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

Optimal. Leaf size=1191 \[ \text{result too large to display} \]

[Out]

(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*
x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2 - 2*a*c + b*
c*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (x*(2*a*b^3*c*d*e - a*b^
2*c*(a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) - 4*a^2*c^2*(c*d^2 - a*e^2)*(1 - 4*n)
- 4*a^2*b*c^2*d*e*(2 - 3*n) - b^4*(c*d^2*(1 - 2*n) + 2*a*e^2*n) + c*(2*a*b^2*c*d
*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*
(1 - 2*n) + 2*a*e^2*n))*x^n))/(2*a^2*c*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n
))) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hyper
geometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2
- 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*
a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*
n) + 2*a*e^2*n)) + (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e
^2*n) - 8*a^2*b*c^2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8
*n^2) + 2*a*b^2*c*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*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*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) - (e^2*(4*a*c*
(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^
(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4
*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3
*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n)) -
(2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e^2*n) - 8*a^2*b*c^
2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*
(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hy
pergeometric2F1[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 + Sqrt[b^2 - 4*a*c])*n^2)

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Rubi [A]  time = 8.98719, antiderivative size = 1191, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\left (-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right ) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) e^2}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt{b^2-4 a c} b-4 a c\right ) n}-\frac{\left (-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right ) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) e^2}{a \left (b^2-4 a c\right ) \left (b^2+\sqrt{b^2-4 a c} b-4 a c\right ) n}+\frac{x \left (b c x^n+b^2-2 a c\right ) e^2}{a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac{\left ((1-n) \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )+\frac{-(1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (8 n^2-6 n+1\right )}{\sqrt{b^2-4 a c}}\right ) x \, _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 \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) n^2}-\frac{\left ((1-n) \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )-\frac{-(1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (8 n^2-6 n+1\right )}{\sqrt{b^2-4 a c}}\right ) x \, _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 \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) n^2}+\frac{x \left (c \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right ) x^n+2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c (1-2 n) d^2+2 a e^2 n\right )\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac{x \left (\left (b c d^2-4 a c e d+a b e^2\right ) x^n+b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )}{2 a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*
x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2 - 2*a*c + b*
c*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (x*(2*a*b^3*c*d*e - a*b^
2*c*(a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) - 4*a^2*c^2*(c*d^2 - a*e^2)*(1 - 4*n)
- 4*a^2*b*c^2*d*e*(2 - 3*n) - b^4*(c*d^2*(1 - 2*n) + 2*a*e^2*n) + c*(2*a*b^2*c*d
*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*
(1 - 2*n) + 2*a*e^2*n))*x^n))/(2*a^2*c*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n
))) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hyper
geometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2
- 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*
a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*
n) + 2*a*e^2*n)) + (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e
^2*n) - 8*a^2*b*c^2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8
*n^2) + 2*a*b^2*c*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*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*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) - (e^2*(4*a*c*
(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^
(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4
*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3
*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n)) -
(2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e^2*n) - 8*a^2*b*c^
2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*
(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hy
pergeometric2F1[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 + 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)**2/(a+b*x**n+c*x**(2*n))**3,x)

[Out]

Timed out

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Mathematica [B]  time = 6.82098, size = 10910, normalized size = 9.16 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{2}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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