3.82 \(\int \frac{x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

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]

-e/(2*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (b*d - a*e)/(4*b^2*(a + b*x^2)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])

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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]

-e/(2*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (b*d - a*e)/(4*b^2*(a + b*x^2)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])

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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]

(2*a + 2*b*x**2)*(d + e*x**2)**2/(8*(a*e - b*d)*(a**2 + 2*a*b*x**2 + b**2*x**4)*
*(3/2))

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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]

(-(a*e) - b*(d + 2*e*x^2))/(4*b^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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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]

-1/4*(b*x^2+a)*(2*b*e*x^2+a*e+b*d)/b^2/((b*x^2+a)^2)^(3/2)

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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]

-1/4*e*(2/(sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)*b^2) - a/((b^2)^(3/2)*(x^2 + a/b)^2*b
)) - 1/4*d/((b^2)^(3/2)*(x^2 + a/b)^2)

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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]

-1/4*(2*b*e*x^2 + b*d + a*e)/(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)

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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]

Integral(x*(d + e*x**2)/((a + b*x**2)**2)**(3/2), x)

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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]

sage0*x