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

Optimal. Leaf size=1194 \[ \text{result too large to display} \]

[Out]

(d*x*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e*x^2*(b^2 - 2*a*c + b*c*x^n))/(a
*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (f*x^3*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*
x^(2*n))) - (c*d*(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) - (c*
d*(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) - (c*e*(4*a*c*(1 - n
) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-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) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2
+ n)/n, (-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) - (2*c*f
*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])
/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (2*c*f*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hyperg
eometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b
^2 - 4*a*c])*n) - (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b -
Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(2 + n)) + (2*b*c^2*e*(2 - n)*x^(2 + n)
*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(
b + Sqrt[b^2 - 4*a*c])*n*(2 + n)) - (2*b*c^2*f*(3 - n)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*
c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(3 + n)) + (2*b*c^2*f*(3 - n
)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/
2)*(b + Sqrt[b^2 - 4*a*c])*n*(3 + n))

________________________________________________________________________________________

Rubi [A]  time = 2.05313, antiderivative size = 1194, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {1796, 1345, 1422, 245, 1384, 1560, 1383, 364} \[ -\frac{2 b c^2 e (2-n) \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) x^{n+2}}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) n (n+2)}+\frac{2 b c^2 e (2-n) \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) x^{n+2}}{a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n (n+2)}-\frac{2 b c^2 f (3-n) \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) x^{n+3}}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) n (n+3)}+\frac{2 b c^2 f (3-n) \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) x^{n+3}}{a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n (n+3)}-\frac{2 c f \left (2 a c (3-2 n)-b^2 (3-n)\right ) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) x^3}{3 a \left (b^2-4 a c\right ) \left (b^2-\sqrt{b^2-4 a c} b-4 a c\right ) n}-\frac{2 c f \left (2 a c (3-2 n)-b^2 (3-n)\right ) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) x^3}{3 a \left (b^2-4 a c\right ) \left (b^2+\sqrt{b^2-4 a c} b-4 a c\right ) n}+\frac{f \left (b c x^n+b^2-2 a c\right ) x^3}{a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac{c e \left (4 a c (1-n)-b^2 (2-n)\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) x^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{c e \left (4 a c (1-n)-b^2 (2-n)\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) x^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{e \left (b c x^n+b^2-2 a c\right ) x^2}{a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac{c d \left (-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) x}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt{b^2-4 a c} b-4 a c\right ) n}-\frac{c d \left (-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) x}{a \left (b^2-4 a c\right ) \left (b^2+\sqrt{b^2-4 a c} b-4 a c\right ) n}+\frac{d \left (b c x^n+b^2-2 a c\right ) x}{a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e*x^2*(b^2 - 2*a*c + b*c*x^n))/(a
*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (f*x^3*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*
x^(2*n))) - (c*d*(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) - (c*
d*(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) - (c*e*(4*a*c*(1 - n
) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-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) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2
+ n)/n, (-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) - (2*c*f
*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])
/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (2*c*f*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hyperg
eometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b
^2 - 4*a*c])*n) - (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b -
Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(2 + n)) + (2*b*c^2*e*(2 - n)*x^(2 + n)
*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(
b + Sqrt[b^2 - 4*a*c])*n*(2 + n)) - (2*b*c^2*f*(3 - n)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*
c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(3 + n)) + (2*b*c^2*f*(3 - n
)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/
2)*(b + Sqrt[b^2 - 4*a*c])*n*(3 + n))

Rule 1796

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

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 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 1384

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

Rule 1560

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0])

