Optimal. Leaf size=51 \[ -\frac{A \log \left (b+c x^2\right )}{2 b^2}+\frac{A \log (x)}{b^2}-\frac{b B-A c}{2 b c \left (b+c x^2\right )} \]
[Out]
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Rubi [A] time = 0.130782, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{A \log \left (b+c x^2\right )}{2 b^2}+\frac{A \log (x)}{b^2}-\frac{b B-A c}{2 b c \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 17.2662, size = 44, normalized size = 0.86 \[ \frac{A \log{\left (x^{2} \right )}}{2 b^{2}} - \frac{A \log{\left (b + c x^{2} \right )}}{2 b^{2}} + \frac{A c - B b}{2 b c \left (b + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x**2+A)/(c*x**4+b*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0508715, size = 46, normalized size = 0.9 \[ \frac{\frac{b (A c-b B)}{c \left (b+c x^2\right )}-A \log \left (b+c x^2\right )+2 A \log (x)}{2 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.017, size = 53, normalized size = 1. \[{\frac{A\ln \left ( x \right ) }{{b}^{2}}}+{\frac{A}{2\,b \left ( c{x}^{2}+b \right ) }}-{\frac{B}{2\,c \left ( c{x}^{2}+b \right ) }}-{\frac{A\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^2+A)/(c*x^4+b*x^2)^2,x)
[Out]
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Maxima [A] time = 1.37628, size = 69, normalized size = 1.35 \[ -\frac{B b - A c}{2 \,{\left (b c^{2} x^{2} + b^{2} c\right )}} - \frac{A \log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac{A \log \left (x^{2}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(c*x^4 + b*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210469, size = 95, normalized size = 1.86 \[ -\frac{B b^{2} - A b c +{\left (A c^{2} x^{2} + A b c\right )} \log \left (c x^{2} + b\right ) - 2 \,{\left (A c^{2} x^{2} + A b c\right )} \log \left (x\right )}{2 \,{\left (b^{2} c^{2} x^{2} + b^{3} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(c*x^4 + b*x^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.09628, size = 46, normalized size = 0.9 \[ \frac{A \log{\left (x \right )}}{b^{2}} - \frac{A \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{2}} - \frac{- A c + B b}{2 b^{2} c + 2 b c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**2+A)/(c*x**4+b*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.213838, size = 70, normalized size = 1.37 \[ -\frac{A{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} - \frac{B b^{2} - A b c}{2 \,{\left (c x^{2} + b\right )} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(c*x^4 + b*x^2)^2,x, algorithm="giac")
[Out]