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

Optimal. Leaf size=543 \[ -\frac{x \left ((1-n) \left (a b e^2-4 a c d e+b c d^2\right )-\frac{b^2 \left (a e^2 (1-3 n)-c d^2 (1-n)\right )+4 a b c d e n+4 a c (1-2 n) \left (c d^2-a e^2\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{x \left (\frac{b^2 \left (a e^2 (1-3 n)-c d^2 (1-n)\right )+4 a b c d e n+4 a c (1-2 n) \left (c d^2-a e^2\right )}{\sqrt{b^2-4 a c}}+(1-n) \left (a b e^2-4 a c d e+b c d^2\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left (x^n \left (a b e^2-4 a c d e+b c d^2\right )-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}-\frac{2 e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

[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))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (2*e^2*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]) - (((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*(1 - n) - (b^2*(a*e^2*(1 - 3*
n) - c*d^2*(1 - n)) + 4*a*c*(c*d^2 - a*e^2)*(1 - 2*n) + 4*a*b*c*d*e*n)/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])])/(a*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) - (2*e^2*x*Hypergeometric2
F1[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]) - (((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*(1 - n) + (b^2*(a*e^2*(1
- 3*n) - c*d^2*(1 - n)) + 4*a*c*(c*d^2 - a*e^2)*(1 - 2*n) + 4*a*b*c*d*e*n)/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])])/(a*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)

_______________________________________________________________________________________

Rubi [A]  time = 5.10893, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{x \left ((1-n) \left (a b e^2-4 a c d e+b c d^2\right )-\frac{b^2 \left (a e^2 (1-3 n)-c d^2 (1-n)\right )+4 a b c d e n+4 a c (1-2 n) \left (c d^2-a e^2\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{x \left (\frac{b^2 \left (a e^2 (1-3 n)-c d^2 (1-n)\right )+4 a b c d e n+4 a c (1-2 n) \left (c d^2-a e^2\right )}{\sqrt{b^2-4 a c}}+(1-n) \left (a b e^2-4 a c d e+b c d^2\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left (x^n \left (a b e^2-4 a c d e+b c d^2\right )-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}-\frac{2 e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^2,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))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (2*e^2*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]) - (((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*(1 - n) - (b^2*(a*e^2*(1 - 3*
n) - c*d^2*(1 - n)) + 4*a*c*(c*d^2 - a*e^2)*(1 - 2*n) + 4*a*b*c*d*e*n)/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])])/(a*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) - (2*e^2*x*Hypergeometric2
F1[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]) - (((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*(1 - n) + (b^2*(a*e^2*(1
- 3*n) - c*d^2*(1 - n)) + 4*a*c*(c*d^2 - a*e^2)*(1 - 2*n) + 4*a*b*c*d*e*n)/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])])/(a*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e*x**n)**2/(a+b*x**n+c*x**(2*n))**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 6.41027, size = 4177, normalized size = 7.69 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

((-(b^2*d^2) + 2*a*c*d^2 + 2*a*b*d*e - 2*a^2*e^2 + b^2*d^2*n - 4*a*c*d^2*n)*x)/(
a^2*(-b^2 + 4*a*c)*n) + ((b^2*d^2 - 2*a*c*d^2 - 2*a*b*d*e + 2*a^2*e^2 - b^2*d^2*
n + 4*a*c*d^2*n)*x)/(a^2*(-b^2 + 4*a*c)*n) - (x*(b^2*d^2 - 2*a*c*d^2 - 2*a*b*d*e
+ 2*a^2*e^2 + b*c*d^2*x^n - 4*a*c*d*e*x^n + a*b*e^2*x^n))/(a*(-b^2 + 4*a*c)*n*(
a + b*x^n + c*x^(2*n))) - (b*c*d^2*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hyper
geometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b
- Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2 - 4
*a*c])/(2*c) + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n,
-(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b
^2 - 4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/(a*(-b^2 + 4
*a*c)) + (4*c*d*e*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeometric2F1[-n^(
-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*
c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^
n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 -
4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/
(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/(-b^2 + 4*a*c) - (b*e^2*x^(1
+ n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/
n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqr
t[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))) + Hypergeo
metric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + S
qrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*
c])/(2*c) + x^n))^n^(-1))))/(-b^2 + 4*a*c) + (b*c*d^2*x^(1 + n)*(x^n)^(n^(-1) -
(1 + n)/n)*(-(Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 -
4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(
-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n
^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*
c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(
-1))))/(a*(-b^2 + 4*a*c)*n) - (4*c*d*e*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(H
ypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-
(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2
- 4*a*c])/(2*c) + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)
/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sq
rt[b^2 - 4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/((-b^2 +
4*a*c)*n) + (b*e^2*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeometric2F1[-n
^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*
a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) +
x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2
- 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^
n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/((-b^2 + 4*a*c)*n) + (b^2*d
^2*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a
*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*
c])/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 -
4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b +
Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + S
qrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((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^2*x*((1 - Hypergeo
metric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - S
qrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n
^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) +
(1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/
(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(
2*c) + x^n))^n^(-1))/((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^2*x*((1 - Hypergeometric2F1[-n^(-1), -n^(
-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c)
+ x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b
^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1
[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 -
4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((
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^2*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n,
-(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-
(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*
c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1)
, (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) +
x^n))]/(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((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) + (2*b
*d*e*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4
*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*
a*c])/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2
- 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b
+ Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b +
Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((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) - (2*a*e^2*x*((1 - Hyperg
eometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b -
Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))
^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c))
+ (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c]
)/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt[b^2 - 4*a*c])
/(2*c) + x^n))^n^(-1))/((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)

_______________________________________________________________________________________

Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{2}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

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))^2,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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))**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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