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

Optimal. Leaf size=64 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0450904, antiderivative size = 64, 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 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[2 + b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.4283, size = 56, normalized size = 0.88 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{2 b} - \frac{\sqrt{x} \sqrt{b x + 2}}{2 b} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

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

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Mathematica [A]  time = 0.0440559, size = 51, normalized size = 0.8 \[ \frac{\sqrt{x} (b x+1) \sqrt{b x+2}}{2 b}-\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(1 + b*x)*Sqrt[2 + b*x])/(2*b) - ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]/b^(
3/2)

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Maple [A]  time = 0.02, size = 75, normalized size = 1.2 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}+{\frac{1}{2\,b}\sqrt{x}\sqrt{bx+2}}-{\frac{1}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(b*x+2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223674, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + 2}{\left (b x + 1\right )} \sqrt{b} \sqrt{x} + \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{2 \, b^{\frac{3}{2}}}, \frac{\sqrt{b x + 2}{\left (b x + 1\right )} \sqrt{-b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{2 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.69718, size = 71, normalized size = 1.11 \[ \frac{b x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} + \frac{3 x^{\frac{3}{2}}}{2 \sqrt{b x + 2}} + \frac{\sqrt{x}}{b \sqrt{b x + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**(5/2)/(2*sqrt(b*x + 2)) + 3*x**(3/2)/(2*sqrt(b*x + 2)) + sqrt(x)/(b*sqrt(b*
x + 2)) - asinh(sqrt(2)*sqrt(b)*sqrt(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(b*x + 2)*sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError