Optimal. Leaf size=67 \[ \frac{3 c \log \left (b+c x^2\right )}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{c}{b^3 \left (b+c x^2\right )}-\frac{1}{2 b^3 x^2}-\frac{c}{4 b^2 \left (b+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.121368, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 c \log \left (b+c x^2\right )}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{c}{b^3 \left (b+c x^2\right )}-\frac{1}{2 b^3 x^2}-\frac{c}{4 b^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^3/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 19.1503, size = 66, normalized size = 0.99 \[ - \frac{c}{4 b^{2} \left (b + c x^{2}\right )^{2}} - \frac{c}{b^{3} \left (b + c x^{2}\right )} - \frac{1}{2 b^{3} x^{2}} - \frac{3 c \log{\left (x^{2} \right )}}{2 b^{4}} + \frac{3 c \log{\left (b + c x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(c*x**4+b*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.0976038, size = 59, normalized size = 0.88 \[ -\frac{\frac{b \left (2 b^2+9 b c x^2+6 c^2 x^4\right )}{x^2 \left (b+c x^2\right )^2}-6 c \log \left (b+c x^2\right )+12 c \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.019, size = 62, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{3}{x}^{2}}}-{\frac{c}{4\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{c}{{b}^{3} \left ( c{x}^{2}+b \right ) }}-3\,{\frac{c\ln \left ( x \right ) }{{b}^{4}}}+{\frac{3\,c\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(c*x^4+b*x^2)^3,x)
[Out]
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Maxima [A] time = 0.676693, size = 104, normalized size = 1.55 \[ -\frac{6 \, c^{2} x^{4} + 9 \, b c x^{2} + 2 \, b^{2}}{4 \,{\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} + \frac{3 \, c \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac{3 \, c \log \left (x^{2}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25849, size = 161, normalized size = 2.4 \[ -\frac{6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3} - 6 \,{\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (c x^{2} + b\right ) + 12 \,{\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (x\right )}{4 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.9944, size = 78, normalized size = 1.16 \[ - \frac{2 b^{2} + 9 b c x^{2} + 6 c^{2} x^{4}}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} - \frac{3 c \log{\left (x \right )}}{b^{4}} + \frac{3 c \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.275326, size = 89, normalized size = 1.33 \[ \frac{3 \, c{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} - \frac{3 \, c{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3}}{4 \,{\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]