3.6 \(\int \frac{1}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx\)

Optimal. Leaf size=283 \[ -\frac{c x \left (-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]

[Out]

(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))) - (c*(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*(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)

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Rubi [A]  time = 0.737543, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{c x \left (-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n + c*x^(2*n))^(-2),x]

[Out]

(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))) - (c*(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*(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)

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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(1/(a+b*x**n+c*x**(2*n))**2,x)

[Out]

Timed out

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Mathematica [B]  time = 6.1722, size = 2170, normalized size = 7.67 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

((-b^2 + 2*a*c + b^2*n - 4*a*c*n)*x)/(a^2*(-b^2 + 4*a*c)*n) + ((b^2 - 2*a*c - b^
2*n + 4*a*c*n)*x)/(a^2*(-b^2 + 4*a*c)*n) + (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))) - (b*c*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 - Sqr
t[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)) + (b*c*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) + (
b^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)) - (4*c*x*((1 - Hypergeomet
ric2F1[-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) - (b^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*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)

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n+c*x^(2*n))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*c*x*x^n + (b^2 - 2*a*c)*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*c*(n - 1)*x^n - 2*a*c*(2
*n - 1) + b^2*(n - 1))/(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. \[{\rm integral}\left (\frac{1}{c^{2} x^{4 \, n} + 2 \, a b x^{n} + a^{2} +{\left (2 \, b c x^{n} + b^{2} + 2 \, a c\right )} x^{2 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(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)

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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(1/(a+b*x**n+c*x**(2*n))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^(2*n) + b*x^n + a)^(-2), x)