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

Optimal. Leaf size=713 \[ \frac{x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}+2 c d \left (8 n^2-6 n+1\right )\right )-2 a b c \left (2 a e \left (-3 n^2-n+1\right )-d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right )+b^3 (1-n) \left (a e-d (1-2 n) \sqrt{b^2-4 a c}\right )+b^4 (-d) \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}-2 c d \left (8 n^2-6 n+1\right )\right )+2 a b c \left (d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}+2 a e \left (-3 n^2-n+1\right )\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right )-b^3 (1-n) \left (d (1-2 n) \sqrt{b^2-4 a c}+a e\right )+b^4 d \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \]

[Out]

(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*
x^n + c*x^(2*n))^2) + (x*(a*b^3*e - 4*a^2*c^2*d*(1 - 4*n) + 5*a*b^2*c*d*(1 - 3*n
) - 2*a^2*b*c*e*(2 - 3*n) - b^4*d*(1 - 2*n) + c*(a*b^2*e + 2*a*b*c*d*(2 - 7*n) -
4*a^2*c*e*(1 - 3*n) - b^3*d*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*n^2*(a + b*
x^n + c*x^(2*n))) + (c*(a*b^2*(Sqrt[b^2 - 4*a*c]*e + 6*c*d*(1 - 3*n))*(1 - n) +
b^3*(a*e - Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) - b^4*d*(1 - 3*n + 2*n^2) - 2*
a*b*c*(2*a*e*(1 - n - 3*n^2) - Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2)) - 4*a^2*c*
(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) + 2*c*d*(1 - 6*n + 8*n^2)))*x*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 -
4*a*c)^2*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n^2) - (c*(a*b^2*(Sqrt[b^2 - 4*a*c
]*e - 6*c*d*(1 - 3*n))*(1 - n) - b^3*(a*e + Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 -
n) + b^4*d*(1 - 3*n + 2*n^2) + 2*a*b*c*(2*a*e*(1 - n - 3*n^2) + Sqrt[b^2 - 4*a*c
]*d*(2 - 9*n + 7*n^2)) - 4*a^2*c*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) - 2*c*d*
(1 - 6*n + 8*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + S
qrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n
^2)

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Rubi [A]  time = 4.99873, antiderivative size = 713, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}+2 c d \left (8 n^2-6 n+1\right )\right )-2 a b c \left (2 a e \left (-3 n^2-n+1\right )-d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right )+b^3 (1-n) \left (a e-d (1-2 n) \sqrt{b^2-4 a c}\right )+b^4 (-d) \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}-2 c d \left (8 n^2-6 n+1\right )\right )+2 a b c \left (d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}+2 a e \left (-3 n^2-n+1\right )\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right )-b^3 (1-n) \left (d (1-2 n) \sqrt{b^2-4 a c}+a e\right )+b^4 d \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*
x^n + c*x^(2*n))^2) + (x*(a*b^3*e - 4*a^2*c^2*d*(1 - 4*n) + 5*a*b^2*c*d*(1 - 3*n
) - 2*a^2*b*c*e*(2 - 3*n) - b^4*d*(1 - 2*n) + c*(a*b^2*e + 2*a*b*c*d*(2 - 7*n) -
4*a^2*c*e*(1 - 3*n) - b^3*d*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*n^2*(a + b*
x^n + c*x^(2*n))) + (c*(a*b^2*(Sqrt[b^2 - 4*a*c]*e + 6*c*d*(1 - 3*n))*(1 - n) +
b^3*(a*e - Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) - b^4*d*(1 - 3*n + 2*n^2) - 2*
a*b*c*(2*a*e*(1 - n - 3*n^2) - Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2)) - 4*a^2*c*
(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) + 2*c*d*(1 - 6*n + 8*n^2)))*x*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 -
4*a*c)^2*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n^2) - (c*(a*b^2*(Sqrt[b^2 - 4*a*c
]*e - 6*c*d*(1 - 3*n))*(1 - n) - b^3*(a*e + Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 -
n) + b^4*d*(1 - 3*n + 2*n^2) + 2*a*b*c*(2*a*e*(1 - n - 3*n^2) + Sqrt[b^2 - 4*a*c
]*d*(2 - 9*n + 7*n^2)) - 4*a^2*c*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) - 2*c*d*
(1 - 6*n + 8*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + S
qrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c + 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)/(a+b*x**n+c*x**(2*n))**3,x)

[Out]

Timed out

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Mathematica [B]  time = 6.6777, size = 8593, normalized size = 12.05 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [F]  time = 0.148, size = 0, normalized size = 0. \[ \int{\frac{d+e{x}^{n}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

1/2*((4*a^2*c^3*e*(3*n - 1) + b^3*c^2*d*(2*n - 1) - (2*b*c^3*d*(7*n - 2) - b^2*c
^2*e)*a)*x*x^(3*n) + (2*b^4*c*d*(2*n - 1) + 2*(b*c^2*e*(9*n - 4) + 2*c^3*d*(4*n
- 1))*a^2 - (b^2*c^2*d*(29*n - 9) - 2*b^3*c*e)*a)*x*x^(2*n) + (4*a^3*c^2*e*(5*n
- 1) + b^5*d*(2*n - 1) + (b^2*c*e*(4*n - 3) - 2*b*c^2*d*n)*a^2 - (4*b^3*c*d*(3*n
- 1) - b^4*e)*a)*x*x^n + (a*b^4*d*(3*n - 1) + 2*(2*c^2*d*(6*n - 1) + b*c*e*(5*n
- 2))*a^3 - (b^2*c*d*(21*n - 5) + b^3*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 +
2*(2*(8*n^2 - 6*n + 1)*c^2*d - b*c*e*(5*n - 2))*a^2 - ((16*n^2 - 21*n + 5)*b^2*c
*d - b^3*e*(n - 1))*a + ((2*n^2 - 3*n + 1)*b^3*c*d + 4*(3*n^2 - 4*n + 1)*a^2*c^2
*e - (2*(7*n^2 - 9*n + 2)*b*c^2*d - b^2*c*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. \[{\rm integral}\left (\frac{e x^{n} + d}{c^{3} x^{6 \, n} + b^{3} x^{3 \, n} + 3 \, a^{2} b x^{n} + a^{3} + 3 \,{\left (b c^{2} x^{n} + b^{2} c + a c^{2}\right )} x^{4 \, n} + 3 \,{\left (2 \, a b c x^{n} + a b^{2} + a^{2} c\right )} x^{2 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^n + d)/(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)/(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. \[ \int \frac{e x^{n} + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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