3.610 \(\int \frac{\sqrt{x}}{\sqrt{2+b x}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

[Out]

(Sqrt[x]*Sqrt[2 + b*x])/b - (2*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(3/2)

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Rubi [A]  time = 0.0329957, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[2 + b*x],x]

[Out]

(Sqrt[x]*Sqrt[2 + b*x])/b - (2*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(3/2)

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Rubi in Sympy [A]  time = 4.93574, size = 39, normalized size = 0.91 \[ \frac{\sqrt{x} \sqrt{b x + 2}}{b} - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*sqrt(b*x + 2)/b - 2*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2)

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Mathematica [A]  time = 0.0288893, size = 43, normalized size = 1. \[ \frac{\sqrt{x} \sqrt{b x+2}}{b}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[2 + b*x])/b - (2*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(3/2)

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Maple [A]  time = 0.006, size = 62, normalized size = 1.4 \[{\frac{1}{b}\sqrt{x}\sqrt{bx+2}}-{1\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x+2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218553, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + 2} \sqrt{b} \sqrt{x} + \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{b^{\frac{3}{2}}}, \frac{\sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{\sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(b*x + 2)*sqrt(b)*sqrt(x) + log(-sqrt(b*x + 2)*b*sqrt(x) + (b*x + 1)*sqrt(
b)))/b^(3/2), (sqrt(b*x + 2)*sqrt(-b)*sqrt(x) - 2*arctan(sqrt(b*x + 2)*sqrt(-b)/
(b*sqrt(x))))/(sqrt(-b)*b)]

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Sympy [A]  time = 6.75506, size = 54, normalized size = 1.26 \[ \frac{x^{\frac{3}{2}}}{\sqrt{b x + 2}} + \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(3/2)/sqrt(b*x + 2) + 2*sqrt(x)/(b*sqrt(b*x + 2)) - 2*asinh(sqrt(2)*sqrt(b)*s
qrt(x)/2)/b**(3/2)

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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(sqrt(x)/sqrt(b*x + 2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError