Optimal. Leaf size=110 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}+\frac{b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac{b x (3 b B-2 A c)}{c^4}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{B x^5}{5 c^2} \]
[Out]
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Rubi [A] time = 0.245605, antiderivative size = 110, 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 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}+\frac{b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac{b x (3 b B-2 A c)}{c^4}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{B x^5}{5 c^2} \]
Antiderivative was successfully verified.
[In] Int[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 49.5373, size = 102, normalized size = 0.93 \[ \frac{B x^{5}}{5 c^{2}} + \frac{b^{\frac{3}{2}} \left (5 A c - 7 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{2 c^{\frac{9}{2}}} - \frac{b^{2} x \left (A c - B b\right )}{2 c^{4} \left (b + c x^{2}\right )} - \frac{b x \left (2 A c - 3 B b\right )}{c^{4}} + \frac{x^{3} \left (A c - 2 B b\right )}{3 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.176229, size = 111, normalized size = 1.01 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}-\frac{x \left (A b^2 c-b^3 B\right )}{2 c^4 \left (b+c x^2\right )}+\frac{b x (3 b B-2 A c)}{c^4}+\frac{x^3 (A c-2 b B)}{3 c^3}+\frac{B x^5}{5 c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.013, size = 132, normalized size = 1.2 \[{\frac{B{x}^{5}}{5\,{c}^{2}}}+{\frac{A{x}^{3}}{3\,{c}^{2}}}-{\frac{2\,B{x}^{3}b}{3\,{c}^{3}}}-2\,{\frac{Abx}{{c}^{3}}}+3\,{\frac{xB{b}^{2}}{{c}^{4}}}-{\frac{A{b}^{2}x}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{Bx{b}^{3}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}+{\frac{5\,{b}^{2}A}{2\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{7\,B{b}^{3}}{2\,{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^10*(B*x^2+A)/(c*x^4+b*x^2)^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^10/(c*x^4 + b*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224261, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, B c^{3} x^{7} - 4 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 20 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} - 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 30 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} x}{60 \,{\left (c^{5} x^{2} + b c^{4}\right )}}, \frac{6 \, B c^{3} x^{7} - 2 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 10 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} - 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) + 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} x}{30 \,{\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^10/(c*x^4 + b*x^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.62263, size = 206, normalized size = 1.87 \[ \frac{B x^{5}}{5 c^{2}} + \frac{x \left (- A b^{2} c + B b^{3}\right )}{2 b c^{4} + 2 c^{5} x^{2}} + \frac{\sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right )}{- 5 A b c + 7 B b^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right )}{- 5 A b c + 7 B b^{2}} + x \right )}}{4} - \frac{x^{3} \left (- A c + 2 B b\right )}{3 c^{3}} + \frac{x \left (- 2 A b c + 3 B b^{2}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210145, size = 155, normalized size = 1.41 \[ -\frac{{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{4}} + \frac{B b^{3} x - A b^{2} c x}{2 \,{\left (c x^{2} + b\right )} c^{4}} + \frac{3 \, B c^{8} x^{5} - 10 \, B b c^{7} x^{3} + 5 \, A c^{8} x^{3} + 45 \, B b^{2} c^{6} x - 30 \, A b c^{7} x}{15 \, c^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^10/(c*x^4 + b*x^2)^2,x, algorithm="giac")
[Out]