3.226 \(\int \frac{\sqrt{b x^2+c x^4}}{x^5} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(3*b*x^6)

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Rubi [A]  time = 0.0672054, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^5,x]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(3*b*x^6)

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Rubi in Sympy [A]  time = 7.73192, size = 20, normalized size = 0.8 \[ - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 b x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(1/2)/x**5,x)

[Out]

-(b*x**2 + c*x**4)**(3/2)/(3*b*x**6)

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Mathematica [A]  time = 0.0248192, size = 25, normalized size = 1. \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 b x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2 + c*x^4]/x^5,x]

[Out]

-(x^2*(b + c*x^2))^(3/2)/(3*b*x^6)

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Maple [A]  time = 0.006, size = 29, normalized size = 1.2 \[ -{\frac{c{x}^{2}+b}{3\,b{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^5,x)

[Out]

-1/3/x^4*(c*x^2+b)/b*(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(sqrt(c*x^4 + b*x^2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262214, size = 38, normalized size = 1.52 \[ -\frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{3 \, b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**5, x)

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GIAC/XCAS [A]  time = 0.287196, size = 85, normalized size = 3.4 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} c^{\frac{3}{2}}{\rm sign}\left (x\right ) + b^{2} c^{\frac{3}{2}}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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