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

Optimal. Leaf size=362 \[ -\frac{c x \left (-b \left (d (1-n) \sqrt{b^2-4 a c}-2 a e n\right )+2 a \left (e (1-n) \sqrt{b^2-4 a c}+2 c d (1-2 n)\right )+b^2 (-(d-d n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (b \left (d (1-n) \sqrt{b^2-4 a c}+2 a e n\right )+2 a \left (c d (2-4 n)-e (1-n) \sqrt{b^2-4 a c}\right )+b^2 (-d) (1-n)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]

[Out]

(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^
n + c*x^(2*n))) - (c*(2*a*(2*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^2*
(d - d*n) - b*(Sqrt[b^2 - 4*a*c]*d*(1 - n) - 2*a*e*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*(2*a*(c*d*(2 - 4*n) - Sqrt[b^2 - 4*a*c]*e*(1
- n)) - b^2*d*(1 - n) + b*(Sqrt[b^2 - 4*a*c]*d*(1 - n) + 2*a*e*n))*x*Hypergeome
tric2F1[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 = 1.38465, antiderivative size = 328, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{c x \left (-(1-n) \sqrt{b^2-4 a c} (b d-2 a e)+2 a b e n+2 a c d (2-4 n)+b^2 (-d) (1-n)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left ((1-n) \sqrt{b^2-4 a c} (b d-2 a e)+2 a b e n+4 a c d (1-2 n)+b^2 (-d) (1-n)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [B]  time = 6.26231, size = 3152, normalized size = 8.71 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

((-(b^2*d) + 2*a*c*d + a*b*e + b^2*d*n - 4*a*c*d*n)*x)/(a^2*(-b^2 + 4*a*c)*n) +
((b^2*d - 2*a*c*d - a*b*e - b^2*d*n + 4*a*c*d*n)*x)/(a^2*(-b^2 + 4*a*c)*n) + (x*
(-(b^2*d) + 2*a*c*d + a*b*e - b*c*d*x^n + 2*a*c*e*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)^(n^(-1) - (1 + n)/n)*(-(Hypergeome
tric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqr
t[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)
) + (2*c*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*c*d*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))) + Hypergeometric2
F1[-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) - (2*c*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)*n) + (b^2*d*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 + S
qrt[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*((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 - Sq
rt[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*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))))/(a*(-b^2 + 4*a*c)*n) + (2*c*d*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 - 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))))/((
-b^2 + 4*a*c)*n) + (b*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)

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{d+e{x}^{n}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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