3.1731 \(\int \frac{x}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15 b x \sqrt{a+\frac{b}{x}}}{4 a^3}+\frac{5 x^2 \sqrt{a+\frac{b}{x}}}{2 a^2}-\frac{2 x^2}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

(-15*b*Sqrt[a + b/x]*x)/(4*a^3) - (2*x^2)/(a*Sqrt[a + b/x]) + (5*Sqrt[a + b/x]*x
^2)/(2*a^2) + (15*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.119663, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15 b x \sqrt{a+\frac{b}{x}}}{4 a^3}+\frac{5 x^2 \sqrt{a+\frac{b}{x}}}{2 a^2}-\frac{2 x^2}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x)^(3/2),x]

[Out]

(-15*b*Sqrt[a + b/x]*x)/(4*a^3) - (2*x^2)/(a*Sqrt[a + b/x]) + (5*Sqrt[a + b/x]*x
^2)/(2*a^2) + (15*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 12.2943, size = 78, normalized size = 0.86 \[ - \frac{2 x^{2}}{a \sqrt{a + \frac{b}{x}}} + \frac{5 x^{2} \sqrt{a + \frac{b}{x}}}{2 a^{2}} - \frac{15 b x \sqrt{a + \frac{b}{x}}}{4 a^{3}} + \frac{15 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**2/(a*sqrt(a + b/x)) + 5*x**2*sqrt(a + b/x)/(2*a**2) - 15*b*x*sqrt(a + b/x)
/(4*a**3) + 15*b**2*atanh(sqrt(a + b/x)/sqrt(a))/(4*a**(7/2))

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Mathematica [A]  time = 0.139778, size = 84, normalized size = 0.92 \[ \frac{15 b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 a^{7/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (2 a^2 x^2-5 a b x-15 b^2\right )}{4 a^3 (a x+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x)^(3/2),x]

[Out]

(Sqrt[a + b/x]*x*(-15*b^2 - 5*a*b*x + 2*a^2*x^2))/(4*a^3*(b + a*x)) + (15*b^2*Lo
g[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(8*a^(7/2))

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Maple [B]  time = 0.018, size = 397, normalized size = 4.4 \[ -{\frac{x}{8\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -4\,{a}^{13/2}\sqrt{a{x}^{2}+bx}{x}^{3}-10\,{a}^{11/2}\sqrt{a{x}^{2}+bx}{x}^{2}b+32\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}b-8\,{a}^{9/2}\sqrt{a{x}^{2}+bx}x{b}^{2}-16\,b{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}+64\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}-2\,{a}^{7/2}\sqrt{a{x}^{2}+bx}{b}^{3}+32\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{5}{b}^{2}-16\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{5}{b}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{3}-32\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{3}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{3}{b}^{4}-16\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{4} \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x)^(3/2),x)

[Out]

-1/8*((a*x+b)/x)^(1/2)*x/a^(13/2)*(-4*a^(13/2)*(a*x^2+b*x)^(1/2)*x^3-10*a^(11/2)
*(a*x^2+b*x)^(1/2)*x^2*b+32*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*b-8*a^(9/2)*(a*x^2+b*
x)^(1/2)*x*b^2-16*b*a^(9/2)*(x*(a*x+b))^(3/2)+64*a^(9/2)*(x*(a*x+b))^(1/2)*x*b^2
-2*a^(7/2)*(a*x^2+b*x)^(1/2)*b^3+32*a^(7/2)*(x*(a*x+b))^(1/2)*b^3+ln(1/2*(2*(a*x
^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^5*b^2-16*ln(1/2*(2*(x*(a*x+b))^(1/
2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^5*b^2+2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2
*a*x+b)/a^(1/2))*x*a^4*b^3-32*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/
2))*x*a^4*b^3+ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*b^4-16*l
n(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*b^4)/(x*(a*x+b))^(1/2)/
(a*x+b)^2

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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/(a + b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243053, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{2} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a^{2} x^{2} - 5 \, a b x - 15 \, b^{2}\right )} \sqrt{a}}{8 \, a^{\frac{7}{2}} \sqrt{\frac{a x + b}{x}}}, -\frac{15 \, b^{2} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (2 \, a^{2} x^{2} - 5 \, a b x - 15 \, b^{2}\right )} \sqrt{-a}}{4 \, \sqrt{-a} a^{3} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*b^2*sqrt((a*x + b)/x)*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)
) + 2*(2*a^2*x^2 - 5*a*b*x - 15*b^2)*sqrt(a))/(a^(7/2)*sqrt((a*x + b)/x)), -1/4*
(15*b^2*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (2*a^2*x^2 -
5*a*b*x - 15*b^2)*sqrt(-a))/(sqrt(-a)*a^3*sqrt((a*x + b)/x))]

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Sympy [A]  time = 18.1049, size = 105, normalized size = 1.15 \[ \frac{x^{\frac{5}{2}}}{2 a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{5 \sqrt{b} x^{\frac{3}{2}}}{4 a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{15 b^{\frac{3}{2}} \sqrt{x}}{4 a^{3} \sqrt{\frac{a x}{b} + 1}} + \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x)**(3/2),x)

[Out]

x**(5/2)/(2*a*sqrt(b)*sqrt(a*x/b + 1)) - 5*sqrt(b)*x**(3/2)/(4*a**2*sqrt(a*x/b +
1)) - 15*b**(3/2)*sqrt(x)/(4*a**3*sqrt(a*x/b + 1)) + 15*b**2*asinh(sqrt(a)*sqrt
(x)/sqrt(b))/(4*a**(7/2))

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GIAC/XCAS [A]  time = 0.260516, size = 142, normalized size = 1.56 \[ -\frac{1}{4} \, b^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{8}{a^{3} \sqrt{\frac{a x + b}{x}}} - \frac{9 \, a \sqrt{\frac{a x + b}{x}} - \frac{7 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (a - \frac{a x + b}{x}\right )}^{2} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a + b/x)^(3/2),x, algorithm="giac")

[Out]

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