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

Optimal. Leaf size=66 \[ \frac{8 b^2 x}{3 a^3 \sqrt{a+b x^2}}+\frac{4 b}{3 a^2 x \sqrt{a+b x^2}}-\frac{1}{3 a x^3 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0592301, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{8 b^2 x}{3 a^3 \sqrt{a+b x^2}}+\frac{4 b}{3 a^2 x \sqrt{a+b x^2}}-\frac{1}{3 a x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.98637, size = 60, normalized size = 0.91 \[ - \frac{1}{3 a x^{3} \sqrt{a + b x^{2}}} + \frac{4 b}{3 a^{2} x \sqrt{a + b x^{2}}} + \frac{8 b^{2} x}{3 a^{3} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a*x**3*sqrt(a + b*x**2)) + 4*b/(3*a**2*x*sqrt(a + b*x**2)) + 8*b**2*x/(3*a
**3*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.0315122, size = 42, normalized size = 0.64 \[ \frac{-a^2+4 a b x^2+8 b^2 x^4}{3 a^3 x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 37, normalized size = 0.6 \[ -{\frac{-8\,{b}^{2}{x}^{4}-4\,ab{x}^{2}+{a}^{2}}{3\,{a}^{3}{x}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^(3/2),x)

[Out]

-1/3*(-8*b^2*x^4-4*a*b*x^2+a^2)/x^3/(b*x^2+a)^(1/2)/a^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229318, size = 68, normalized size = 1.03 \[ \frac{{\left (8 \, b^{2} x^{4} + 4 \, a b x^{2} - a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*x^4),x, algorithm="fricas")

[Out]

1/3*(8*b^2*x^4 + 4*a*b*x^2 - a^2)*sqrt(b*x^2 + a)/(a^3*b*x^5 + a^4*x^3)

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Sympy [A]  time = 4.23255, size = 233, normalized size = 3.53 \[ - \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{12 a b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{8 b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3*b**(9/2)*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**
3*b**6*x**6) + 3*a**2*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*
a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 12*a*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(3
*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 8*b**(15/2)*x**6*sqrt(a
/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6)

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GIAC/XCAS [A]  time = 0.223298, size = 143, normalized size = 2.17 \[ \frac{b^{2} x}{\sqrt{b x^{2} + a} a^{3}} - \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{3}{2}} - 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a b^{\frac{3}{2}} + 5 \, a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*x^4),x, algorithm="giac")

[Out]

b^2*x/(sqrt(b*x^2 + a)*a^3) - 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(3/2) - 1
2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2) + 5*a^2*b^(3/2))/(((sqrt(b)*x - sqrt
(b*x^2 + a))^2 - a)^3*a^2)