Optimal. Leaf size=27 \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
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Rubi [A] time = 0.0133516, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x*(b + 2*c*x^2)*(-a + b*x^2 + c*x^4)^p,x]
[Out]
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Rubi in Sympy [A] time = 6.06874, size = 19, normalized size = 0.7 \[ \frac{\left (- a + b x^{2} + c x^{4}\right )^{p + 1}}{2 \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2-a)**p,x)
[Out]
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Mathematica [A] time = 0.0381282, size = 26, normalized size = 0.96 \[ \frac{\left (-a+b x^2+c x^4\right )^{p+1}}{2 p+2} \]
Antiderivative was successfully verified.
[In] Integrate[x*(b + 2*c*x^2)*(-a + b*x^2 + c*x^4)^p,x]
[Out]
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Maple [A] time = 0.005, size = 26, normalized size = 1. \[{\frac{ \left ( c{x}^{4}+b{x}^{2}-a \right ) ^{1+p}}{2+2\,p}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(2*c*x^2+b)*(c*x^4+b*x^2-a)^p,x)
[Out]
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Maxima [A] time = 0.831103, size = 50, normalized size = 1.85 \[ \frac{{\left (c x^{4} + b x^{2} - a\right )}{\left (c x^{4} + b x^{2} - a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)*(c*x^4 + b*x^2 - a)^p*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287448, size = 50, normalized size = 1.85 \[ \frac{{\left (c x^{4} + b x^{2} - a\right )}{\left (c x^{4} + b x^{2} - a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)*(c*x^4 + b*x^2 - a)^p*x,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(x*(2*c*x**2+b)*(c*x**4+b*x**2-a)**p,x)
[Out]
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GIAC/XCAS [A] time = 0.2727, size = 101, normalized size = 3.74 \[ \frac{c x^{4} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} - a\right )\right )} + b x^{2} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} - a\right )\right )} - a e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} - a\right )\right )}}{2 \,{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)*(c*x^4 + b*x^2 - a)^p*x,x, algorithm="giac")
[Out]