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

Optimal. Leaf size=82 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 x \sqrt{a+\frac{b}{x}}}{a^3}-\frac{10 x}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(-2*x)/(3*a*(a + b/x)^(3/2)) - (10*x)/(3*a^2*Sqrt[a + b/x]) + (5*Sqrt[a + b/x]*x
)/a^3 - (5*b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)

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Rubi [A]  time = 0.112652, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 x \sqrt{a+\frac{b}{x}}}{a^3}-\frac{10 x}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(-5/2),x]

[Out]

(-2*x)/(3*a*(a + b/x)^(3/2)) - (10*x)/(3*a^2*Sqrt[a + b/x]) + (5*Sqrt[a + b/x]*x
)/a^3 - (5*b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)

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Rubi in Sympy [A]  time = 11.5393, size = 70, normalized size = 0.85 \[ - \frac{2 x}{3 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{10 x}{3 a^{2} \sqrt{a + \frac{b}{x}}} + \frac{5 x \sqrt{a + \frac{b}{x}}}{a^{3}} - \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.133915, size = 82, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2+20 a b x+15 b^2\right )}{3 a^3 (a x+b)^2}-\frac{5 b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(-5/2),x]

[Out]

(Sqrt[a + b/x]*x*(15*b^2 + 20*a*b*x + 3*a^2*x^2))/(3*a^3*(b + a*x)^2) - (5*b*Log
[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2*a^(7/2))

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Maple [B]  time = 0.006, size = 276, normalized size = 3.4 \[ -{\frac{x}{6\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -30\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}+24\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}x-90\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}b+20\,b{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}-90\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{6}b-30\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{5}{b}^{2}+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{3}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{4} \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2),x)

[Out]

-1/6*((a*x+b)/x)^(1/2)*x/a^(13/2)*(-30*a^(13/2)*(x*(a*x+b))^(1/2)*x^3+24*a^(11/2
)*(x*(a*x+b))^(3/2)*x-90*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*b+20*b*a^(9/2)*(x*(a*x+b
))^(3/2)-90*a^(9/2)*(x*(a*x+b))^(1/2)*x*b^2+15*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/
2)+2*a*x+b)/a^(1/2))*x^3*a^6*b-30*a^(7/2)*(x*(a*x+b))^(1/2)*b^3+45*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+45*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+15*ln(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)^3

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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((a + b/x)^(-5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242169, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (3 \, a^{2} x^{2} + 20 \, a b x + 15 \, b^{2}\right )} \sqrt{a}}{6 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} x^{2} + 20 \, a b x + 15 \, b^{2}\right )} \sqrt{-a}}{3 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*(a*b*x + b^2)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x +
b)*sqrt(a)) + 2*(3*a^2*x^2 + 20*a*b*x + 15*b^2)*sqrt(a))/((a^4*x + a^3*b)*sqrt(
a)*sqrt((a*x + b)/x)), 1/3*(15*(a*b*x + b^2)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a
)*sqrt((a*x + b)/x))) + (3*a^2*x^2 + 20*a*b*x + 15*b^2)*sqrt(-a))/((a^4*x + a^3*
b)*sqrt(-a)*sqrt((a*x + b)/x))]

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Sympy [A]  time = 17.8996, size = 774, normalized size = 9.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(5/2),x)

[Out]

6*a**17*x**4*sqrt(1 + b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(
35/2)*b**2*x + 6*a**(33/2)*b**3) + 46*a**16*b*x**3*sqrt(1 + b/(a*x))/(6*a**(39/2
)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 15*a**1
6*b*x**3*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**
2*x + 6*a**(33/2)*b**3) - 30*a**16*b*x**3*log(sqrt(1 + b/(a*x)) + 1)/(6*a**(39/2
)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 70*a**1
5*b**2*x**2*sqrt(1 + b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(3
5/2)*b**2*x + 6*a**(33/2)*b**3) + 45*a**15*b**2*x**2*log(b/(a*x))/(6*a**(39/2)*x
**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 90*a**15*b
**2*x**2*log(sqrt(1 + b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18
*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 30*a**14*b**3*x*sqrt(1 + b/(a*x))/(6*a**
(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 45
*a**14*b**3*x*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2
)*b**2*x + 6*a**(33/2)*b**3) - 90*a**14*b**3*x*log(sqrt(1 + b/(a*x)) + 1)/(6*a**
(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 15
*a**13*b**4*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*
b**2*x + 6*a**(33/2)*b**3) - 30*a**13*b**4*log(sqrt(1 + b/(a*x)) + 1)/(6*a**(39/
2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3)

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GIAC/XCAS [A]  time = 0.254127, size = 132, normalized size = 1.61 \[ \frac{1}{3} \, b{\left (\frac{2 \,{\left (a + \frac{6 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{3} \sqrt{\frac{a x + b}{x}}} + \frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{3 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b*(2*(a + 6*(a*x + b)/x)*x/((a*x + b)*a^3*sqrt((a*x + b)/x)) + 15*arctan(sqr
t((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3) - 3*sqrt((a*x + b)/x)/((a - (a*x + b)/x)
*a^3))