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

Optimal. Leaf size=34 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{b}} \]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a*x^2 + b*x^3]])/Sqrt[b]

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Rubi [A]  time = 0.0773505, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a*x^2 + b*x^3]])/Sqrt[b]

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Rubi in Sympy [A]  time = 8.40381, size = 31, normalized size = 0.91 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a x^{2} + b x^{3}}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(sqrt(b)*x**(3/2)/sqrt(a*x**2 + b*x**3))/sqrt(b)

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Mathematica [A]  time = 0.0224151, size = 54, normalized size = 1.59 \[ \frac{2 x \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b} \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 58, normalized size = 1.7 \[{1\sqrt{x}\sqrt{x \left ( bx+a \right ) }\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225469, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, \sqrt{b x^{3} + a x^{2}} b \sqrt{x} +{\left (2 \, b x^{2} + a x\right )} \sqrt{b}}{x}\right )}{\sqrt{b}}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-b}}{b x^{\frac{3}{2}}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[log((2*sqrt(b*x^3 + a*x^2)*b*sqrt(x) + (2*b*x^2 + a*x)*sqrt(b))/x)/sqrt(b), -2*
sqrt(-b)*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-b)/(b*x^(3/2)))/b]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223977, size = 31, normalized size = 0.91 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*ln(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/sqrt(b)