3.656 \(\int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=213 \[ \frac{7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \]

[Out]

(x*(a + b*x^2))/(8*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (7*x*(a + b*x^2)^2)/(4
8*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*x*(a + b*x^2)^3)/(192*a^3*(a^2 +
2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*x*(a + b*x^2)^4)/(128*a^4*(a^2 + 2*a*b*x^2 + b
^2*x^4)^(5/2)) + (35*(a + b*x^2)^5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(9/2)*Sqr
t[b]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))

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Rubi [A]  time = 0.1777, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^2))/(8*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (7*x*(a + b*x^2)^2)/(4
8*a^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*x*(a + b*x^2)^3)/(192*a^3*(a^2 +
2*a*b*x^2 + b^2*x^4)^(5/2)) + (35*x*(a + b*x^2)^4)/(128*a^4*(a^2 + 2*a*b*x^2 + b
^2*x^4)^(5/2)) + (35*(a + b*x^2)^5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(9/2)*Sqr
t[b]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))

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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)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0725017, size = 105, normalized size = 0.49 \[ \frac{\sqrt{a} \sqrt{b} x \left (279 a^3+511 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right )+105 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{384 a^{9/2} \sqrt{b} \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[b]*x*(279*a^3 + 511*a^2*b*x^2 + 385*a*b^2*x^4 + 105*b^3*x^6) + 105
*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(9/2)*Sqrt[b]*(a + b*x^2)^3*S
qrt[(a + b*x^2)^2])

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Maple [A]  time = 0.012, size = 169, normalized size = 0.8 \[{\frac{b{x}^{2}+a}{384\,{a}^{4}} \left ( 105\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{8}{b}^{4}+105\,\sqrt{ab}{x}^{7}{b}^{3}+420\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{6}a{b}^{3}+385\,\sqrt{ab}{x}^{5}a{b}^{2}+630\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{2}+511\,\sqrt{ab}{x}^{3}{a}^{2}b+420\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}b+279\,\sqrt{ab}x{a}^{3}+105\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/384*(105*arctan(x*b/(a*b)^(1/2))*x^8*b^4+105*(a*b)^(1/2)*x^7*b^3+420*arctan(x*
b/(a*b)^(1/2))*x^6*a*b^3+385*(a*b)^(1/2)*x^5*a*b^2+630*arctan(x*b/(a*b)^(1/2))*x
^4*a^2*b^2+511*(a*b)^(1/2)*x^3*a^2*b+420*arctan(x*b/(a*b)^(1/2))*x^2*a^3*b+279*(
a*b)^(1/2)*x*a^3+105*arctan(x*b/(a*b)^(1/2))*a^4)*(b*x^2+a)/(a*b)^(1/2)/a^4/((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((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 = 0.269975, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (105 \, b^{3} x^{7} + 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} + 279 \, a^{3} x\right )} \sqrt{-a b}}{768 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \sqrt{-a b}}, \frac{105 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (105 \, b^{3} x^{7} + 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} + 279 \, a^{3} x\right )} \sqrt{a b}}{384 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(105*(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*(105*b^3*x^7 + 385*a*b^2*x^5 + 5
11*a^2*b*x^3 + 279*a^3*x)*sqrt(-a*b))/((a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*
x^4 + 4*a^7*b*x^2 + a^8)*sqrt(-a*b)), 1/384*(105*(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) + (105*b^3*x^7 + 385*a*b^2*x^
5 + 511*a^2*b*x^3 + 279*a^3*x)*sqrt(a*b))/((a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*
b^2*x^4 + 4*a^7*b*x^2 + a^8)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(-5/2), x)

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GIAC/XCAS [A]  time = 0.60368, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x