Optimal. Leaf size=141 \[ \frac{e x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)}+\frac{d x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)} \]
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Rubi [A] time = 0.288941, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{e x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)}+\frac{d x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^n)^3/(a + c*x^(2*n)),x]
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Rubi in Sympy [A] time = 27.7449, size = 151, normalized size = 1.07 \[ \frac{d^{3} x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a} + \frac{3 d^{2} e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (n + 1\right )} + \frac{3 d e^{2} x^{2 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (2 n + 1\right )} + \frac{e^{3} x^{3 n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{3 n + 1}{2 n} \\ \frac{5 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (3 n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d+e*x**n)**3/(a+c*x**(2*n)),x)
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Mathematica [A] time = 0.188088, size = 128, normalized size = 0.91 \[ \frac{d (n+1) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+e x \left (x^n \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+a e \left (3 d (n+1)+e x^n\right )\right )}{a c (n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^n)^3/(a + c*x^(2*n)),x]
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Maple [F] time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{a+c{x}^{2\,n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d+e*x^n)^3/(a+c*x^(2*n)),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 \, d e^{2}{\left (n + 1\right )} x + e^{3} x x^{n}}{c{\left (n + 1\right )}} - \int -\frac{c d^{3} - 3 \, a d e^{2} +{\left (3 \, c d^{2} e - a e^{3}\right )} x^{n}}{c^{2} x^{2 \, n} + a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)^3/(c*x^(2*n) + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c x^{2 \, n} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)^3/(c*x^(2*n) + a),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d+e*x**n)**3/(a+c*x**(2*n)),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)^3/(c*x^(2*n) + a),x, algorithm="giac")
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