3.538 \(\int \frac{(2+b x)^{3/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=60 \[ 2 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 (b x+2)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{b x+2}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.044396, antiderivative size = 60, 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 \[ 2 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 (b x+2)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + b*x)^(3/2)/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.90151, size = 58, normalized size = 0.97 \[ 2 b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{2 b \sqrt{b x + 2}}{\sqrt{x}} - \frac{2 \left (b x + 2\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+2)**(3/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0596666, size = 49, normalized size = 0.82 \[ 2 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{4 \sqrt{b x+2} (2 b x+1)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + b*x)^(3/2)/x^(5/2),x]

[Out]

(-4*Sqrt[2 + b*x]*(1 + 2*b*x))/(3*x^(3/2)) + 2*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[x])
/Sqrt[2]]

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Maple [A]  time = 0.028, size = 73, normalized size = 1.2 \[ -{\frac{8\,{b}^{2}{x}^{2}+20\,bx+8}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}}+{1{b}^{{\frac{3}{2}}}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(3/2)/x^(5/2),x)

[Out]

-4/3*(2*b^2*x^2+5*b*x+2)/x^(3/2)/(b*x+2)^(1/2)+b^(3/2)*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^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232047, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b^{\frac{3}{2}} x^{2} \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) - 4 \,{\left (2 \, b x + 1\right )} \sqrt{b x + 2} \sqrt{x}}{3 \, x^{2}}, \frac{2 \,{\left (3 \, \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) - 2 \,{\left (2 \, b x + 1\right )} \sqrt{b x + 2} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + 2)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 27.0085, size = 70, normalized size = 1.17 \[ - \frac{8 b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{3} - b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{2}{b x}} + 1 \right )} - \frac{4 \sqrt{b} \sqrt{1 + \frac{2}{b x}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+2)**(3/2)/x**(5/2),x)

[Out]

-8*b**(3/2)*sqrt(1 + 2/(b*x))/3 - b**(3/2)*log(1/(b*x)) + 2*b**(3/2)*log(sqrt(1
+ 2/(b*x)) + 1) - 4*sqrt(b)*sqrt(1 + 2/(b*x))/(3*x)

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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^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError