n + 9*b^2*c^3*d^4*e
^2*n + 2*b^3*c^2*d^3*e^3*n - 3*b^4*c*d^2*e^4*n)*a^2 - (b^2*c^4*d^6*n - 3*b^3*c^3*d^5*e*n + 3*b^4*c^2*d^4*e^2*n
- b^5*c*d^3*e^3*n)*a)*x^(2*n) + (4*a^5*b*c*e^6*n + (12*b*c^2*d^2*e^4*n - 12*b^2*c*d*e^5*n - b^3*e^6*n)*a^4 +
3*(4*b*c^3*d^4*e^2*n - 8*b^2*c^2*d^3*e^3*n + 3*b^3*c*d^2*e^4*n + b^4*d*e^5*n)*a^3 + (4*b*c^4*d^6*n - 12*b^2*c^
3*d^5*e*n + 9*b^3*c^2*d^4*e^2*n + 2*b^4*c*d^3*e^3*n - 3*b^5*d^2*e^4*n)*a^2 - (b^3*c^3*d^6*n - 3*b^4*c^2*d^5*e*
n + 3*b^5*c*d^4*e^2*n - b^6*d^3*e^3*n)*a)*x^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^2),x)

[Out]

int(1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^2), 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(1/(d+e*x**n)**2/(a+b*x**n+c*x**(2*n))**2,x)

[Out]

Timed out

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