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

Optimal. Leaf size=203 \[ -\frac{(1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1)}+\frac{x \left (-a e^2+c d^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c} \]

[Out]

(x*(c*d^2 - a*e^2 + 2*c*d*e*x^n))/(2*a*c*n*(a + c*x^(2*n))) + (e^2*x*Hypergeomet
ric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) - ((c*d^2 - a*e^2)*(
1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a
^2*c*n) - (d*e*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1)
)/2, -((c*x^(2*n))/a)])/(a^2*n*(1 + n))

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Rubi [A]  time = 0.345091, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{(1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1)}+\frac{x \left (-a e^2+c d^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*d^2 - a*e^2 + 2*c*d*e*x^n))/(2*a*c*n*(a + c*x^(2*n))) + (e^2*x*Hypergeomet
ric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) - ((c*d^2 - a*e^2)*(
1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a
^2*c*n) - (d*e*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1)
)/2, -((c*x^(2*n))/a)])/(a^2*n*(1 + n))

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Rubi in Sympy [A]  time = 18.8567, size = 109, normalized size = 0.54 \[ \frac{d^{2} x{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2}} + \frac{2 d e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2} \left (n + 1\right )} + \frac{e^{2} x^{2 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2} \left (2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*x*hyper((2, 1/(2*n)), ((n + 1/2)/n,), -c*x**(2*n)/a)/a**2 + 2*d*e*x**(n + 1
)*hyper((2, (n + 1)/(2*n)), ((3*n + 1)/(2*n),), -c*x**(2*n)/a)/(a**2*(n + 1)) +
e**2*x**(2*n + 1)*hyper((2, (n + 1/2)/n), (2 + 1/(2*n),), -c*x**(2*n)/a)/(a**2*(
2*n + 1))

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Mathematica [A]  time = 0.464009, size = 142, normalized size = 0.7 \[ \frac{x \left (\frac{\left (a e^2+c d^2 (2 n-1)\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{c}+\frac{a \left (c d \left (d+2 e x^n\right )-a e^2\right )}{c \left (a+c x^{2 n}\right )}+\frac{2 d e (n-1) x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}\right )}{2 a^2 n} \]

Antiderivative was successfully verified.

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

[Out]

(x*((a*(-(a*e^2) + c*d*(d + 2*e*x^n)))/(c*(a + c*x^(2*n))) + ((a*e^2 + c*d^2*(-1
+ 2*n))*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/c + (2*d*
e*(-1 + n)*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))
/a)])/(1 + n)))/(2*a^2*n)

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{2}}{ \left ( a+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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