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

Optimal. Leaf size=86 \[ -\frac{10 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{5 \sqrt{x} \sqrt{b x+2}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{b x+2}}-\frac{2 x^{5/2}}{3 b (b x+2)^{3/2}} \]

[Out]

(-2*x^(5/2))/(3*b*(2 + b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[2 + b*x]) + (5*Sqr
t[x]*Sqrt[2 + b*x])/b^3 - (10*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0651031, antiderivative size = 86, 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 \[ -\frac{10 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{5 \sqrt{x} \sqrt{b x+2}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{b x+2}}-\frac{2 x^{5/2}}{3 b (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(2 + b*x)^(5/2),x]

[Out]

(-2*x^(5/2))/(3*b*(2 + b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[2 + b*x]) + (5*Sqr
t[x]*Sqrt[2 + b*x])/b^3 - (10*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.4665, size = 82, normalized size = 0.95 \[ - \frac{2 x^{\frac{5}{2}}}{3 b \left (b x + 2\right )^{\frac{3}{2}}} - \frac{10 x^{\frac{3}{2}}}{3 b^{2} \sqrt{b x + 2}} + \frac{5 \sqrt{x} \sqrt{b x + 2}}{b^{3}} - \frac{10 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(5/2)/(3*b*(b*x + 2)**(3/2)) - 10*x**(3/2)/(3*b**2*sqrt(b*x + 2)) + 5*sqrt
(x)*sqrt(b*x + 2)/b**3 - 10*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(7/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.107493, size = 60, normalized size = 0.7 \[ \frac{\sqrt{x} \left (3 b^2 x^2+40 b x+60\right )}{3 b^3 (b x+2)^{3/2}}-\frac{10 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(60 + 40*b*x + 3*b^2*x^2))/(3*b^3*(2 + b*x)^(3/2)) - (10*ArcSinh[(Sqrt[
b]*Sqrt[x])/Sqrt[2]])/b^(7/2)

_______________________________________________________________________________________

Maple [B]  time = 0.047, size = 136, normalized size = 1.6 \[{\frac{1}{{b}^{3}}\sqrt{x}\sqrt{bx+2}}+{1 \left ( -5\,{\frac{1}{{b}^{7/2}}\ln \left ({\frac{bx+1}{\sqrt{b}}}+\sqrt{b{x}^{2}+2\,x} \right ) }-{\frac{8}{3\,{b}^{5}}\sqrt{b \left ( x+2\,{b}^{-1} \right ) ^{2}-2\,x-4\,{b}^{-1}} \left ( x+2\,{b}^{-1} \right ) ^{-2}}+{\frac{28}{3\,{b}^{4}}\sqrt{b \left ( x+2\,{b}^{-1} \right ) ^{2}-2\,x-4\,{b}^{-1}} \left ( x+2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x+2)^(5/2),x)

[Out]

x^(1/2)*(b*x+2)^(1/2)/b^3+(-5/b^(7/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*x)^(1/2))-8/3/
b^5/(x+2/b)^2*(b*(x+2/b)^2-2*x-4/b)^(1/2)+28/3/b^4/(x+2/b)*(b*(x+2/b)^2-2*x-4/b)
^(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(x^(5/2)/(b*x + 2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25207, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b x + 2\right )}^{\frac{3}{2}} \sqrt{x} \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right ) +{\left (3 \, b^{2} x^{3} + 40 \, b x^{2} + 60 \, x\right )} \sqrt{b}}{3 \,{\left (b^{4} x + 2 \, b^{3}\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x}}, -\frac{30 \,{\left (b x + 2\right )}^{\frac{3}{2}} \sqrt{x} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{3} + 40 \, b x^{2} + 60 \, x\right )} \sqrt{-b}}{3 \,{\left (b^{4} x + 2 \, b^{3}\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(15*(b*x + 2)^(3/2)*sqrt(x)*log(-sqrt(b*x + 2)*b*sqrt(x) + (b*x + 1)*sqrt(b
)) + (3*b^2*x^3 + 40*b*x^2 + 60*x)*sqrt(b))/((b^4*x + 2*b^3)*sqrt(b*x + 2)*sqrt(
b)*sqrt(x)), -1/3*(30*(b*x + 2)^(3/2)*sqrt(x)*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*s
qrt(x))) - (3*b^2*x^3 + 40*b*x^2 + 60*x)*sqrt(-b))/((b^4*x + 2*b^3)*sqrt(b*x + 2
)*sqrt(-b)*sqrt(x))]

_______________________________________________________________________________________

Sympy [A]  time = 79.2109, size = 308, normalized size = 3.58 \[ \frac{3 b^{\frac{23}{2}} x^{15}}{3 b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{b x + 2} + 6 b^{\frac{25}{2}} x^{\frac{25}{2}} \sqrt{b x + 2}} + \frac{40 b^{\frac{21}{2}} x^{14}}{3 b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{b x + 2} + 6 b^{\frac{25}{2}} x^{\frac{25}{2}} \sqrt{b x + 2}} + \frac{60 b^{\frac{19}{2}} x^{13}}{3 b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{b x + 2} + 6 b^{\frac{25}{2}} x^{\frac{25}{2}} \sqrt{b x + 2}} - \frac{30 b^{10} x^{\frac{27}{2}} \sqrt{b x + 2} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{3 b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{b x + 2} + 6 b^{\frac{25}{2}} x^{\frac{25}{2}} \sqrt{b x + 2}} - \frac{60 b^{9} x^{\frac{25}{2}} \sqrt{b x + 2} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{3 b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{b x + 2} + 6 b^{\frac{25}{2}} x^{\frac{25}{2}} \sqrt{b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b**(23/2)*x**15/(3*b**(27/2)*x**(27/2)*sqrt(b*x + 2) + 6*b**(25/2)*x**(25/2)*s
qrt(b*x + 2)) + 40*b**(21/2)*x**14/(3*b**(27/2)*x**(27/2)*sqrt(b*x + 2) + 6*b**(
25/2)*x**(25/2)*sqrt(b*x + 2)) + 60*b**(19/2)*x**13/(3*b**(27/2)*x**(27/2)*sqrt(
b*x + 2) + 6*b**(25/2)*x**(25/2)*sqrt(b*x + 2)) - 30*b**10*x**(27/2)*sqrt(b*x +
2)*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/(3*b**(27/2)*x**(27/2)*sqrt(b*x + 2) + 6*b**
(25/2)*x**(25/2)*sqrt(b*x + 2)) - 60*b**9*x**(25/2)*sqrt(b*x + 2)*asinh(sqrt(2)*
sqrt(b)*sqrt(x)/2)/(3*b**(27/2)*x**(27/2)*sqrt(b*x + 2) + 6*b**(25/2)*x**(25/2)*
sqrt(b*x + 2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239039, size = 246, normalized size = 2.86 \[ \frac{{\left (\frac{15 \,{\rm ln}\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{b^{\frac{5}{2}}} + \frac{3 \, \sqrt{{\left (b x + 2\right )} b - 2 \, b} \sqrt{b x + 2}}{b^{3}} + \frac{16 \,{\left (9 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt{b} + 24 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac{3}{2}} + 28 \, b^{\frac{5}{2}}\right )}}{{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} b^{2}}\right )}{\left | b \right |}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(15*ln((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2)/b^(5/2) + 3*sqrt
((b*x + 2)*b - 2*b)*sqrt(b*x + 2)/b^3 + 16*(9*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x
+ 2)*b - 2*b))^4*sqrt(b) + 24*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))
^2*b^(3/2) + 28*b^(5/2))/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 +
2*b)^3*b^2))*abs(b)/b^2