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

Optimal. Leaf size=288 \[ -\frac{e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac{x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{3 d e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.522391, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac{x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{3 d e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.384, size = 158, normalized size = 0.55 \[ \frac{d^{3} x{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2}} + \frac{3 d^{2} e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2} \left (n + 1\right )} + \frac{3 d e^{2} x^{2 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2} \left (2 n + 1\right )} + \frac{e^{3} x^{3 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{3 n + 1}{2 n} \\ \frac{5 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{2} \left (3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*x*hyper((2, 1/(2*n)), ((n + 1/2)/n,), -c*x**(2*n)/a)/a**2 + 3*d**2*e*x**(n
+ 1)*hyper((2, (n + 1)/(2*n)), ((3*n + 1)/(2*n),), -c*x**(2*n)/a)/(a**2*(n + 1))
+ 3*d*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)) + e**3*x**(3*n + 1)*hyper((2, (3*n + 1)/(2*n)), ((5*n + 1)/(2*n
),), -c*x**(2*n)/a)/(a**2*(3*n + 1))

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Mathematica [A]  time = 0.492747, size = 165, normalized size = 0.57 \[ \frac{x \left (\left (3 a d e^2+c d^3 (2 n-1)\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+\frac{e x^n \left (a e^2 (n+1)+3 c d^2 (n-1)\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+\frac{a \left (c d^2 \left (d+3 e x^n\right )-a e^2 \left (3 d+e x^n\right )\right )}{a+c x^{2 n}}\right )}{2 a^2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{ \left ( a+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((3*c*d^2*e - a*e^3)*x*x^n + (c*d^3 - 3*a*d*e^2)*x)/(a*c^2*n*x^(2*n) + a^2*c
*n) + integrate(1/2*(c*d^3*(2*n - 1) + 3*a*d*e^2 + (a*e^3*(n + 1) + 3*c*d^2*e*(n
- 1))*x^n)/(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. \[{\rm integral}\left (\frac{e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c^{2} x^{4 \, n} + 2 \, a c x^{2 \, n} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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