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

Optimal. Leaf size=48 \[ -\frac{2 b c x \sqrt{\frac{c}{a+b x^2}}}{a^2}-\frac{c \sqrt{\frac{c}{a+b x^2}}}{a x} \]

[Out]

-((c*Sqrt[c/(a + b*x^2)])/(a*x)) - (2*b*c*x*Sqrt[c/(a + b*x^2)])/a^2

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Rubi [A]  time = 0.192181, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 b c x \sqrt{\frac{c}{a+b x^2}}}{a^2}-\frac{c \sqrt{\frac{c}{a+b x^2}}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-((c*Sqrt[c/(a + b*x^2)])/(a*x)) - (2*b*c*x*Sqrt[c/(a + b*x^2)])/a^2

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Rubi in Sympy [A]  time = 6.2987, size = 39, normalized size = 0.81 \[ - \frac{c \sqrt{\frac{c}{a + b x^{2}}}}{a x} - \frac{2 b c x \sqrt{\frac{c}{a + b x^{2}}}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*sqrt(c/(a + b*x**2))/(a*x) - 2*b*c*x*sqrt(c/(a + b*x**2))/a**2

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Mathematica [A]  time = 0.0244448, size = 32, normalized size = 0.67 \[ -\frac{c \left (a+2 b x^2\right ) \sqrt{\frac{c}{a+b x^2}}}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

-((c*Sqrt[c/(a + b*x^2)]*(a + 2*b*x^2))/(a^2*x))

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Maple [A]  time = 0.007, size = 37, normalized size = 0.8 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 2\,b{x}^{2}+a \right ) }{{a}^{2}x} \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x^2+a))^(3/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.685968, size = 62, normalized size = 1.29 \[ -\frac{2 \, b^{2} c^{\frac{3}{2}} x^{4} + 3 \, a b c^{\frac{3}{2}} x^{2} + a^{2} c^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.278843, size = 43, normalized size = 0.9 \[ -\frac{{\left (2 \, b c x^{2} + a c\right )} \sqrt{\frac{c}{b x^{2} + a}}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.296351, size = 109, normalized size = 2.27 \[ -{\left (\frac{b c x{\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c x^{2} + a c} a^{2}} - \frac{2 \, \sqrt{b c} c{\rm sign}\left (b x^{2} + a\right )}{{\left ({\left (\sqrt{b c} x - \sqrt{b c x^{2} + a c}\right )}^{2} - a c\right )} a}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*c*x*sign(b*x^2 + a)/(sqrt(b*c*x^2 + a*c)*a^2) - 2*sqrt(b*c)*c*sign(b*x^2 + a
)/(((sqrt(b*c)*x - sqrt(b*c*x^2 + a*c))^2 - a*c)*a))*c