Optimal. Leaf size=25 \[ \frac{\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
[Out]
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Rubi [A] time = 0.0138469, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\left (a+b x^2+c x^4\right )^{p+1}}{2 (p+1)} \]
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 = 5.55645, size = 19, normalized size = 0.76 \[ \frac{\left (a + b x^{2} + c x^{4}\right )^{p + 1}}{2 \left (p + 1\right )} \]
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 = 0.0336139, size = 24, 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 = 24, 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 = 0.856237, size = 45, normalized size = 1.8 \[ \frac{{\left (c x^{4} + b x^{2} + a\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]
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 = 0.278075, size = 45, normalized size = 1.8 \[ \frac{{\left (c x^{4} + b x^{2} + a\right )}{\left (c x^{4} + b x^{2} + a\right )}^{p}}{2 \,{\left (p + 1\right )}} \]
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 = 0.272383, size = 92, normalized size = 3.68 \[ \frac{c x^{4} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} + b x^{2} e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )} + a e^{\left (p{\rm ln}\left (c x^{4} + b x^{2} + a\right )\right )}}{2 \,{\left (p + 1\right )}} \]
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]