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

Optimal. Leaf size=424 \[ \frac{e (1-3 n) (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 c n^2 (n+1)}+\frac{d (1-4 n) (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 c n^2}-\frac{x \left (e (1-3 n) x^n \left (3 c d^2-a e^2\right )+d (1-4 n) \left (c d^2-3 a e^2\right )\right )}{8 a^2 c n^2 \left (a+c x^{2 n}\right )}-\frac{3 d e^2 (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac{e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}+\frac{x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c n \left (a+c x^{2 n}\right )^2}+\frac{e^2 x \left (3 d+e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )} \]

[Out]

(x*(d*(c*d^2 - 3*a*e^2) + e*(3*c*d^2 - a*e^2)*x^n))/(4*a*c*n*(a + c*x^(2*n))^2)
+ (e^2*x*(3*d + e*x^n))/(2*a*c*n*(a + c*x^(2*n))) - (x*(d*(c*d^2 - 3*a*e^2)*(1 -
4*n) + e*(3*c*d^2 - a*e^2)*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + c*x^(2*n))) + (d*(
c*d^2 - 3*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) - (3*d*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*c*d^2 - a*e^2)
*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2,
-((c*x^(2*n))/a)])/(8*a^3*c*n^2*(1 + n)) - (e^3*(1 - n)*x^(1 + n)*Hypergeometri
c2F1[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 = 0.829315, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{e (1-3 n) (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 c n^2 (n+1)}+\frac{d (1-4 n) (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 c n^2}-\frac{x \left (e (1-3 n) x^n \left (3 c d^2-a e^2\right )+d (1-4 n) \left (c d^2-3 a e^2\right )\right )}{8 a^2 c n^2 \left (a+c x^{2 n}\right )}-\frac{3 d e^2 (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac{e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}+\frac{x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c n \left (a+c x^{2 n}\right )^2}+\frac{e^2 x \left (3 d+e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d*(c*d^2 - 3*a*e^2) + e*(3*c*d^2 - a*e^2)*x^n))/(4*a*c*n*(a + c*x^(2*n))^2)
+ (e^2*x*(3*d + e*x^n))/(2*a*c*n*(a + c*x^(2*n))) - (x*(d*(c*d^2 - 3*a*e^2)*(1 -
4*n) + e*(3*c*d^2 - a*e^2)*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + c*x^(2*n))) + (d*(
c*d^2 - 3*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) - (3*d*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*c*d^2 - a*e^2)
*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2,
-((c*x^(2*n))/a)])/(8*a^3*c*n^2*(1 + n)) - (e^3*(1 - n)*x^(1 + n)*Hypergeometri
c2F1[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 = 27.8458, size = 158, normalized size = 0.37 \[ \frac{d^{3} x{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3}} + \frac{3 d^{2} e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3} \left (n + 1\right )} + \frac{3 d e^{2} x^{2 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3} \left (2 n + 1\right )} + \frac{e^{3} x^{3 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{3 n + 1}{2 n} \\ \frac{5 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a^{3} \left (3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.22617, size = 252, normalized size = 0.59 \[ \frac{x \left (\frac{a \left (-a^2 e^2 \left (d (6 n-3)+e (n-1) x^n\right )+a c \left (d^3 (6 n-1)+3 d^2 e (5 n-1) x^n+3 d e^2 x^{2 n}+e^3 (n+1) x^{3 n}\right )+c^2 d^2 x^{2 n} \left (d (4 n-1)+3 e (3 n-1) x^n\right )\right )}{\left (a+c x^{2 n}\right )^2}+d (2 n-1) \left (3 a e^2+c d^2 (4 n-1)\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+\frac{e (n-1) x^n \left (a e^2 (n+1)+3 c d^2 (3 n-1)\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}\right )}{8 a^3 c n^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{ \left ( a+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((3*c^2*d^2*e*(3*n - 1) + a*c*e^3*(n + 1))*x*x^(3*n) + (c^2*d^3*(4*n - 1) +
3*a*c*d*e^2)*x*x^(2*n) + (3*a*c*d^2*e*(5*n - 1) - a^2*e^3*(n - 1))*x*x^n + (a*c*
d^3*(6*n - 1) - 3*a^2*d*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*((8*n^2 - 6*n + 1)*c*d^3 + 3*a*d*e^2*(2*n -
1) + (3*(3*n^2 - 4*n + 1)*c*d^2*e + (n^2 - 1)*a*e^3)*x^n)/(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. \[{\rm integral}\left (\frac{e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c^{3} x^{6 \, n} + 3 \, a c^{2} x^{4 \, n} + 3 \, a^{2} c x^{2 \, n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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