Optimal. Leaf size=117 \[ -\frac{5 \sqrt{c} (3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{9/2}}-\frac{c x (7 b B-11 A c)}{8 b^4 \left (b+c x^2\right )}-\frac{b B-3 A c}{b^4 x}-\frac{c x (b B-A c)}{4 b^3 \left (b+c x^2\right )^2}-\frac{A}{3 b^3 x^3} \]
[Out]
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Rubi [A] time = 0.432406, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{5 \sqrt{c} (3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{9/2}}-\frac{c x (7 b B-11 A c)}{8 b^4 \left (b+c x^2\right )}-\frac{b B-3 A c}{b^4 x}-\frac{c x (b B-A c)}{4 b^3 \left (b+c x^2\right )^2}-\frac{A}{3 b^3 x^3} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 61.4887, size = 109, normalized size = 0.93 \[ - \frac{A}{3 b^{3} x^{3}} + \frac{c x \left (A c - B b\right )}{4 b^{3} \left (b + c x^{2}\right )^{2}} + \frac{c x \left (11 A c - 7 B b\right )}{8 b^{4} \left (b + c x^{2}\right )} + \frac{3 A c - B b}{b^{4} x} + \frac{5 \sqrt{c} \left (7 A c - 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.117052, size = 119, normalized size = 1.02 \[ -\frac{5 \sqrt{c} (3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{9/2}}-\frac{x \left (7 b B c-11 A c^2\right )}{8 b^4 \left (b+c x^2\right )}+\frac{3 A c-b B}{b^4 x}-\frac{c x (b B-A c)}{4 b^3 \left (b+c x^2\right )^2}-\frac{A}{3 b^3 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.023, size = 152, normalized size = 1.3 \[ -{\frac{A}{3\,{b}^{3}{x}^{3}}}+3\,{\frac{Ac}{{b}^{4}x}}-{\frac{B}{{b}^{3}x}}+{\frac{11\,A{x}^{3}{c}^{3}}{8\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{7\,B{c}^{2}{x}^{3}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{13\,Ax{c}^{2}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,Bcx}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{35\,A{c}^{2}}{8\,{b}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{15\,Bc}{8\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(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^2/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222551, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \,{\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 50 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} + 16 \, A b^{3} + 16 \,{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \,{\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + 2 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5} +{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{3}\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} + 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right )}{48 \,{\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}}, -\frac{15 \,{\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 25 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 8 \,{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \,{\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + 2 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5} +{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{3}\right )} \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right )}{24 \,{\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^2/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.31821, size = 226, normalized size = 1.93 \[ \frac{5 \sqrt{- \frac{c}{b^{9}}} \left (- 7 A c + 3 B b\right ) \log{\left (- \frac{5 b^{5} \sqrt{- \frac{c}{b^{9}}} \left (- 7 A c + 3 B b\right )}{- 35 A c^{2} + 15 B b c} + x \right )}}{16} - \frac{5 \sqrt{- \frac{c}{b^{9}}} \left (- 7 A c + 3 B b\right ) \log{\left (\frac{5 b^{5} \sqrt{- \frac{c}{b^{9}}} \left (- 7 A c + 3 B b\right )}{- 35 A c^{2} + 15 B b c} + x \right )}}{16} - \frac{8 A b^{3} + x^{6} \left (- 105 A c^{3} + 45 B b c^{2}\right ) + x^{4} \left (- 175 A b c^{2} + 75 B b^{2} c\right ) + x^{2} \left (- 56 A b^{2} c + 24 B b^{3}\right )}{24 b^{6} x^{3} + 48 b^{5} c x^{5} + 24 b^{4} c^{2} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.214415, size = 146, normalized size = 1.25 \[ -\frac{5 \,{\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{4}} - \frac{7 \, B b c^{2} x^{3} - 11 \, A c^{3} x^{3} + 9 \, B b^{2} c x - 13 \, A b c^{2} x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{4}} - \frac{3 \, B b x^{2} - 9 \, A c x^{2} + A b}{3 \, b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^2/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]