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

Optimal. Leaf size=79 \[ -\frac{2 (b x+2)^{5/2}}{\sqrt{x}}+\frac{5}{2} b \sqrt{x} (b x+2)^{3/2}+\frac{15}{2} b \sqrt{x} \sqrt{b x+2}+15 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

[Out]

(15*b*Sqrt[x]*Sqrt[2 + b*x])/2 + (5*b*Sqrt[x]*(2 + b*x)^(3/2))/2 - (2*(2 + b*x)^
(5/2))/Sqrt[x] + 15*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0555846, antiderivative size = 79, 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 \[ -\frac{2 (b x+2)^{5/2}}{\sqrt{x}}+\frac{5}{2} b \sqrt{x} (b x+2)^{3/2}+\frac{15}{2} b \sqrt{x} \sqrt{b x+2}+15 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + b*x)^(5/2)/x^(3/2),x]

[Out]

(15*b*Sqrt[x]*Sqrt[2 + b*x])/2 + (5*b*Sqrt[x]*(2 + b*x)^(3/2))/2 - (2*(2 + b*x)^
(5/2))/Sqrt[x] + 15*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.25842, size = 76, normalized size = 0.96 \[ 15 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{5 b \sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{2} + \frac{15 b \sqrt{x} \sqrt{b x + 2}}{2} - \frac{2 \left (b x + 2\right )^{\frac{5}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*sqrt(b)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2) + 5*b*sqrt(x)*(b*x + 2)**(3/2)/2 + 1
5*b*sqrt(x)*sqrt(b*x + 2)/2 - 2*(b*x + 2)**(5/2)/sqrt(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.049166, size = 56, normalized size = 0.71 \[ \frac{\sqrt{b x+2} \left (b^2 x^2+9 b x-16\right )}{2 \sqrt{x}}+15 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + b*x]*(-16 + 9*b*x + b^2*x^2))/(2*Sqrt[x]) + 15*Sqrt[b]*ArcSinh[(Sqrt[b
]*Sqrt[x])/Sqrt[2]]

_______________________________________________________________________________________

Maple [A]  time = 0.024, size = 81, normalized size = 1. \[{\frac{{b}^{3}{x}^{3}+11\,{b}^{2}{x}^{2}+2\,bx-32}{2}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}}+{\frac{15}{2}\sqrt{b}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(5/2)/x^(3/2),x)

[Out]

1/2*(b^3*x^3+11*b^2*x^2+2*b*x-32)/x^(1/2)/(b*x+2)^(1/2)+15/2*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)

_______________________________________________________________________________________

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^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.253828, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) +{\left (b^{2} x^{2} + 9 \, b x - 16\right )} \sqrt{b x + 2} \sqrt{x}}{2 \, x}, \frac{30 \, \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (b^{2} x^{2} + 9 \, b x - 16\right )} \sqrt{b x + 2} \sqrt{x}}{2 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(15*sqrt(b)*x*log(b*x + sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1) + (b^2*x^2 + 9*b
*x - 16)*sqrt(b*x + 2)*sqrt(x))/x, 1/2*(30*sqrt(-b)*x*arctan(sqrt(b*x + 2)/(sqrt
(-b)*sqrt(x))) + (b^2*x^2 + 9*b*x - 16)*sqrt(b*x + 2)*sqrt(x))/x]

_______________________________________________________________________________________

Sympy [A]  time = 76.2539, size = 94, normalized size = 1.19 \[ 15 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} + \frac{11 b^{2} x^{\frac{3}{2}}}{2 \sqrt{b x + 2}} + \frac{b \sqrt{x}}{\sqrt{b x + 2}} - \frac{16}{\sqrt{x} \sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*sqrt(b)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2) + b**3*x**(5/2)/(2*sqrt(b*x + 2)) +
11*b**2*x**(3/2)/(2*sqrt(b*x + 2)) + b*sqrt(x)/sqrt(b*x + 2) - 16/(sqrt(x)*sqrt(
b*x + 2))

_______________________________________________________________________________________

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^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError