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

Optimal. Leaf size=58 \[ -\frac{2 (b x+2)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{b x+2}+6 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

[Out]

3*b*Sqrt[x]*Sqrt[2 + b*x] - (2*(2 + b*x)^(3/2))/Sqrt[x] + 6*Sqrt[b]*ArcSinh[(Sqr
t[b]*Sqrt[x])/Sqrt[2]]

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Rubi [A]  time = 0.0408551, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 (b x+2)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{b x+2}+6 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + b*x)^(3/2)/x^(3/2),x]

[Out]

3*b*Sqrt[x]*Sqrt[2 + b*x] - (2*(2 + b*x)^(3/2))/Sqrt[x] + 6*Sqrt[b]*ArcSinh[(Sqr
t[b]*Sqrt[x])/Sqrt[2]]

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Rubi in Sympy [A]  time = 6.71353, size = 56, normalized size = 0.97 \[ 6 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + 3 b \sqrt{x} \sqrt{b x + 2} - \frac{2 \left (b x + 2\right )^{\frac{3}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0481059, size = 45, normalized size = 0.78 \[ \frac{\sqrt{b x+2} (b x-4)}{\sqrt{x}}+6 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-4 + b*x)*Sqrt[2 + b*x])/Sqrt[x] + 6*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]
]

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Maple [A]  time = 0.025, size = 72, normalized size = 1.2 \[{({b}^{2}{x}^{2}-2\,bx-8){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}}+3\,{\frac{\sqrt{b}\sqrt{x \left ( bx+2 \right ) }}{\sqrt{x}\sqrt{bx+2}}\ln \left ({\frac{bx+1}{\sqrt{b}}}+\sqrt{b{x}^{2}+2\,x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(3/2)/x^(3/2),x)

[Out]

(b^2*x^2-2*b*x-8)/x^(1/2)/(b*x+2)^(1/2)+3*b^(1/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*x)
^(1/2))*(x*(b*x+2))^(1/2)/x^(1/2)/(b*x+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((b*x + 2)^(3/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222048, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) + \sqrt{b x + 2}{\left (b x - 4\right )} \sqrt{x}}{x}, \frac{6 \, \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) + \sqrt{b x + 2}{\left (b x - 4\right )} \sqrt{x}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(3*sqrt(b)*x*log(b*x + sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1) + sqrt(b*x + 2)*(b*x
- 4)*sqrt(x))/x, (6*sqrt(-b)*x*arctan(sqrt(b*x + 2)/(sqrt(-b)*sqrt(x))) + sqrt(b
*x + 2)*(b*x - 4)*sqrt(x))/x]

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Sympy [A]  time = 12.6304, size = 73, normalized size = 1.26 \[ 6 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{b x + 2}} - \frac{2 b \sqrt{x}}{\sqrt{b x + 2}} - \frac{8}{\sqrt{x} \sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(b)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2) + b**2*x**(3/2)/sqrt(b*x + 2) - 2*b*s
qrt(x)/sqrt(b*x + 2) - 8/(sqrt(x)*sqrt(b*x + 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((b*x + 2)^(3/2)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError