3.1772 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+2 \sqrt{x} \left (a+\frac{b}{x}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x}\right )^{3/2}}{2 \sqrt{x}}-\frac{15 a b \sqrt{a+\frac{b}{x}}}{4 \sqrt{x}} \]

[Out]

(-15*a*b*Sqrt[a + b/x])/(4*Sqrt[x]) - (5*b*(a + b/x)^(3/2))/(2*Sqrt[x]) + 2*(a +
b/x)^(5/2)*Sqrt[x] - (15*a^2*Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/
4

_______________________________________________________________________________________

Rubi [A]  time = 0.123352, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+2 \sqrt{x} \left (a+\frac{b}{x}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x}\right )^{3/2}}{2 \sqrt{x}}-\frac{15 a b \sqrt{a+\frac{b}{x}}}{4 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/Sqrt[x],x]

[Out]

(-15*a*b*Sqrt[a + b/x])/(4*Sqrt[x]) - (5*b*(a + b/x)^(3/2))/(2*Sqrt[x]) + 2*(a +
b/x)^(5/2)*Sqrt[x] - (15*a^2*Sqrt[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/
4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.3093, size = 85, normalized size = 0.88 \[ - \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4} - \frac{15 a b \sqrt{a + \frac{b}{x}}}{4 \sqrt{x}} - \frac{5 b \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{2 \sqrt{x}} + 2 \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(5/2)/x**(1/2),x)

[Out]

-15*a**2*sqrt(b)*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/4 - 15*a*b*sqrt(a + b/x)
/(4*sqrt(x)) - 5*b*(a + b/x)**(3/2)/(2*sqrt(x)) + 2*sqrt(x)*(a + b/x)**(5/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.232231, size = 91, normalized size = 0.94 \[ \frac{\sqrt{a+\frac{b}{x}} \left (8 a^2 x^2-9 a b x-2 b^2\right )}{4 x^{3/2}}-\frac{15}{4} a^2 \sqrt{b} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\frac{15}{8} a^2 \sqrt{b} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(-2*b^2 - 9*a*b*x + 8*a^2*x^2))/(4*x^(3/2)) - (15*a^2*Sqrt[b]*Log
[b + Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]])/4 + (15*a^2*Sqrt[b]*Log[x])/8

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 93, normalized size = 1. \[ -{\frac{1}{4}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{2}b{x}^{2}-8\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b}+9\,xa{b}^{3/2}\sqrt{ax+b}+2\,{b}^{5/2}\sqrt{ax+b} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x^(1/2),x)

[Out]

-1/4*((a*x+b)/x)^(1/2)/x^(3/2)*(15*arctanh((a*x+b)^(1/2)/b^(1/2))*a^2*b*x^2-8*x^
2*a^2*b^(1/2)*(a*x+b)^(1/2)+9*x*a*b^(3/2)*(a*x+b)^(1/2)+2*b^(5/2)*(a*x+b)^(1/2))
/(a*x+b)^(1/2)/b^(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((a + b/x)^(5/2)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.246082, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x^{2} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \,{\left (8 \, a^{2} x^{2} - 9 \, a b x - 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \, x^{2}}, -\frac{15 \, a^{2} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) -{\left (8 \, a^{2} x^{2} - 9 \, a b x - 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*a^2*sqrt(b)*x^2*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x
) + 2*(8*a^2*x^2 - 9*a*b*x - 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/x^2, -1/4*(15*a^2
*sqrt(-b)*x^2*arctan(sqrt(x)*sqrt((a*x + b)/x)/sqrt(-b)) - (8*a^2*x^2 - 9*a*b*x
- 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/x^2]

_______________________________________________________________________________________

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((a+b/x)**(5/2)/x**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.294607, size = 92, normalized size = 0.95 \[ \frac{1}{4} \,{\left (\frac{15 \, b \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 8 \, \sqrt{a x + b} - \frac{9 \,{\left (a x + b\right )}^{\frac{3}{2}} b - 7 \, \sqrt{a x + b} b^{2}}{a^{2} x^{2}}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(15*b*arctan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) + 8*sqrt(a*x + b) - (9*(a*x +
b)^(3/2)*b - 7*sqrt(a*x + b)*b^2)/(a^2*x^2))*a^2