Rule 1383

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

Rule 364

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

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx &=\int \left (\frac{d}{\left (a+b x^n+c x^{2 n}\right )^2}+\frac{e x}{\left (a+b x^n+c x^{2 n}\right )^2}+\frac{f x^2}{\left (a+b x^n+c x^{2 n}\right )^2}\right ) \, dx\\ &=d \int \frac{1}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx+e \int \frac{x}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx+f \int \frac{x^2}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{f x^3 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{d \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 \left (b^2-4 a c\right ) n}-\frac{e \int \frac{x \left (-4 a c (1-n)+b^2 (2-n)+b c (2-n) x^n\right )}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}-\frac{f \int \frac{x^2 \left (-2 a c (3-2 n)+b^2 (3-n)+b c (3-n) x^n\right )}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{f x^3 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{e \int \left (-\frac{b^2 \left (1-\frac{4 a c (-1+n)}{b^2 (-2+n)}\right ) (-2+n) x}{a+b x^n+c x^{2 n}}-\frac{b c (-2+n) x^{1+n}}{a+b x^n+c x^{2 n}}\right ) \, dx}{a \left (b^2-4 a c\right ) n}-\frac{f \int \left (-\frac{b^2 (-3+n) \left (1-\frac{2 a c (-3+2 n)}{b^2 (-3+n)}\right ) x^2}{a+b x^n+c x^{2 n}}-\frac{b c (-3+n) x^{2+n}}{a+b x^n+c x^{2 n}}\right ) \, dx}{a \left (b^2-4 a c\right ) n}+\frac{\left (c d \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 (c d \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{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{f x^3 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c d \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{c d \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 (e \left (4 a c (1-n)-b^2 (2-n)\right )\right ) \int \frac{x}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}+\frac{\left (f \left (2 a c (3-2 n)-b^2 (3-n)\right )\right ) \int \frac{x^2}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}-\frac{(b c e (2-n)) \int \frac{x^{1+n}}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}-\frac{(b c f (3-n)) \int \frac{x^{2+n}}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{f x^3 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c d \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{c d \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 (2 c e \left (4 a c (1-n)-b^2 (2-n)\right )\right ) \int \frac{x}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (2 c e \left (4 a c (1-n)-b^2 (2-n)\right )\right ) \int \frac{x}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}+\frac{\left (2 c f \left (2 a c (3-2 n)-b^2 (3-n)\right )\right ) \int \frac{x^2}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (2 c f \left (2 a c (3-2 n)-b^2 (3-n)\right )\right ) \int \frac{x^2}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (2 b c^2 e (2-n)\right ) \int \frac{x^{1+n}}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}+\frac{\left (2 b c^2 e (2-n)\right ) \int \frac{x^{1+n}}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (2 b c^2 f (3-n)\right ) \int \frac{x^{2+n}}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}+\frac{\left (2 b c^2 f (3-n)\right ) \int \frac{x^{2+n}}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{f x^3 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c d \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{c d \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{c e \left (4 a c (1-n)-b^2 (2-n)\right ) x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{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{c e \left (4 a c (1-n)-b^2 (2-n)\right ) x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{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{2 c f \left (2 a c (3-2 n)-b^2 (3-n)\right ) x^3 \, _2F_1\left (1,\frac{3}{n};\frac{3+n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{3 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) n}-\frac{2 c f \left (2 a c (3-2 n)-b^2 (3-n)\right ) x^3 \, _2F_1\left (1,\frac{3}{n};\frac{3+n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n}-\frac{2 b c^2 e (2-n) x^{2+n} \, _2F_1\left (1,\frac{2+n}{n};2 \left (1+\frac{1}{n}\right );-\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 (2+n)}+\frac{2 b c^2 e (2-n) x^{2+n} \, _2F_1\left (1,\frac{2+n}{n};2 \left (1+\frac{1}{n}\right );-\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 (2+n)}-\frac{2 b c^2 f (3-n) x^{3+n} \, _2F_1\left (1,\frac{3+n}{n};2+\frac{3}{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 (3+n)}+\frac{2 b c^2 f (3-n) x^{3+n} \, _2F_1\left (1,\frac{3+n}{n};2+\frac{3}{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 (3+n)}\\ \end{align*}

