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

Optimal. Leaf size=126 \[ -\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{x^{3/2} \sqrt{b x+2}}{8 b^2}+\frac{1}{5} x^{7/2} (b x+2)^{3/2}+\frac{3}{20} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{20 b} \]

[Out]

(3*Sqrt[x]*Sqrt[2 + b*x])/(8*b^3) - (x^(3/2)*Sqrt[2 + b*x])/(8*b^2) + (x^(5/2)*S
qrt[2 + b*x])/(20*b) + (3*x^(7/2)*Sqrt[2 + b*x])/20 + (x^(7/2)*(2 + b*x)^(3/2))/
5 - (3*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.10299, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{x^{3/2} \sqrt{b x+2}}{8 b^2}+\frac{1}{5} x^{7/2} (b x+2)^{3/2}+\frac{3}{20} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{20 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(2 + b*x)^(3/2),x]

[Out]

(3*Sqrt[x]*Sqrt[2 + b*x])/(8*b^3) - (x^(3/2)*Sqrt[2 + b*x])/(8*b^2) + (x^(5/2)*S
qrt[2 + b*x])/(20*b) + (3*x^(7/2)*Sqrt[2 + b*x])/20 + (x^(7/2)*(2 + b*x)^(3/2))/
5 - (3*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.2817, size = 119, normalized size = 0.94 \[ \frac{x^{\frac{5}{2}} \left (b x + 2\right )^{\frac{5}{2}}}{5 b} - \frac{x^{\frac{3}{2}} \left (b x + 2\right )^{\frac{5}{2}}}{4 b^{2}} + \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{4 b^{3}} - \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{8 b^{3}} - \frac{3 \sqrt{x} \sqrt{b x + 2}}{8 b^{3}} - \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(5/2)*(b*x + 2)**(5/2)/(5*b) - x**(3/2)*(b*x + 2)**(5/2)/(4*b**2) + sqrt(x)*(
b*x + 2)**(5/2)/(4*b**3) - sqrt(x)*(b*x + 2)**(3/2)/(8*b**3) - 3*sqrt(x)*sqrt(b*
x + 2)/(8*b**3) - 3*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(4*b**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0857385, size = 78, normalized size = 0.62 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (8 b^4 x^4+22 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{40 b^3}-\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(15 - 5*b*x + 2*b^2*x^2 + 22*b^3*x^3 + 8*b^4*x^4))/(40*b^
3) - (3*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 123, normalized size = 1. \[{\frac{1}{5\,b}{x}^{{\frac{5}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{4\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{1}{8\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{3}{8\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}-{\frac{3}{8}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+2)^(3/2),x)

[Out]

1/5/b*x^(5/2)*(b*x+2)^(5/2)-1/4/b^2*x^(3/2)*(b*x+2)^(5/2)+1/4/b^3*x^(1/2)*(b*x+2
)^(5/2)-1/8/b^3*x^(1/2)*(b*x+2)^(3/2)-3/8*x^(1/2)*(b*x+2)^(1/2)/b^3-3/8/b^(7/2)*
(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))

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.220873, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (8 \, b^{4} x^{4} + 22 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - 5 \, b x + 15\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 15 \, \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{40 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, b^{4} x^{4} + 22 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - 5 \, b x + 15\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 30 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{40 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/40*((8*b^4*x^4 + 22*b^3*x^3 + 2*b^2*x^2 - 5*b*x + 15)*sqrt(b*x + 2)*sqrt(b)*s
qrt(x) + 15*log(-sqrt(b*x + 2)*b*sqrt(x) + (b*x + 1)*sqrt(b)))/b^(7/2), 1/40*((8
*b^4*x^4 + 22*b^3*x^3 + 2*b^2*x^2 - 5*b*x + 15)*sqrt(b*x + 2)*sqrt(-b)*sqrt(x) -
30*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^3)]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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