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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]