3.1801 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{11/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{9/2}}+\frac{35 a \sqrt{a+\frac{b}{x}}}{4 b^4 \sqrt{x}}-\frac{35 \sqrt{a+\frac{b}{x}}}{6 b^3 x^{3/2}}+\frac{14}{3 b^2 x^{5/2} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

2/(3*b*(a + b/x)^(3/2)*x^(7/2)) + 14/(3*b^2*Sqrt[a + b/x]*x^(5/2)) - (35*Sqrt[a
+ b/x])/(6*b^3*x^(3/2)) + (35*a*Sqrt[a + b/x])/(4*b^4*Sqrt[x]) - (35*a^2*ArcTanh
[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.198109, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{9/2}}+\frac{35 a \sqrt{a+\frac{b}{x}}}{4 b^4 \sqrt{x}}-\frac{35 \sqrt{a+\frac{b}{x}}}{6 b^3 x^{3/2}}+\frac{14}{3 b^2 x^{5/2} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^(11/2)),x]

[Out]

2/(3*b*(a + b/x)^(3/2)*x^(7/2)) + 14/(3*b^2*Sqrt[a + b/x]*x^(5/2)) - (35*Sqrt[a
+ b/x])/(6*b^3*x^(3/2)) + (35*a*Sqrt[a + b/x])/(4*b^4*Sqrt[x]) - (35*a^2*ArcTanh
[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.9956, size = 112, normalized size = 0.87 \[ - \frac{35 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4 b^{\frac{9}{2}}} + \frac{35 a \sqrt{a + \frac{b}{x}}}{4 b^{4} \sqrt{x}} + \frac{2}{3 b x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{14}{3 b^{2} x^{\frac{5}{2}} \sqrt{a + \frac{b}{x}}} - \frac{35 \sqrt{a + \frac{b}{x}}}{6 b^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**(11/2),x)

[Out]

-35*a**2*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/(4*b**(9/2)) + 35*a*sqrt(a + b/x
)/(4*b**4*sqrt(x)) + 2/(3*b*x**(7/2)*(a + b/x)**(3/2)) + 14/(3*b**2*x**(5/2)*sqr
t(a + b/x)) - 35*sqrt(a + b/x)/(6*b**3*x**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.32784, size = 107, normalized size = 0.83 \[ \frac{-210 a^2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+105 a^2 \log (x)+\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (105 a^3 x^3+140 a^2 b x^2+21 a b^2 x-6 b^3\right )}{x^{3/2} (a x+b)^2}}{24 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^(11/2)),x]

[Out]

((2*Sqrt[b]*Sqrt[a + b/x]*(-6*b^3 + 21*a*b^2*x + 140*a^2*b*x^2 + 105*a^3*x^3))/(
x^(3/2)*(b + a*x)^2) - 210*a^2*Log[b + Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] + 105*a^2*
Log[x])/(24*b^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.031, size = 117, normalized size = 0.9 \[ -{\frac{1}{12\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 105\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}{x}^{3}{a}^{3}+105\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){x}^{2}{a}^{2}b\sqrt{ax+b}-105\,{x}^{3}{a}^{3}\sqrt{b}-140\,{b}^{3/2}{x}^{2}{a}^{2}-21\,{b}^{5/2}xa+6\,{b}^{7/2} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^(11/2),x)

[Out]

-1/12*((a*x+b)/x)^(1/2)/x^(3/2)*(105*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2
)*x^3*a^3+105*arctanh((a*x+b)^(1/2)/b^(1/2))*x^2*a^2*b*(a*x+b)^(1/2)-105*x^3*a^3
*b^(1/2)-140*b^(3/2)*x^2*a^2-21*b^(5/2)*x*a+6*b^(7/2))/(a*x+b)^2/b^(9/2)

_______________________________________________________________________________________

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(1/((a + b/x)^(5/2)*x^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.254303, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (105 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 21 \, a b^{2} x - 6 \, b^{3}\right )} \sqrt{b}}{24 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}, \frac{105 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (105 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 21 \, a b^{2} x - 6 \, b^{3}\right )} \sqrt{-b}}{12 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^(11/2)),x, algorithm="fricas")

[Out]

[1/24*(105*(a^3*x^3 + a^2*b*x^2)*sqrt(x)*sqrt((a*x + b)/x)*log(-(2*b*sqrt(x)*sqr
t((a*x + b)/x) - (a*x + 2*b)*sqrt(b))/x) + 2*(105*a^3*x^3 + 140*a^2*b*x^2 + 21*a
*b^2*x - 6*b^3)*sqrt(b))/((a*b^4*x^3 + b^5*x^2)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x
)), 1/12*(105*(a^3*x^3 + a^2*b*x^2)*sqrt(x)*sqrt((a*x + b)/x)*arctan(b/(sqrt(-b)
*sqrt(x)*sqrt((a*x + b)/x))) + (105*a^3*x^3 + 140*a^2*b*x^2 + 21*a*b^2*x - 6*b^3
)*sqrt(-b))/((a*b^4*x^3 + b^5*x^2)*sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))]

_______________________________________________________________________________________

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(1/(a+b/x)**(5/2)/x**(11/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268367, size = 109, normalized size = 0.84 \[ \frac{1}{12} \, a^{2}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{8 \,{\left (9 \, a x + 10 \, b\right )}}{{\left (a x + b\right )}^{\frac{3}{2}} b^{4}} + \frac{3 \,{\left (11 \,{\left (a x + b\right )}^{\frac{3}{2}} - 13 \, \sqrt{a x + b} b\right )}}{a^{2} b^{4} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^(11/2)),x, algorithm="giac")

[Out]

1/12*a^2*(105*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^4) + 8*(9*a*x + 10*b)/(
(a*x + b)^(3/2)*b^4) + 3*(11*(a*x + b)^(3/2) - 13*sqrt(a*x + b)*b)/(a^2*b^4*x^2)
)