3.273 \(\int \frac{1}{\sqrt{x} \sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}} \]

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(a*x^(3/2))

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Rubi [A]  time = 0.0660682, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{2 \sqrt{a x^2+b x^3}}{a x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(a*x^(3/2))

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Rubi in Sympy [A]  time = 7.31027, size = 22, normalized size = 0.88 \[ - \frac{2 \sqrt{a x^{2} + b x^{3}}}{a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a*x**2 + b*x**3)/(a*x**(3/2))

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Mathematica [A]  time = 0.022267, size = 23, normalized size = 0.92 \[ -\frac{2 \sqrt{x^2 (a+b x)}}{a x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x^2*(a + b*x)])/(a*x^(3/2))

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Maple [A]  time = 0.006, size = 27, normalized size = 1.1 \[ -2\,{\frac{\sqrt{x} \left ( bx+a \right ) }{a\sqrt{b{x}^{3}+a{x}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.42923, size = 20, normalized size = 0.8 \[ -\frac{2 \, \sqrt{b x + a}}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

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Fricas [A]  time = 0.227814, size = 28, normalized size = 1.12 \[ -\frac{2 \, \sqrt{b x^{3} + a x^{2}}}{a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x^3 + a*x^2)/(a*x^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219929, size = 41, normalized size = 1.64 \[ \frac{4 \, \sqrt{b}}{{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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