Optimal. Leaf size=67 \[ \frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{3 \sqrt{x} \sqrt{b x+2}}{2 b^2}+\frac{x^{3/2} \sqrt{b x+2}}{2 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.047697, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}-\frac{3 \sqrt{x} \sqrt{b x+2}}{2 b^2}+\frac{x^{3/2} \sqrt{b x+2}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/Sqrt[2 + b*x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.11934, size = 61, normalized size = 0.91 \[ \frac{x^{\frac{3}{2}} \sqrt{b x + 2}}{2 b} - \frac{3 \sqrt{x} \sqrt{b x + 2}}{2 b^{2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(b*x+2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.051179, size = 51, normalized size = 0.76 \[ \frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{\sqrt{x} \sqrt{b x+2} (b x-3)}{2 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/Sqrt[2 + b*x],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 78, normalized size = 1.2 \[{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}-{\frac{3}{2\,{b}^{2}}\sqrt{x}\sqrt{bx+2}}+{\frac{3}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(b*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(x^(3/2)/sqrt(b*x + 2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.232883, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{b x + 2}{\left (b x - 3\right )} \sqrt{b} \sqrt{x} + 3 \, \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{2 \, b^{\frac{5}{2}}}, \frac{\sqrt{b x + 2}{\left (b x - 3\right )} \sqrt{-b} \sqrt{x} + 6 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{2 \, \sqrt{-b} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x + 2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 15.0076, size = 75, normalized size = 1.12 \[ \frac{x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} - \frac{x^{\frac{3}{2}}}{2 b \sqrt{b x + 2}} - \frac{3 \sqrt{x}}{b^{2} \sqrt{b x + 2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(b*x+2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x + 2),x, algorithm="giac")
[Out]