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

Optimal. Leaf size=81 \[ 10 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+5 b^2 \sqrt{x} \sqrt{b x+2}-\frac{2 (b x+2)^{5/2}}{3 x^{3/2}}-\frac{10 b (b x+2)^{3/2}}{3 \sqrt{x}} \]

[Out]

5*b^2*Sqrt[x]*Sqrt[2 + b*x] - (10*b*(2 + b*x)^(3/2))/(3*Sqrt[x]) - (2*(2 + b*x)^
(5/2))/(3*x^(3/2)) + 10*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi [A]  time = 0.0579604, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ 10 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+5 b^2 \sqrt{x} \sqrt{b x+2}-\frac{2 (b x+2)^{5/2}}{3 x^{3/2}}-\frac{10 b (b x+2)^{3/2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + b*x)^(5/2)/x^(5/2),x]

[Out]

5*b^2*Sqrt[x]*Sqrt[2 + b*x] - (10*b*(2 + b*x)^(3/2))/(3*Sqrt[x]) - (2*(2 + b*x)^
(5/2))/(3*x^(3/2)) + 10*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi in Sympy [A]  time = 9.09954, size = 78, normalized size = 0.96 \[ 10 b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + 5 b^{2} \sqrt{x} \sqrt{b x + 2} - \frac{10 b \left (b x + 2\right )^{\frac{3}{2}}}{3 \sqrt{x}} - \frac{2 \left (b x + 2\right )^{\frac{5}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*b**(3/2)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2) + 5*b**2*sqrt(x)*sqrt(b*x + 2) - 10
*b*(b*x + 2)**(3/2)/(3*sqrt(x)) - 2*(b*x + 2)**(5/2)/(3*x**(3/2))

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Mathematica [A]  time = 0.0538381, size = 57, normalized size = 0.7 \[ 10 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )+\frac{\sqrt{b x+2} \left (3 b^2 x^2-28 b x-8\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + b*x]*(-8 - 28*b*x + 3*b^2*x^2))/(3*x^(3/2)) + 10*b^(3/2)*ArcSinh[(Sqrt
[b]*Sqrt[x])/Sqrt[2]]

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Maple [A]  time = 0.027, size = 82, normalized size = 1. \[{\frac{3\,{b}^{3}{x}^{3}-22\,{b}^{2}{x}^{2}-64\,bx-16}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}}+5\,{\frac{{b}^{3/2}\sqrt{x \left ( bx+2 \right ) }}{\sqrt{x}\sqrt{bx+2}}\ln \left ({\frac{bx+1}{\sqrt{b}}}+\sqrt{b{x}^{2}+2\,x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(5/2)/x^(5/2),x)

[Out]

1/3*(3*b^3*x^3-22*b^2*x^2-64*b*x-16)/x^(3/2)/(b*x+2)^(1/2)+5*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)^(5/2)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226717, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{\frac{3}{2}} x^{2} \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) +{\left (3 \, b^{2} x^{2} - 28 \, b x - 8\right )} \sqrt{b x + 2} \sqrt{x}}{3 \, x^{2}}, \frac{30 \, \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (3 \, b^{2} x^{2} - 28 \, b x - 8\right )} \sqrt{b x + 2} \sqrt{x}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(15*b^(3/2)*x^2*log(b*x + sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1) + (3*b^2*x^2 -
28*b*x - 8)*sqrt(b*x + 2)*sqrt(x))/x^2, 1/3*(30*sqrt(-b)*b*x^2*arctan(sqrt(b*x
+ 2)/(sqrt(-b)*sqrt(x))) + (3*b^2*x^2 - 28*b*x - 8)*sqrt(b*x + 2)*sqrt(x))/x^2]

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Sympy [A]  time = 75.7673, size = 88, normalized size = 1.09 \[ b^{\frac{5}{2}} x \sqrt{1 + \frac{2}{b x}} - \frac{28 b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{3} - 5 b^{\frac{3}{2}} \log{\left (\frac{1}{b x} \right )} + 10 b^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{2}{b x}} + 1 \right )} - \frac{8 \sqrt{b} \sqrt{1 + \frac{2}{b x}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(5/2)*x*sqrt(1 + 2/(b*x)) - 28*b**(3/2)*sqrt(1 + 2/(b*x))/3 - 5*b**(3/2)*log(
1/(b*x)) + 10*b**(3/2)*log(sqrt(1 + 2/(b*x)) + 1) - 8*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)^(5/2)/x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError