Optimal. Leaf size=65 \[ \frac{b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac{3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac{3 b \log \left (b+c x^2\right )}{2 c^4}+\frac{x^2}{2 c^3} \]
[Out]
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Rubi [A] time = 0.119345, antiderivative size = 65, 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 \[ \frac{b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac{3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac{3 b \log \left (b+c x^2\right )}{2 c^4}+\frac{x^2}{2 c^3} \]
Antiderivative was successfully verified.
[In] Int[x^13/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{3}}{4 c^{4} \left (b + c x^{2}\right )^{2}} - \frac{3 b^{2}}{2 c^{4} \left (b + c x^{2}\right )} - \frac{3 b \log{\left (b + c x^{2} \right )}}{2 c^{4}} + \frac{\int ^{x^{2}} \frac{1}{c^{3}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**13/(c*x**4+b*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.104605, size = 48, normalized size = 0.74 \[ -\frac{\frac{b^2 \left (5 b+6 c x^2\right )}{\left (b+c x^2\right )^2}+6 b \log \left (b+c x^2\right )-2 c x^2}{4 c^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^13/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.015, size = 58, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{c}^{3}}}+{\frac{{b}^{3}}{4\,{c}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{3\,{b}^{2}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{3\,b\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^13/(c*x^4+b*x^2)^3,x)
[Out]
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Maxima [A] time = 0.699089, size = 89, normalized size = 1.37 \[ -\frac{6 \, b^{2} c x^{2} + 5 \, b^{3}}{4 \,{\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac{x^{2}}{2 \, c^{3}} - \frac{3 \, b \log \left (c x^{2} + b\right )}{2 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253847, size = 123, normalized size = 1.89 \[ \frac{2 \, c^{3} x^{6} + 4 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} - 5 \, b^{3} - 6 \,{\left (b c^{2} x^{4} + 2 \, b^{2} c x^{2} + b^{3}\right )} \log \left (c x^{2} + b\right )}{4 \,{\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.06777, size = 66, normalized size = 1.02 \[ - \frac{3 b \log{\left (b + c x^{2} \right )}}{2 c^{4}} - \frac{5 b^{3} + 6 b^{2} c x^{2}}{4 b^{2} c^{4} + 8 b c^{5} x^{2} + 4 c^{6} x^{4}} + \frac{x^{2}}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**13/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.273912, size = 84, normalized size = 1.29 \[ \frac{x^{2}}{2 \, c^{3}} - \frac{3 \, b{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac{9 \, b c^{2} x^{4} + 12 \, b^{2} c x^{2} + 4 \, b^{3}}{4 \,{\left (c x^{2} + b\right )}^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]