Optimal. Leaf size=58 \[ \frac{\sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{x (b B-A c)}{c^2}+\frac{B x^3}{3 c} \]
[Out]
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Rubi [A] time = 0.11399, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{x (b B-A c)}{c^2}+\frac{B x^3}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(A + B*x^2))/(b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 16.9807, size = 49, normalized size = 0.84 \[ \frac{B x^{3}}{3 c} - \frac{\sqrt{b} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{5}{2}}} + \frac{x \left (A c - B b\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(B*x**2+A)/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.064945, size = 57, normalized size = 0.98 \[ \frac{\sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}+\frac{x (A c-b B)}{c^2}+\frac{B x^3}{3 c} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(A + B*x^2))/(b*x^2 + c*x^4),x]
[Out]
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Maple [A] time = 0.005, size = 68, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,c}}+{\frac{Ax}{c}}-{\frac{xBb}{{c}^{2}}}-{\frac{Ab}{c}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{{b}^{2}B}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(B*x^2+A)/(c*x^4+b*x^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217328, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, B c x^{3} - 3 \,{\left (B b - A c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 6 \,{\left (B b - A c\right )} x}{6 \, c^{2}}, \frac{B c x^{3} + 3 \,{\left (B b - A c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) - 3 \,{\left (B b - A c\right )} x}{3 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.87534, size = 90, normalized size = 1.55 \[ \frac{B x^{3}}{3 c} - \frac{\sqrt{- \frac{b}{c^{5}}} \left (- A c + B b\right ) \log{\left (- c^{2} \sqrt{- \frac{b}{c^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{b}{c^{5}}} \left (- A c + B b\right ) \log{\left (c^{2} \sqrt{- \frac{b}{c^{5}}} + x \right )}}{2} - \frac{x \left (- A c + B b\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(B*x**2+A)/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.208004, size = 77, normalized size = 1.33 \[ \frac{{\left (B b^{2} - A b c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{2}} + \frac{B c^{2} x^{3} - 3 \, B b c x + 3 \, A c^{2} x}{3 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]