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

Optimal. Leaf size=44 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{b x+2}} \]

[Out]

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

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Rubi [A]  time = 0.0325928, antiderivative size = 44, 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 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(2 + b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.46941, size = 41, normalized size = 0.93 \[ - \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} + \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0324645, size = 44, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{2 \sqrt{x}}{b \sqrt{b x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.114, size = 48, normalized size = 1.1 \[ 2\,{\frac{1}{{b}^{3/2}\sqrt{\pi }} \left ( -1/2\,{\frac{\sqrt{\pi }\sqrt{x}\sqrt{2}\sqrt{b}}{\sqrt{1/2\,bx+1}}}+\sqrt{\pi }{\it Arcsinh} \left ( 1/2\,\sqrt{b}\sqrt{x}\sqrt{2} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x+2)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220105, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + 2} \sqrt{x} \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right ) - 2 \, \sqrt{b} x}{\sqrt{b x + 2} b^{\frac{3}{2}} \sqrt{x}}, \frac{2 \,{\left (\sqrt{b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) - \sqrt{-b} x\right )}}{\sqrt{b x + 2} \sqrt{-b} b \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.95379, size = 41, normalized size = 0.93 \[ - \frac{2 \sqrt{x}}{b \sqrt{b x + 2}} + \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222615, size = 111, normalized size = 2.52 \[ -\frac{{\left (\frac{{\rm ln}\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{\sqrt{b}} + \frac{8 \, \sqrt{b}}{{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b}\right )}{\left | b \right |}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(ln((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2)/sqrt(b) + 8*sqrt(b)/((
sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b))*abs(b)/b^2