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

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]

(-3*Sqrt[x]*Sqrt[2 + b*x])/(2*b^2) + (x^(3/2)*Sqrt[2 + b*x])/(2*b) + (3*ArcSinh[
(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(5/2)

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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]

(-3*Sqrt[x]*Sqrt[2 + b*x])/(2*b^2) + (x^(3/2)*Sqrt[2 + b*x])/(2*b) + (3*ArcSinh[
(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(5/2)

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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]

x**(3/2)*sqrt(b*x + 2)/(2*b) - 3*sqrt(x)*sqrt(b*x + 2)/(2*b**2) + 3*asinh(sqrt(2
)*sqrt(b)*sqrt(x)/2)/b**(5/2)

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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]

(Sqrt[x]*(-3 + b*x)*Sqrt[2 + b*x])/(2*b^2) + (3*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2
]])/b^(5/2)

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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]

1/2*x^(3/2)*(b*x+2)^(1/2)/b-3/2*x^(1/2)*(b*x+2)^(1/2)/b^2+3/2/b^(5/2)*(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(x^(3/2)/sqrt(b*x + 2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

[1/2*(sqrt(b*x + 2)*(b*x - 3)*sqrt(b)*sqrt(x) + 3*log(sqrt(b*x + 2)*b*sqrt(x) +
(b*x + 1)*sqrt(b)))/b^(5/2), 1/2*(sqrt(b*x + 2)*(b*x - 3)*sqrt(-b)*sqrt(x) + 6*a
rctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^2)]

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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]

x**(5/2)/(2*sqrt(b*x + 2)) - x**(3/2)/(2*b*sqrt(b*x + 2)) - 3*sqrt(x)/(b**2*sqrt
(b*x + 2)) + 3*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(5/2)

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

[Out]

Exception raised: NotImplementedError