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

Optimal. Leaf size=1707 \[ \text{result too large to display} \]

[Out]

(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b^2*e^3
+ 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(2*a*c*(b^2 - 4*a*
c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(3*b^2*c*d - 6*a*c^2*d - b^3*e + a*b*c*
e + c*(3*b*c*d - b^2*e - 2*a*c*e)*x^n))/(a*c^2*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^
(2*n))) - (x*(a*b^2*c^2*d*(3*a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) + 4*a^2*c^3*d*
(c*d^2 - 3*a*e^2)*(1 - 4*n) - 2*a*b^5*e^3*n + 2*a^2*b*c^2*e*(3*c*d^2*(2 - 3*n) -
5*a*e^2*n) - 3*a*b^3*c*e*(c*d^2 - 3*a*e^2*n) + b^4*c*d*(c*d^2*(1 - 2*n) + 6*a*e
^2*n) + c*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d
*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c
*e*(3*c*d^2 - a*e^2*(1 + 2*n)))*x^n))/(2*a^2*c^2*(b^2 - 4*a*c)^2*n^2*(a + b*x^n
+ c*x^(2*n))) + (e^2*(b*c*(2*a*e*(2 - 5*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*
a*c*(6*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d
- Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*
c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 -
4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n
- 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2
*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) - (2*a*b^5*e^3*(1 - n)*n - b^4*c*d*
(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 6*n +
8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)) -
4*a^2*b*c^2*e*(3*c*d^2*(1 - n - 3*n^2) + a*e^2*(1 - 11*n + 19*n^2)) + a*b^3*c*e*
(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometr
ic2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*c*(b^2
- 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) + (e^2*(b*c*(2*a*e*(2 - 5*n) - 3*Sqrt[b^
2 - 4*a*c]*d*(1 - n)) - 2*a*c*(6*c*d*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*e*(1 - n)) -
b^3*e*(1 - n) + b^2*(3*c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1
, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b
^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*
(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*
(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) + (2*a*b^
5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c
*d^2 - 3*a*e^2)*(1 - 6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e
^2*(1 - 10*n + 15*n^2)) - 4*a^2*b*c^2*e*(3*c*d^2*(1 - n - 3*n^2) + a*e^2*(1 - 11
*n + 19*n^2)) + a*b^3*c*e*(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^
2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2
- 4*a*c])])/(2*a^2*c*(b^2 - 4*a*c)^2*(b + Sqrt[b^2 - 4*a*c])*n^2)

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Rubi [A]  time = 13.9279, antiderivative size = 1707, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b^2*e^3
+ 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(2*a*c*(b^2 - 4*a*
c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(3*b^2*c*d - 6*a*c^2*d - b^3*e + a*b*c*
e + c*(3*b*c*d - b^2*e - 2*a*c*e)*x^n))/(a*c^2*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^
(2*n))) - (x*(a*b^2*c^2*d*(3*a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) + 4*a^2*c^3*d*
(c*d^2 - 3*a*e^2)*(1 - 4*n) - 2*a*b^5*e^3*n + 2*a^2*b*c^2*e*(3*c*d^2*(2 - 3*n) -
5*a*e^2*n) - 3*a*b^3*c*e*(c*d^2 - 3*a*e^2*n) + b^4*c*d*(c*d^2*(1 - 2*n) + 6*a*e
^2*n) + c*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d
*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c
*e*(3*c*d^2 - a*e^2*(1 + 2*n)))*x^n))/(2*a^2*c^2*(b^2 - 4*a*c)^2*n^2*(a + b*x^n
+ c*x^(2*n))) + (e^2*(b*c*(2*a*e*(2 - 5*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*
a*c*(6*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d
- Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*
c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 -
4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n
- 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2
*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) - (2*a*b^5*e^3*(1 - n)*n - b^4*c*d*
(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 6*n +
8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)) -
4*a^2*b*c^2*e*(3*c*d^2*(1 - n - 3*n^2) + a*e^2*(1 - 11*n + 19*n^2)) + a*b^3*c*e*
(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometr
ic2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*c*(b^2
- 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) + (e^2*(b*c*(2*a*e*(2 - 5*n) - 3*Sqrt[b^
2 - 4*a*c]*d*(1 - n)) - 2*a*c*(6*c*d*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*e*(1 - n)) -
b^3*e*(1 - n) + b^2*(3*c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1
, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b
^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*
(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*
(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) + (2*a*b^
5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c
*d^2 - 3*a*e^2)*(1 - 6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e
^2*(1 - 10*n + 15*n^2)) - 4*a^2*b*c^2*e*(3*c*d^2*(1 - n - 3*n^2) + a*e^2*(1 - 11
*n + 19*n^2)) + a*b^3*c*e*(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^
2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2
- 4*a*c])])/(2*a^2*c*(b^2 - 4*a*c)^2*(b + Sqrt[b^2 - 4*a*c])*n^2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 7.79723, size = 13018, normalized size = 7.63 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [F]  time = 0.188, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((b^3*c^2*d^3*(2*n - 1) + 4*a^3*c^2*e^3*(n + 1) + (12*c^3*d^2*e*(3*n - 1) +
b^2*c*e^3*(2*n - 1) - 18*b*c^2*d*e^2*n)*a^2 - (2*b*c^3*d^3*(7*n - 2) - 3*b^2*c^2
*d^2*e)*a)*x*x^(3*n) + (2*b^4*c*d^3*(2*n - 1) + 2*(b*c*e^3*(3*n + 2) + 6*c^2*d*e
^2)*a^3 - (3*b^2*c*d*e^2*(9*n + 1) - 6*b*c^2*d^2*e*(9*n - 4) - 4*c^3*d^3*(4*n -
1) - b^3*e^3*(3*n - 1))*a^2 - (b^2*c^2*d^3*(29*n - 9) - 6*b^3*c*d^2*e)*a)*x*x^(2
*n) + (b^5*d^3*(2*n - 1) - 4*a^4*c*e^3*(n - 1) + (b^2*e^3*(10*n - 1) + 12*c^2*d^
2*e*(5*n - 1) - 6*b*c*d*e^2*(5*n - 2))*a^3 + (3*b^2*c*d^2*e*(4*n - 3) - 3*b^3*d*
e^2*(2*n + 1) - 2*b*c^2*d^3*n)*a^2 - (4*b^3*c*d^3*(3*n - 1) - 3*b^4*d^2*e)*a)*x*
x^n + (a*b^4*d^3*(3*n - 1) - 6*(2*c*d*e^2*(2*n - 1) - b*e^3*n)*a^4 + (4*c^2*d^3*
(6*n - 1) + 6*b*c*d^2*e*(5*n - 2) - 3*b^2*d*e^2*(n + 1))*a^3 - (b^2*c*d^3*(21*n
- 5) + 3*b^3*d^2*e*(n - 1))*a^2)*x)/(a^4*b^4*n^2 - 8*a^5*b^2*c*n^2 + 16*a^6*c^2*
n^2 + (a^2*b^4*c^2*n^2 - 8*a^3*b^2*c^3*n^2 + 16*a^4*c^4*n^2)*x^(4*n) + 2*(a^2*b^
5*c*n^2 - 8*a^3*b^3*c^2*n^2 + 16*a^4*b*c^3*n^2)*x^(3*n) + (a^2*b^6*n^2 - 6*a^3*b
^4*c*n^2 + 32*a^5*c^3*n^2)*x^(2*n) + 2*(a^3*b^5*n^2 - 8*a^4*b^3*c*n^2 + 16*a^5*b
*c^2*n^2)*x^n) + integrate(1/2*((2*n^2 - 3*n + 1)*b^4*d^3 + 6*(2*c*d*e^2*(2*n -
1) - b*e^3*n)*a^3 + (4*(8*n^2 - 6*n + 1)*c^2*d^3 - 6*b*c*d^2*e*(5*n - 2) + 3*b^2
*d*e^2*(n + 1))*a^2 - ((16*n^2 - 21*n + 5)*b^2*c*d^3 - 3*b^3*d^2*e*(n - 1))*a +
((2*n^2 - 3*n + 1)*b^3*c*d^3 + 4*(n^2 - 1)*a^3*c*e^3 + (12*(3*n^2 - 4*n + 1)*c^2
*d^2*e - 18*(n^2 - n)*b*c*d*e^2 + (2*n^2 - 3*n + 1)*b^2*e^3)*a^2 - (2*(7*n^2 - 9
*n + 2)*b*c^2*d^3 - 3*b^2*c*d^2*e*(n - 1))*a)*x^n)/(a^3*b^4*n^2 - 8*a^4*b^2*c*n^
2 + 16*a^5*c^2*n^2 + (a^2*b^4*c*n^2 - 8*a^3*b^2*c^2*n^2 + 16*a^4*c^3*n^2)*x^(2*n
) + (a^2*b^5*n^2 - 8*a^3*b^3*c*n^2 + 16*a^4*b*c^2*n^2)*x^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^3/(c*x^(2*n) + b*x^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) + b^3*
x^(3*n) + 3*a^2*b*x^n + a^3 + 3*(b*c^2*x^n + b^2*c + a*c^2)*x^(4*n) + 3*(2*a*b*c
*x^n + a*b^2 + a^2*c)*x^(2*n)), 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+b*x**n+c*x**(2*n))**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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