3.365 \(\int \frac{\sqrt{a+b x^2}}{x^4} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

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Rubi [A]  time = 0.0219531, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/x^4,x]

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

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Rubi in Sympy [A]  time = 3.1768, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**2)**(3/2)/(3*a*x**3)

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Mathematica [A]  time = 0.0151685, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]/x^4,x]

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

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Maple [A]  time = 0.006, size = 18, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x^4,x)

[Out]

-1/3*(b*x^2+a)^(3/2)/a/x^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(sqrt(b*x^2 + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234115, size = 23, normalized size = 1.1 \[ -\frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b*x^2 + a)^(3/2)/(a*x^3)

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Sympy [A]  time = 2.0858, size = 42, normalized size = 2. \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211832, size = 80, normalized size = 3.81 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{3}{2}} + a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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