Mathematica [B]  time = 6.45798, size = 4238, normalized size = 3.55 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*e*x^2)/(2*a^2*(-b^2 + 4*a*c)) - (2*c*e*x^2)/(a*(-b^2 + 4*a*c)) - (b^2*e*x^2)/(a^2*(-b^2 + 4*a*c)*n) + (2*
c*e*x^2)/(a*(-b^2 + 4*a*c)*n) + (e*(2*b^2 - 4*a*c - b^2*n + 4*a*c*n)*x^2)/(2*a^2*(-b^2 + 4*a*c)*n) + (b^2*f*x^
3)/(3*a^2*(-b^2 + 4*a*c)) - (4*c*f*x^3)/(3*a*(-b^2 + 4*a*c)) - (b^2*f*x^3)/(a^2*(-b^2 + 4*a*c)*n) + (2*c*f*x^3
)/(a*(-b^2 + 4*a*c)*n) + (f*(3*b^2 - 6*a*c - b^2*n + 4*a*c*n)*x^3)/(3*a^2*(-b^2 + 4*a*c)*n) + (x*(d + e*x + f*
x^2)*(-b^2 + 2*a*c - b*c*x^n))/(a*(-b^2 + 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (b*c*d*x^(1 + n)*(x^n)^(1 + n^(-
1) - (1 + n)/n)*(Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sq
rt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), (2*
c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*
(-b^2 + 4*a*c)*(1 + n^(-1))*n^2) + (b*c*d*x^(1 + n)*(x^n)^(1 + n^(-1) - (1 + n)/n)*(Hypergeometric2F1[1, 1 + n
^(-1), 2 + n^(-1), (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 -
4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b +
Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*(1 + n^(-1))*n) + (b^2*d*x*(
Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*
c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4
*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)) - (4*c*d*
x*(Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/
(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2
- 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(-b^2 + 4*a*c) - (b^2*d*x
*(Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(
2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 -
4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*n) + (2*
c*d*x*(Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c
]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[
b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*n) +
(b^2*e*x^2*(Hypergeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]
))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4
*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(2*a*(-b^2 + 4*a*c)) - (2*c*
e*x^2*(Hypergeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2
*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c]
)]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(-b^2 + 4*a*c) - (b^2*e*x^2*(Hype
rgeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b -
Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b
+ Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*n) + (2*c*e*x^2*(Hypergeom
etric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt
[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 2/n, 1 + 2/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqr
t[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*n) + (b^2*f*x^3*(Hypergeometric2F
1[1, 3/n, 1 + 3/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 -
4*a*c])^2/(2*c)) + Hypergeometric2F1[1, 3/n, 1 + 3/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 -
4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(3*a*(-b^2 + 4*a*c)) - (4*c*f*x^3*(Hypergeometric2F1[1,
3/n, 1 + 3/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c
])^2/(2*c)) + Hypergeometric2F1[1, 3/n, 1 + 3/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*
c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(3*(-b^2 + 4*a*c)) - (b^2*f*x^3*(Hypergeometric2F1[1, 3/n, 1
+ 3/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2
*c)) + Hypergeometric2F1[1, 3/n, 1 + 3/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2
*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*n) + (2*c*f*x^3*(Hypergeometric2F1[1, 3/n, 1 + 3/n
, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c))
+ Hypergeometric2F1[1, 3/n, 1 + 3/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) +
(-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*n) + (b*c*e*x^(2 + n)*(Hypergeometric2F1[1, (2 + n)/n, 2
+ 2/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2
*c)) + Hypergeometric2F1[1, (2 + n)/n, 2 + 2/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c
]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*(2 + n)) - (2*b*c*e*x^(2 + n)*(Hypergeometri
c2F1[1, (2 + n)/n, 2 + 2/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sq
rt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, (2 + n)/n, 2 + 2/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(
-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*n*(2 + n)) + (b*c*f*x^(
3 + n)*(Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*(-b - Sqrt[b^2 - 4*a*
c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (2*c*x^n)/(-b + Sqrt
[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*(3
+ n)) - (3*b*c*f*x^(3 + n)*(Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (2*c*x^n)/(-b - Sqrt[b^2 - 4*a*c])]/((b*
(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n,
(2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))
/(a*(-b^2 + 4*a*c)*n*(3 + n))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{f{x}^{2}+ex+d}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{2} f - 2 \, a c f\right )} x^{3} +{\left (b^{2} e - 2 \, a c e\right )} x^{2} +{\left (b c f x^{3} + b c e x^{2} + b c d x\right )} x^{n} +{\left (b^{2} d - 2 \, a c d\right )} x}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}} - \int \frac{2 \, a c d{\left (2 \, n - 1\right )} - b^{2} d{\left (n - 1\right )} +{\left (2 \, a c f{\left (2 \, n - 3\right )} - b^{2} f{\left (n - 3\right )}\right )} x^{2} -{\left (b c f{\left (n - 3\right )} x^{2} + b c e{\left (n - 2\right )} x + b c d{\left (n - 1\right )}\right )} x^{n} +{\left (4 \, a c e{\left (n - 1\right )} - b^{2} e{\left (n - 2\right )}\right )} x}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{f x^{2} + e x + d}{c^{2} x^{4 \, n} + b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 2 \,{\left (b c x^{n} + a c\right )} x^{2 \, n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x^{2} + e x + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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