3.54 \(\int \frac{\left (d+e x^n\right )^2}{\left (a+c x^{2 n}\right )^3} \, dx\)

Optimal. Leaf size=272 \[ \frac{(1-4 n) (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 c n^2}+\frac{d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{4 a^3 n^2 (n+1)}-\frac{x \left ((1-4 n) \left (c d^2-a e^2\right )+2 c d e (1-3 n) x^n\right )}{8 a^2 c n^2 \left (a+c x^{2 n}\right )}+\frac{e^2 x \, _2F_1\left (2,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 c}+\frac{x \left (-a e^2+c d^2+2 c d e x^n\right )}{4 a c n \left (a+c x^{2 n}\right )^2} \]

[Out]

(x*(c*d^2 - a*e^2 + 2*c*d*e*x^n))/(4*a*c*n*(a + c*x^(2*n))^2) - (x*((c*d^2 - a*e
^2)*(1 - 4*n) + 2*c*d*e*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + c*x^(2*n))) + ((c*d^2
- a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((
c*x^(2*n))/a)])/(8*a^3*c*n^2) + (d*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2
F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(4*a^3*n^2*(1 + n)) + (e
^2*x*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*c)

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Rubi [A]  time = 0.516454, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{(1-4 n) (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 c n^2}+\frac{d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{4 a^3 n^2 (n+1)}-\frac{x \left ((1-4 n) \left (c d^2-a e^2\right )+2 c d e (1-3 n) x^n\right )}{8 a^2 c n^2 \left (a+c x^{2 n}\right )}+\frac{e^2 x \, _2F_1\left (2,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 c}+\frac{x \left (-a e^2+c d^2+2 c d e x^n\right )}{4 a c n \left (a+c x^{2 n}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^n)^2/(a + c*x^(2*n))^3,x]

[Out]

(x*(c*d^2 - a*e^2 + 2*c*d*e*x^n))/(4*a*c*n*(a + c*x^(2*n))^2) - (x*((c*d^2 - a*e
^2)*(1 - 4*n) + 2*c*d*e*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + c*x^(2*n))) + ((c*d^2
- a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((
c*x^(2*n))/a)])/(8*a^3*c*n^2) + (d*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2
F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(4*a^3*n^2*(1 + n)) + (e
^2*x*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*c)

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Rubi in Sympy [A]  time = 19.3914, size = 109, normalized size = 0.4 \[ \frac{d^{2} x{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3}} + \frac{2 d e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3} \left (n + 1\right )} + \frac{e^{2} x^{2 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3} \left (2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e*x**n)**2/(a+c*x**(2*n))**3,x)

[Out]

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

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Mathematica [A]  time = 0.825264, size = 212, normalized size = 0.78 \[ \frac{x \left (\frac{a \left (a^2 e^2 (1-2 n)+a c \left (d^2 (6 n-1)+2 d e (5 n-1) x^n+e^2 x^{2 n}\right )+c^2 d x^{2 n} \left (d (4 n-1)+2 e (3 n-1) x^n\right )\right )}{c \left (a+c x^{2 n}\right )^2}+\frac{(2 n-1) \left (a e^2+c d^2 (4 n-1)\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{c}+\frac{2 d e \left (3 n^2-4 n+1\right ) x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}\right )}{8 a^3 n^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^n)^2/(a + c*x^(2*n))^3,x]

[Out]

(x*((a*(a^2*e^2*(1 - 2*n) + c^2*d*x^(2*n)*(d*(-1 + 4*n) + 2*e*(-1 + 3*n)*x^n) +
a*c*(d^2*(-1 + 6*n) + 2*d*e*(-1 + 5*n)*x^n + e^2*x^(2*n))))/(c*(a + c*x^(2*n))^2
) + ((-1 + 2*n)*(a*e^2 + c*d^2*(-1 + 4*n))*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(
2*n), -((c*x^(2*n))/a)])/c + (2*d*e*(1 - 4*n + 3*n^2)*x^n*Hypergeometric2F1[1, (
1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n)))/(8*a^3*n^2)

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{2}}{ \left ( a+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e*x^n)^2/(a+c*x^(2*n))^3,x)

[Out]

int((d+e*x^n)^2/(a+c*x^(2*n))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 \, c^{2} d e{\left (3 \, n - 1\right )} x x^{3 \, n} + 2 \, a c d e{\left (5 \, n - 1\right )} x x^{n} +{\left (c^{2} d^{2}{\left (4 \, n - 1\right )} + a c e^{2}\right )} x x^{2 \, n} +{\left (a c d^{2}{\left (6 \, n - 1\right )} - a^{2} e^{2}{\left (2 \, n - 1\right )}\right )} x}{8 \,{\left (a^{2} c^{3} n^{2} x^{4 \, n} + 2 \, a^{3} c^{2} n^{2} x^{2 \, n} + a^{4} c n^{2}\right )}} + \int \frac{2 \,{\left (3 \, n^{2} - 4 \, n + 1\right )} c d e x^{n} +{\left (8 \, n^{2} - 6 \, n + 1\right )} c d^{2} + a e^{2}{\left (2 \, n - 1\right )}}{8 \,{\left (a^{2} c^{2} n^{2} x^{2 \, n} + a^{3} c n^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2/(c*x^(2*n) + a)^3,x, algorithm="maxima")

[Out]

1/8*(2*c^2*d*e*(3*n - 1)*x*x^(3*n) + 2*a*c*d*e*(5*n - 1)*x*x^n + (c^2*d^2*(4*n -
1) + a*c*e^2)*x*x^(2*n) + (a*c*d^2*(6*n - 1) - a^2*e^2*(2*n - 1))*x)/(a^2*c^3*n
^2*x^(4*n) + 2*a^3*c^2*n^2*x^(2*n) + a^4*c*n^2) + integrate(1/8*(2*(3*n^2 - 4*n
+ 1)*c*d*e*x^n + (8*n^2 - 6*n + 1)*c*d^2 + a*e^2*(2*n - 1))/(a^2*c^2*n^2*x^(2*n)
+ a^3*c*n^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}{c^{3} x^{6 \, n} + 3 \, a c^{2} x^{4 \, n} + 3 \, a^{2} c x^{2 \, n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2/(c*x^(2*n) + a)^3,x, algorithm="fricas")

[Out]

integral((e^2*x^(2*n) + 2*d*e*x^n + d^2)/(c^3*x^(6*n) + 3*a*c^2*x^(4*n) + 3*a^2*
c*x^(2*n) + a^3), 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))**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{n} + d\right )}^{2}}{{\left (c x^{2 \, n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2/(c*x^(2*n) + a)^3,x, algorithm="giac")

[Out]

integrate((e*x^n + d)^2/(c*x^(2*n) + a)^3, x)