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

Optimal. Leaf size=169 \[ \frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac{15 \sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 \left (a+b x^2\right )}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

5/(8*a^2*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*x*(a + b*x^2)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4]) - (15*(a + b*x^2))/(8*a^3*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (15*Sqrt[b]*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.185066, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac{15 \sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 \left (a+b x^2\right )}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

5/(8*a^2*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*x*(a + b*x^2)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4]) - (15*(a + b*x^2))/(8*a^3*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (15*Sqrt[b]*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])

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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/x**2/(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 = 0.0631704, size = 93, normalized size = 0.55 \[ \frac{-\sqrt{a} \left (8 a^2+25 a b x^2+15 b^2 x^4\right )-15 \sqrt{b} x \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} x \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[a]*(8*a^2 + 25*a*b*x^2 + 15*b^2*x^4)) - 15*Sqrt[b]*x*(a + b*x^2)^2*ArcTa
n[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*x*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.024, size = 119, normalized size = 0.7 \[ -{\frac{b{x}^{2}+a}{8\,{a}^{3}x} \left ( 15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{5}{b}^{3}+15\,\sqrt{ab}{x}^{4}{b}^{2}+30\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}a{b}^{2}+25\,\sqrt{ab}{x}^{2}ab+15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) x{a}^{2}b+8\,\sqrt{ab}{a}^{2} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(15*arctan(x*b/(a*b)^(1/2))*x^5*b^3+15*(a*b)^(1/2)*x^4*b^2+30*arctan(x*b/(a
*b)^(1/2))*x^3*a*b^2+25*(a*b)^(1/2)*x^2*a*b+15*arctan(x*b/(a*b)^(1/2))*x*a^2*b+8
*(a*b)^(1/2)*a^2)*(b*x^2+a)/x/(a*b)^(1/2)/a^3/((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(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273984, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} x^{4} + 50 \, a b x^{2} - 15 \,{\left (b^{2} x^{5} + 2 \, a b x^{3} + a^{2} x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 16 \, a^{2}}{16 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}, -\frac{15 \, b^{2} x^{4} + 25 \, a b x^{2} + 15 \,{\left (b^{2} x^{5} + 2 \, a b x^{3} + a^{2} x\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 8 \, a^{2}}{8 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}\right ] \]

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^2),x, algorithm="fricas")

[Out]

[-1/16*(30*b^2*x^4 + 50*a*b*x^2 - 15*(b^2*x^5 + 2*a*b*x^3 + a^2*x)*sqrt(-b/a)*lo
g((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 16*a^2)/(a^3*b^2*x^5 + 2*a^4*b*x
^3 + a^5*x), -1/8*(15*b^2*x^4 + 25*a*b*x^2 + 15*(b^2*x^5 + 2*a*b*x^3 + a^2*x)*sq
rt(b/a)*arctan(b*x/(a*sqrt(b/a))) + 8*a^2)/(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*((a + b*x**2)**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.617207, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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^2),x, algorithm="giac")

[Out]

sage0*x