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

Optimal. Leaf size=263 \[ -\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.517118, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.3245, size = 231, normalized size = 0.88 \[ - \frac{2 c d x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}} - \frac{2 c d x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}} - \frac{c e x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}} - \frac{c e x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 4.23679, size = 525, normalized size = 2. \[ c x \left (-2 d \left (\frac{1-\left (\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-2^{-1/n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )}\right )-e x \left (\frac{1-\left (\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right )^{-2/n} \, _2F_1\left (-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-4^{-1/n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-2/n} \, _2F_1\left (-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

c*x*(-(e*x*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4*a*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))^(2/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-2/n, -2
/n, (-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(4^n^
(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)))/(Sqrt[b^2 - 4*a*c]*(b +
Sqrt[b^2 - 4*a*c])))) - 2*d*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/
n, (b - Sqrt[b^2 - 4*a*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^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - H
ypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt
[b^2 - 4*a*c] + 2*c*x^n)]/(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^
n^(-1)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))

_______________________________________________________________________________________

Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{ex+d}{a+b{x}^{n}+c{x}^{2\,n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x + d}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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