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

Optimal. Leaf size=128 \[ -\frac{3 x}{8 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x^3}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.151702, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{3 x}{8 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x^3}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*x)/(8*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - x^3/(4*b*(a + b*x^2)*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) + (3*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*
b^(5/2)*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(x**4/(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 = 0.0607869, size = 84, normalized size = 0.66 \[ \frac{3 \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{a} \sqrt{b} x \left (3 a+5 b x^2\right )}{8 \sqrt{a} b^{5/2} \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 97, normalized size = 0.8 \[ -{\frac{b{x}^{2}+a}{8\,{b}^{2}} \left ( -3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{b}^{2}+5\,\sqrt{ab}{x}^{3}b-6\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}ab+3\,\sqrt{ab}xa-3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(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+5*(a*b)^(1/2)*x^3*b-6*arctan(x*b/(a*b)^
(1/2))*x^2*a*b+3*(a*b)^(1/2)*x*a-3*a^2*arctan(x*b/(a*b)^(1/2)))*(b*x^2+a)/(a*b)^
(1/2)/b^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(x^4/(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 = 0.268007, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (5 \, b x^{3} + 3 \, a x\right )} \sqrt{-a b}}{16 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{-a b}}, \frac{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (5 \, b x^{3} + 3 \, a x\right )} \sqrt{a b}}{8 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.694764, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x