Optimal. Leaf size=27 \[ \frac{x^{2 (p+1)} \left (b+c x^2\right )^{p+1}}{2 (p+1)} \]
[Out]
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Rubi [A] time = 0.0210792, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ \frac{x^{2 (p+1)} \left (b+c x^2\right )^{p+1}}{2 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*(1 + p))*(b + c*x^2)^p*(b + 2*c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 6.80083, size = 20, normalized size = 0.74 \[ \frac{x^{2 p + 2} \left (b + c x^{2}\right )^{p + 1}}{2 \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1+2*p)*(c*x**2+b)**p*(2*c*x**2+b),x)
[Out]
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Mathematica [A] time = 0.0537443, size = 26, normalized size = 0.96 \[ \frac{x^{2 p+2} \left (b+c x^2\right )^{p+1}}{2 p+2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*(1 + p))*(b + c*x^2)^p*(b + 2*c*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 26, normalized size = 1. \[{\frac{{x}^{2\,p+2} \left ( c{x}^{2}+b \right ) ^{1+p}}{2\,p+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1+2*p)*(c*x^2+b)^p*(2*c*x^2+b),x)
[Out]
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Maxima [A] time = 1.57925, size = 47, normalized size = 1.74 \[ \frac{{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)*(c*x^2 + b)^p*x^(2*p + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234278, size = 43, normalized size = 1.59 \[ \frac{{\left (c x^{3} + b x\right )}{\left (c x^{2} + b\right )}^{p} x^{2 \, p + 1}}{2 \,{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)*(c*x^2 + b)^p*x^(2*p + 1),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**(1+2*p)*(c*x**2+b)**p*(2*c*x**2+b),x)
[Out]
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GIAC/XCAS [A] time = 0.216161, size = 73, normalized size = 2.7 \[ \frac{c x^{3} e^{\left (p{\rm ln}\left (c x^{2} + b\right ) + 2 \, p{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )} + b x e^{\left (p{\rm ln}\left (c x^{2} + b\right ) + 2 \, p{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x^2 + b)*(c*x^2 + b)^p*x^(2*p + 1),x, algorithm="giac")
[Out]