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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]