Optimal. Leaf size=96 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.279585, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 30.6541, size = 88, normalized size = 0.92 \[ - \frac{A}{b^{3} x} - \frac{x \left (A c - B b\right )}{4 b^{2} \left (b + c x^{2}\right )^{2}} - \frac{x \left (7 A c - 3 B b\right )}{8 b^{3} \left (b + c x^{2}\right )} - \frac{3 \left (5 A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{7}{2}} \sqrt{c}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.107797, size = 96, normalized size = 1. \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.016, size = 125, normalized size = 1.3 \[ -{\frac{A}{{b}^{3}x}}-{\frac{7\,A{x}^{3}{c}^{2}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{3\,Bc{x}^{3}}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,Axc}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{5\,Bx}{8\,b \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,Ac}{8\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{8\,{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(B*x^2+A)/(c*x^4+b*x^2)^3,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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222152, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \log \left (-\frac{2 \, b c x -{\left (c x^{2} - b\right )} \sqrt{-b c}}{c x^{2} + b}\right ) - 2 \,{\left (3 \,{\left (B b c - 5 \, A c^{2}\right )} x^{4} - 8 \, A b^{2} + 5 \,{\left (B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt{-b c}}{16 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{-b c}}, \frac{3 \,{\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) +{\left (3 \,{\left (B b c - 5 \, A c^{2}\right )} x^{4} - 8 \, A b^{2} + 5 \,{\left (B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt{b c}}{8 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{b c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.71002, size = 194, normalized size = 2.02 \[ - \frac{3 \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right ) \log{\left (- \frac{3 b^{4} \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right ) \log{\left (\frac{3 b^{4} \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac{- 8 A b^{2} + x^{4} \left (- 15 A c^{2} + 3 B b c\right ) + x^{2} \left (- 25 A b c + 5 B b^{2}\right )}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.213915, size = 111, normalized size = 1.16 \[ \frac{3 \,{\left (B b - 5 \, A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{3}} - \frac{A}{b^{3} x} + \frac{3 \, B b c x^{3} - 7 \, A c^{2} x^{3} + 5 \, B b^{2} x - 9 \, A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]