Optimal. Leaf size=77 \[ -\frac{b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{e}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
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Rubi [A] time = 0.180047, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{e}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[(x*(d + e*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.0144, size = 48, normalized size = 0.62 \[ \frac{\left (2 a + 2 b x^{2}\right ) \left (d + e x^{2}\right )^{2}}{8 \left (a e - b d\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x**2+d)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0415956, size = 45, normalized size = 0.58 \[ \frac{-a e-b \left (d+2 e x^2\right )}{4 b^2 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(d + e*x^2))/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 38, normalized size = 0.5 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 2\,{x}^{2}be+ae+bd \right ) }{4\,{b}^{2}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x^2+d)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.709258, size = 96, normalized size = 1.25 \[ -\frac{1}{4} \, e{\left (\frac{2}{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}} b^{2}} - \frac{a}{{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2} b}\right )} - \frac{d}{4 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)*x/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258708, size = 57, normalized size = 0.74 \[ -\frac{2 \, b e x^{2} + b d + a e}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)*x/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (d + e x^{2}\right )}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x**2+d)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.629919, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)*x/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")
[Out]