3.642 \(\int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{3 x \left (a+b x^2\right )^2}{8 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{3 \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]

[Out]

(x*(a + b*x^2))/(4*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + (3*x*(a + b*x^2)^2)/(8
*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + (3*(a + b*x^2)^3*ArcTan[(Sqrt[b]*x)/Sq
rt[a]])/(8*a^(5/2)*Sqrt[b]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))

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Rubi [A]  time = 0.101372, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{3 x \left (a+b x^2\right )^2}{8 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x \left (a+b x^2\right )}{4 a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{3 \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3/2),x]

[Out]

(x*(a + b*x^2))/(4*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + (3*x*(a + b*x^2)^2)/(8
*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)) + (3*(a + b*x^2)^3*ArcTan[(Sqrt[b]*x)/Sq
rt[a]])/(8*a^(5/2)*Sqrt[b]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0485606, size = 83, normalized size = 0.61 \[ \frac{\sqrt{a} \sqrt{b} x \left (5 a+3 b x^2\right )+3 \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-3/2),x]

[Out]

(Sqrt[a]*Sqrt[b]*x*(5*a + 3*b*x^2) + 3*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]
)/(8*a^(5/2)*Sqrt[b]*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.011, size = 97, normalized size = 0.7 \[{\frac{b{x}^{2}+a}{8\,{a}^{2}} \left ( 3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{b}^{2}+3\,\sqrt{ab}{x}^{3}b+6\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}ab+5\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/8*(3*arctan(x*b/(a*b)^(1/2))*x^4*b^2+3*(a*b)^(1/2)*x^3*b+6*arctan(x*b/(a*b)^(1
/2))*x^2*a*b+5*(a*b)^(1/2)*x*a+3*a^2*arctan(x*b/(a*b)^(1/2)))*(b*x^2+a)/(a*b)^(1
/2)/a^2/((b*x^2+a)^2)^(3/2)

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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^2*x^4 + 2*a*b*x^2 + a^2)^(-3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269753, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \, b x^{3} + 5 \, a x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-a b}}, \frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, b x^{3} + 5 \, a x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-3/2),x, algorithm="fricas")

[Out]

[1/16*(3*(b^2*x^4 + 2*a*b*x^2 + a^2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x
^2 + a)) + 2*(3*b*x^3 + 5*a*x)*sqrt(-a*b))/((a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*sq
rt(-a*b)), 1/8*(3*(b^2*x^4 + 2*a*b*x^2 + a^2)*arctan(sqrt(a*b)*x/a) + (3*b*x^3 +
5*a*x)*sqrt(a*b))/((a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.622283, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-3/2),x, algorithm="giac")

[Out]

sage0*x