Optimal. Leaf size=37 \[ -\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.197311, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A b-x^2 (b B-2 A c)}{b^2 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 14.7607, size = 37, normalized size = 1. \[ - \frac{2 A b + x^{2} \left (4 A c - 2 B b\right )}{2 b^{2} \sqrt{b x^{2} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0582011, size = 37, normalized size = 1. \[ \frac{b B x^2-A \left (b+2 c x^2\right )}{b^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 47, normalized size = 1.3 \[ -{\frac{ \left ( c{x}^{2}+b \right ){x}^{2} \left ( 2\,A{x}^{2}c-Bb{x}^{2}+Ab \right ) }{{b}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.39912, size = 88, normalized size = 2.38 \[ -A{\left (\frac{2 \, c x^{2}}{\sqrt{c x^{4} + b x^{2}} b^{2}} + \frac{1}{\sqrt{c x^{4} + b x^{2}} b}\right )} + \frac{B x^{2}}{\sqrt{c x^{4} + b x^{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227884, size = 66, normalized size = 1.78 \[ \frac{\sqrt{c x^{4} + b x^{2}}{\left ({\left (B b - 2 \, A c\right )} x^{2} - A b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232652, size = 49, normalized size = 1.32 \[ \frac{\frac{{\left (B b - 2 \, A c\right )} x^{2}}{b^{2}} - \frac{A}{b}}{\sqrt{c x^{4} + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")
[Out]