Optimal. Leaf size=299 \[ d^3 x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{3 d^2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+\frac{3 d e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right ),-p;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 n+1}+\frac{e^3 x^{3 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (3+\frac{1}{n}\right ),-p;\frac{1}{2} \left (5+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{3 n+1} \]
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Rubi [A] time = 0.318933, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ d^3 x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{3 d^2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+\frac{3 d e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right ),-p;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 n+1}+\frac{e^3 x^{3 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (3+\frac{1}{n}\right ),-p;\frac{1}{2} \left (5+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{3 n+1} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^n)^3*(a + c*x^(2*n))^p,x]
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Rubi in Sympy [A] time = 41.2103, size = 240, normalized size = 0.8 \[ d^{3} x \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )} + \frac{3 d^{2} e x^{n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{n + 1} + \frac{3 d e^{2} x^{2 n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{2 n + 1} + \frac{e^{3} x^{3 n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3 n + 1}{2 n} \\ \frac{5 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{3 n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d+e*x**n)**3*(a+c*x**(2*n))**p,x)
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Mathematica [A] time = 0.396314, size = 213, normalized size = 0.71 \[ x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (d^2 \left (d \, _2F_1\left (\frac{1}{2 n},-p;1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+\frac{3 e x^n \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}\right )+\frac{3 d e^2 x^{2 n} \, _2F_1\left (1+\frac{1}{2 n},-p;2+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{2 n+1}+\frac{e^3 x^{3 n} \, _2F_1\left (\frac{1}{2} \left (3+\frac{1}{n}\right ),-p;\frac{1}{2} \left (5+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{3 n+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^n)^3*(a + c*x^(2*n))^p,x]
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Maple [F] time = 0.169, size = 0, normalized size = 0. \[ \int \left ( d+e{x}^{n} \right ) ^{3} \left ( a+c{x}^{2\,n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d+e*x^n)^3*(a+c*x^(2*n))^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{n} + d\right )}^{3}{\left (c x^{2 \, n} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)^3*(c*x^(2*n) + a)^p,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}\right )}{\left (c x^{2 \, n} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)^3*(c*x^(2*n) + a)^p,x, algorithm="fricas")
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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+c*x**(2*n))**p,x)
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GIAC/XCAS [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((e*x^n + d)^3*(c*x^(2*n) + a)^p,x, algorithm="giac")
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