3.655 \(\int \frac{x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=213 \[ \frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(5*x)/(128*a^3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - x/(8*b*(a + b*x^2)^3*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4]) + x/(48*a*b*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x
^4]) + (5*x)/(192*a^2*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*(a + b
*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(7/2)*b^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])

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Rubi [A]  time = 0.22966, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(5*x)/(128*a^3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - x/(8*b*(a + b*x^2)^3*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4]) + x/(48*a*b*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x
^4]) + (5*x)/(192*a^2*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*(a + b
*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(7/2)*b^(3/2)*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**2/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0682047, size = 105, normalized size = 0.49 \[ \frac{\sqrt{a} \sqrt{b} x \left (-15 a^3+73 a^2 b x^2+55 a b^2 x^4+15 b^3 x^6\right )+15 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{384 a^{7/2} b^{3/2} \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(Sqrt[a]*Sqrt[b]*x*(-15*a^3 + 73*a^2*b*x^2 + 55*a*b^2*x^4 + 15*b^3*x^6) + 15*(a
+ b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(7/2)*b^(3/2)*(a + b*x^2)^3*Sqrt[
(a + b*x^2)^2])

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Maple [A]  time = 0.023, size = 172, normalized size = 0.8 \[{\frac{b{x}^{2}+a}{384\,{a}^{3}b} \left ( 15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{8}{b}^{4}+15\,\sqrt{ab}{x}^{7}{b}^{3}+60\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{6}a{b}^{3}+55\,\sqrt{ab}{x}^{5}a{b}^{2}+90\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{2}+73\,\sqrt{ab}{x}^{3}{a}^{2}b+60\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}b-15\,\sqrt{ab}x{a}^{3}+15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/384*(15*arctan(x*b/(a*b)^(1/2))*x^8*b^4+15*(a*b)^(1/2)*x^7*b^3+60*arctan(x*b/(
a*b)^(1/2))*x^6*a*b^3+55*(a*b)^(1/2)*x^5*a*b^2+90*arctan(x*b/(a*b)^(1/2))*x^4*a^
2*b^2+73*(a*b)^(1/2)*x^3*a^2*b+60*arctan(x*b/(a*b)^(1/2))*x^2*a^3*b-15*(a*b)^(1/
2)*x*a^3+15*arctan(x*b/(a*b)^(1/2))*a^4)*(b*x^2+a)/(a*b)^(1/2)/b/a^3/((b*x^2+a)^
2)^(5/2)

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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^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270827, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (15 \, b^{3} x^{7} + 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} - 15 \, a^{3} x\right )} \sqrt{-a b}}{768 \,{\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \sqrt{-a b}}, \frac{15 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (15 \, b^{3} x^{7} + 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} - 15 \, a^{3} x\right )} \sqrt{a b}}{384 \,{\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/768*(15*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*log((2*a*
b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(15*b^3*x^7 + 55*a*b^2*x^5 + 73*a
^2*b*x^3 - 15*a^3*x)*sqrt(-a*b))/((a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 +
4*a^6*b^2*x^2 + a^7*b)*sqrt(-a*b)), 1/384*(15*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^
2*x^4 + 4*a^3*b*x^2 + a^4)*arctan(sqrt(a*b)*x/a) + (15*b^3*x^7 + 55*a*b^2*x^5 +
73*a^2*b*x^3 - 15*a^3*x)*sqrt(a*b))/((a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^
4 + 4*a^6*b^2*x^2 + a^7*b)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**2/((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.612447, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x