Optimal. Leaf size=90 \[ \frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{3/2} c^{5/2}}-\frac{x (5 b B-A c)}{8 b c^2 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 c^2 \left (b+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.180611, antiderivative size = 90, 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 \[ \frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{3/2} c^{5/2}}-\frac{x (5 b B-A c)}{8 b c^2 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 c^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 23.6155, size = 78, normalized size = 0.87 \[ - \frac{x \left (A c - B b\right )}{4 c^{2} \left (b + c x^{2}\right )^{2}} + \frac{x \left (A c - 5 B b\right )}{8 b c^{2} \left (b + c x^{2}\right )} + \frac{\left (A c + 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{3}{2}} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.137071, size = 83, normalized size = 0.92 \[ \frac{\frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\sqrt{c} x \left (-b c \left (A+5 B x^2\right )+A c^2 x^2-3 b^2 B\right )}{b \left (b+c x^2\right )^2}}{8 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.013, size = 89, normalized size = 1. \[{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ({\frac{ \left ( Ac-5\,Bb \right ){x}^{3}}{8\,bc}}-{\frac{ \left ( Ac+3\,Bb \right ) x}{8\,{c}^{2}}} \right ) }+{\frac{A}{8\,bc}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(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^8/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223054, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 3 \, B b^{3} + A b^{2} c + 2 \,{\left (3 \, B b^{2} c + A b c^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, b c x +{\left (c x^{2} - b\right )} \sqrt{-b c}}{c x^{2} + b}\right ) - 2 \,{\left ({\left (5 \, B b c - A c^{2}\right )} x^{3} +{\left (3 \, B b^{2} + A b c\right )} x\right )} \sqrt{-b c}}{16 \,{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \sqrt{-b c}}, \frac{{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 3 \, B b^{3} + A b^{2} c + 2 \,{\left (3 \, B b^{2} c + A b c^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) -{\left ({\left (5 \, B b c - A c^{2}\right )} x^{3} +{\left (3 \, B b^{2} + A b c\right )} x\right )} \sqrt{b c}}{8 \,{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \sqrt{b c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^8/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.70804, size = 153, normalized size = 1.7 \[ - \frac{\sqrt{- \frac{1}{b^{3} c^{5}}} \left (A c + 3 B b\right ) \log{\left (- b^{2} c^{2} \sqrt{- \frac{1}{b^{3} c^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{b^{3} c^{5}}} \left (A c + 3 B b\right ) \log{\left (b^{2} c^{2} \sqrt{- \frac{1}{b^{3} c^{5}}} + x \right )}}{16} - \frac{x^{3} \left (- A c^{2} + 5 B b c\right ) + x \left (A b c + 3 B b^{2}\right )}{8 b^{3} c^{2} + 16 b^{2} c^{3} x^{2} + 8 b c^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.217001, size = 105, normalized size = 1.17 \[ \frac{{\left (3 \, B b + A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b c^{2}} - \frac{5 \, B b c x^{3} - A c^{2} x^{3} + 3 \, B b^{2} x + A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^8/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]