Optimal. Leaf size=166 \[ \frac{d (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1} \]
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Rubi [A] time = 0.216018, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{d (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1} \]
Antiderivative was successfully verified.
[In] Int[(f*x)^m*(d + e*x^n)*(a + c*x^(2*n))^p,x]
[Out]
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Rubi in Sympy [A] time = 27.4712, size = 136, normalized size = 0.82 \[ \frac{d \left (f x\right )^{m + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{2 n} \\ 1 + \frac{m + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{f \left (m + 1\right )} + \frac{e x^{n} \left (f x\right )^{- n} \left (f x\right )^{m + n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + n + 1}{2 n} \\ \frac{m + 3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{f \left (m + n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x)**m*(d+e*x**n)*(a+c*x**(2*n))**p,x)
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Mathematica [A] time = 0.147541, size = 136, normalized size = 0.82 \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (d (m+n+1) \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )+e (m+1) x^n \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )\right )}{(m+1) (m+n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(f*x)^m*(d + e*x^n)*(a + c*x^(2*n))^p,x]
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Maple [F] time = 0.105, size = 0, normalized size = 0. \[ \int \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) \left ( a+c{x}^{2\,n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x)^m*(d+e*x^n)*(a+c*x^(2*n))^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{n} + d\right )}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)*(c*x^(2*n) + a)^p*(f*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{n} + d\right )}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)*(c*x^(2*n) + a)^p*(f*x)^m,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((f*x)**m*(d+e*x**n)*(a+c*x**(2*n))**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{n} + d\right )}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)*(c*x^(2*n) + a)^p*(f*x)^m,x, algorithm="giac")
[Out]