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

Optimal. Leaf size=750 \[ \frac{x \left (x^n \left (-\left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )}{a c n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}+\frac{e^2 x \left (\frac{6 c d-3 b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}+\frac{e^2 x \left (e-\frac{3 (2 c d-b e)}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{c \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+\frac{-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )-\frac{-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}+b\right )} \]

[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))/(a*c*(b^2 - 4*a*c)
*n*(a + b*x^n + c*x^(2*n))) + (e^2*(e + (6*c*d - 3*b*e)/Sqrt[b^2 - 4*a*c])*x*Hyp
ergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(c*(b
- Sqrt[b^2 - 4*a*c])) + (((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))*(1 - n) + (b^2*c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*
(1 - 3*n) + 4*a*c^2*d*(c*d^2 - 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) +
3*c*d^2*n))/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])])/(a*c*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) +
(e^2*(e - (3*(2*c*d - b*e))/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])])/(c*(b + Sqrt[b^2 - 4*a*c])) + (((
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))*(1 - n) - (b^2*
c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*(1 - 3*n) + 4*a*c^2*d*(c*d^2
- 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) + 3*c*d^2*n))/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]
)])/(a*c*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)

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Rubi [A]  time = 5.68353, antiderivative size = 750, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x \left (x^n \left (-\left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )}{a c n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}+\frac{e^2 x \left (\frac{6 c d-3 b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}+\frac{e^2 x \left (e-\frac{3 (2 c d-b e)}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{c \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+\frac{-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left ((1-n) \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )-\frac{-a b^3 e^3 (1-3 n)+b^2 c d \left (3 a e^2 (1-3 n)-c d^2 (1-n)\right )+2 a b c e \left (a e^2 (2-5 n)+3 c d^2 n\right )+4 a c^2 d (1-2 n) \left (c d^2-3 a e^2\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a c n \left (b^2-4 a c\right ) \left (\sqrt{b^2-4 a c}+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2,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))/(a*c*(b^2 - 4*a*c)
*n*(a + b*x^n + c*x^(2*n))) + (e^2*(e + (6*c*d - 3*b*e)/Sqrt[b^2 - 4*a*c])*x*Hyp
ergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(c*(b
- Sqrt[b^2 - 4*a*c])) + (((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))*(1 - n) + (b^2*c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*
(1 - 3*n) + 4*a*c^2*d*(c*d^2 - 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) +
3*c*d^2*n))/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])])/(a*c*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) +
(e^2*(e - (3*(2*c*d - b*e))/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])])/(c*(b + Sqrt[b^2 - 4*a*c])) + (((
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))*(1 - n) - (b^2*
c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*(1 - 3*n) + 4*a*c^2*d*(c*d^2
- 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) + 3*c*d^2*n))/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]
)])/(a*c*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)

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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))**2,x)

[Out]

Timed out

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Mathematica [B]  time = 6.48779, size = 5537, normalized size = 7.38 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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

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)^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*
b*x^n + a^2 + (2*b*c*x^n + b^2 + 2*a*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))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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