Optimal. Leaf size=97 \[ -\frac{(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}+\frac{\log (x) (b B-3 A c)}{b^4}+\frac{b B-2 A c}{2 b^3 \left (b+c x^2\right )}-\frac{A}{2 b^3 x^2}+\frac{b B-A c}{4 b^2 \left (b+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.244392, antiderivative size = 97, 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 \[ -\frac{(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}+\frac{\log (x) (b B-3 A c)}{b^4}+\frac{b B-2 A c}{2 b^3 \left (b+c x^2\right )}-\frac{A}{2 b^3 x^2}+\frac{b B-A c}{4 b^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 26.5305, size = 90, normalized size = 0.93 \[ - \frac{A}{2 b^{3} x^{2}} - \frac{A c - B b}{4 b^{2} \left (b + c x^{2}\right )^{2}} - \frac{2 A c - B b}{2 b^{3} \left (b + c x^{2}\right )} - \frac{\left (3 A c - B b\right ) \log{\left (x^{2} \right )}}{2 b^{4}} + \frac{\left (3 A c - B b\right ) \log{\left (b + c x^{2} \right )}}{2 b^{4}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.101317, size = 86, normalized size = 0.89 \[ \frac{\frac{b^2 (b B-A c)}{\left (b+c x^2\right )^2}+\frac{2 b (b B-2 A c)}{b+c x^2}-2 (b B-3 A c) \log \left (b+c x^2\right )+4 \log (x) (b B-3 A c)-\frac{2 A b}{x^2}}{4 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.027, size = 118, normalized size = 1.2 \[ -{\frac{A}{2\,{b}^{3}{x}^{2}}}-3\,{\frac{A\ln \left ( x \right ) c}{{b}^{4}}}+{\frac{\ln \left ( x \right ) B}{{b}^{3}}}-{\frac{Ac}{{b}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{B}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}-{\frac{Ac}{4\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{B}{4\,b \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{3\,c\ln \left ( c{x}^{2}+b \right ) A}{2\,{b}^{4}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) B}{2\,{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^2+A)/(c*x^4+b*x^2)^3,x)
[Out]
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Maxima [A] time = 1.38141, size = 147, normalized size = 1.52 \[ \frac{2 \,{\left (B b c - 3 \, A c^{2}\right )} x^{4} - 2 \, A b^{2} + 3 \,{\left (B b^{2} - 3 \, A b c\right )} x^{2}}{4 \,{\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} - \frac{{\left (B b - 3 \, A c\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac{{\left (B b - 3 \, A c\right )} \log \left (x^{2}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212027, size = 266, normalized size = 2.74 \[ \frac{2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} - 2 \, A b^{3} + 3 \,{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2} - 2 \,{\left ({\left (B b c^{2} - 3 \, A c^{3}\right )} x^{6} + 2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} +{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \,{\left ({\left (B b c^{2} - 3 \, A c^{3}\right )} x^{6} + 2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} +{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.4998, size = 107, normalized size = 1.1 \[ \frac{- 2 A b^{2} + x^{4} \left (- 6 A c^{2} + 2 B b c\right ) + x^{2} \left (- 9 A b c + 3 B b^{2}\right )}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} + \frac{\left (- 3 A c + B b\right ) \log{\left (x \right )}}{b^{4}} - \frac{\left (- 3 A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.217474, size = 142, normalized size = 1.46 \[ \frac{{\left (B b - 3 \, A c\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} - \frac{{\left (B b c - 3 \, A c^{2}\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} + \frac{2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} - 2 \, A b^{3} + 3 \,{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}}{4 \,{\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]