3.265 \(\int \frac{1}{x^3 \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}-\frac{\sqrt{b x^2+c x^4}}{3 b x^4} \]

[Out]

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

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Rubi [A]  time = 0.141434, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 c \sqrt{b x^2+c x^4}}{3 b^2 x^2}-\frac{\sqrt{b x^2+c x^4}}{3 b x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.284, size = 44, normalized size = 0.85 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{3 b x^{4}} + \frac{2 c \sqrt{b x^{2} + c x^{4}}}{3 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0375158, size = 35, normalized size = 0.67 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (2 c x^2-b\right )}{3 b^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 37, normalized size = 0.7 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -2\,c{x}^{2}+b \right ) }{3\,{b}^{2}{x}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264754, size = 42, normalized size = 0.81 \[ \frac{\sqrt{c x^{4} + b x^{2}}{\left (2 \, c x^{2} - b\right )}}{3 \, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(x**2*(b + c*x**2))), x)

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GIAC/XCAS [A]  time = 0.278295, size = 36, normalized size = 0.69 \[ -\frac{{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{c + \frac{b}{x^{2}}} c}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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