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

Optimal. Leaf size=308 \[ \frac{x \left (\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)} \]

[Out]

(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^(1 + n))/(c*(1 + n)) + ((3*c^2*d^2*e - 3*b*c*
d*e^2 + b^2*e^3 - a*c*e^3 + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*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^2*(b - Sqrt[b^2 - 4*a*c])) + ((3*c^2*d^2*e - 3*b*c*d*e
^2 + b^2*e^3 - a*c*e^3 - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))
/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + S
qrt[b^2 - 4*a*c])])/(c^2*(b + Sqrt[b^2 - 4*a*c]))

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Rubi [A]  time = 1.50681, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{x \left (\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^(1 + n))/(c*(1 + n)) + ((3*c^2*d^2*e - 3*b*c*
d*e^2 + b^2*e^3 - a*c*e^3 + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*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^2*(b - Sqrt[b^2 - 4*a*c])) + ((3*c^2*d^2*e - 3*b*c*d*e
^2 + b^2*e^3 - a*c*e^3 - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))
/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + S
qrt[b^2 - 4*a*c])])/(c^2*(b + Sqrt[b^2 - 4*a*c]))

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Rubi in Sympy [A]  time = 177.191, size = 566, normalized size = 1.84 \[ \text{result too large to display} \]

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)),x)

[Out]

-2*c*d**3*x*hyper((1, 1/n), (1 + 1/n,), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(-4
*a*c + b**2 + b*sqrt(-4*a*c + b**2)) - 2*c*d**3*x*hyper((1, 1/n), (1 + 1/n,), -2
*c*x**n/(b - sqrt(-4*a*c + b**2)))/(-4*a*c + b**2 - b*sqrt(-4*a*c + b**2)) - 6*c
*d**2*e*x**(n + 1)*hyper((1, (n + 1)/n), (2 + 1/n,), -2*c*x**n/(b + sqrt(-4*a*c
+ b**2)))/((b + sqrt(-4*a*c + b**2))*(n + 1)*sqrt(-4*a*c + b**2)) + 6*c*d**2*e*x
**(n + 1)*hyper((1, (n + 1)/n), (2 + 1/n,), -2*c*x**n/(b - sqrt(-4*a*c + b**2)))
/((b - sqrt(-4*a*c + b**2))*(n + 1)*sqrt(-4*a*c + b**2)) - 6*c*d*e**2*x**(2*n +
1)*hyper((1, 2 + 1/n), (3 + 1/n,), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/((b + sq
rt(-4*a*c + b**2))*(2*n + 1)*sqrt(-4*a*c + b**2)) + 6*c*d*e**2*x**(2*n + 1)*hype
r((1, 2 + 1/n), (3 + 1/n,), -2*c*x**n/(b - sqrt(-4*a*c + b**2)))/((b - sqrt(-4*a
*c + b**2))*(2*n + 1)*sqrt(-4*a*c + b**2)) - 2*c*e**3*x**(3*n + 1)*hyper((1, 3 +
1/n), (4 + 1/n,), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/((b + sqrt(-4*a*c + b**2
))*(3*n + 1)*sqrt(-4*a*c + b**2)) + 2*c*e**3*x**(3*n + 1)*hyper((1, 3 + 1/n), (4
+ 1/n,), -2*c*x**n/(b - sqrt(-4*a*c + b**2)))/((b - sqrt(-4*a*c + b**2))*(3*n +
1)*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 3.90376, size = 455, normalized size = 1.48 \[ -\frac{2^{-\frac{n+1}{n}} x \left (\left (b \left (a e^3 \sqrt{b^2-4 a c}+3 a c d e^2+c^2 d^3\right )+c \left (c d^2 \left (d \sqrt{b^2-4 a c}-6 a e\right )+a e^2 \left (2 a e-3 d \sqrt{b^2-4 a c}\right )\right )-a b^2 e^3\right ) \left (\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )+\left (b \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2-c^2 d^3\right )+c \left (c d^2 \left (d \sqrt{b^2-4 a c}+6 a e\right )-a e^2 \left (3 d \sqrt{b^2-4 a c}+2 a e\right )\right )+a b^2 e^3\right ) \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )-\frac{c 2^{\frac{1}{n}+1} \sqrt{b^2-4 a c} \left (a e^3 x^n+c d^3 (n+1)\right )}{n+1}\right )}{a c^2 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

-((x*(-((2^(1 + n^(-1))*c*Sqrt[b^2 - 4*a*c]*(c*d^3*(1 + n) + a*e^3*x^n))/(1 + n)
) + ((-(a*b^2*e^3) + b*(c^2*d^3 + 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + c*(c*
d^2*(Sqrt[b^2 - 4*a*c]*d - 6*a*e) + a*e^2*(-3*Sqrt[b^2 - 4*a*c]*d + 2*a*e)))*Hyp
ergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b
^2 - 4*a*c] + 2*c*x^n)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((
a*b^2*e^3 + b*(-(c^2*d^3) - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + c*(-(a*e^2*
(3*Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 6*a*e)))*Hyperge
ometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 -
4*a*c] + 2*c*x^n)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/(2^((1
+ n)/n)*a*c^2*Sqrt[b^2 - 4*a*c]))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{a+b{x}^{n}+c{x}^{2\,n}}}\, dx \]

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)),x)

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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

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),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*x^(2*n) + b*x^n
+ a), 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)),x)

[Out]

Timed out

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

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),x, algorithm="giac")

[Out]

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