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

Optimal. Leaf size=61 \[ \frac{1}{2} \sqrt{x} (b x+2)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{b x+2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0398299, antiderivative size = 61, 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 \[ \frac{1}{2} \sqrt{x} (b x+2)^{3/2}+\frac{3}{2} \sqrt{x} \sqrt{b x+2}+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + b*x)^(3/2)/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.3803, size = 56, normalized size = 0.92 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{2} + \frac{3 \sqrt{x} \sqrt{b x + 2}}{2} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.041608, size = 48, normalized size = 0.79 \[ \frac{1}{2} \sqrt{x} \sqrt{b x+2} (b x+5)+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(5 + b*x))/2 + (3*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/Sqr
t[b]

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Maple [A]  time = 0.008, size = 72, normalized size = 1.2 \[{\frac{1}{2} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3}{2}\sqrt{x}\sqrt{bx+2}}+{\frac{3}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(3/2)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224266, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x + 5\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 3 \, \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{2 \, \sqrt{b}}, \frac{{\left (b x + 5\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} + 6 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{2 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.5101, size = 76, normalized size = 1.25 \[ \frac{b^{2} x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} + \frac{7 b x^{\frac{3}{2}}}{2 \sqrt{b x + 2}} + \frac{5 \sqrt{x}}{\sqrt{b x + 2}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**(5/2)/(2*sqrt(b*x + 2)) + 7*b*x**(3/2)/(2*sqrt(b*x + 2)) + 5*sqrt(x)/sqr
t(b*x + 2) + 3*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b)

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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)/sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError