3.1673 \(\int \frac{\sqrt{x}}{\left (a+\frac{b}{x}\right )^2} \, dx\)

Optimal. Leaf size=70 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (a x+b)} \]

[Out]

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

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Rubi [A]  time = 0.0777431, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + b/x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.9459, size = 63, normalized size = 0.9 \[ - \frac{x^{\frac{5}{2}}}{a \left (a x + b\right )} + \frac{5 x^{\frac{3}{2}}}{3 a^{2}} - \frac{5 b \sqrt{x}}{a^{3}} + \frac{5 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(5/2)/(a*(a*x + b)) + 5*x**(3/2)/(3*a**2) - 5*b*sqrt(x)/a**3 + 5*b**(3/2)*at
an(sqrt(a)*sqrt(x)/sqrt(b))/a**(7/2)

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Mathematica [A]  time = 0.0743033, size = 68, normalized size = 0.97 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}+\frac{\sqrt{x} \left (2 a^2 x^2-10 a b x-15 b^2\right )}{3 a^3 (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-15*b^2 - 10*a*b*x + 2*a^2*x^2))/(3*a^3*(b + a*x)) + (5*b^(3/2)*ArcTan
[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(7/2)

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Maple [A]  time = 0.016, size = 61, normalized size = 0.9 \[{\frac{2}{3\,{a}^{2}}{x}^{{\frac{3}{2}}}}-4\,{\frac{b\sqrt{x}}{{a}^{3}}}-{\frac{{b}^{2}}{{a}^{3} \left ( ax+b \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(a+b/x)^2,x)

[Out]

2/3*x^(3/2)/a^2-4*b*x^(1/2)/a^3-1/a^3*b^2*x^(1/2)/(a*x+b)+5/a^3*b^2/(a*b)^(1/2)*
arctan(a*x^(1/2)/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242955, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (2 \, a^{2} x^{2} - 10 \, a b x - 15 \, b^{2}\right )} \sqrt{x}}{6 \,{\left (a^{4} x + a^{3} b\right )}}, \frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) +{\left (2 \, a^{2} x^{2} - 10 \, a b x - 15 \, b^{2}\right )} \sqrt{x}}{3 \,{\left (a^{4} x + a^{3} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*(a*b*x + b^2)*sqrt(-b/a)*log((a*x + 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x +
b)) + 2*(2*a^2*x^2 - 10*a*b*x - 15*b^2)*sqrt(x))/(a^4*x + a^3*b), 1/3*(15*(a*b*x
+ b^2)*sqrt(b/a)*arctan(sqrt(x)/sqrt(b/a)) + (2*a^2*x^2 - 10*a*b*x - 15*b^2)*sq
rt(x))/(a^4*x + a^3*b)]

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Sympy [A]  time = 32.0277, size = 479, normalized size = 6.84 \[ \begin{cases} \tilde{\infty } x^{\frac{7}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a^{2}} & \text{for}\: b = 0 \\\frac{2 x^{\frac{7}{2}}}{7 b^{2}} & \text{for}\: a = 0 \\\frac{4 i a^{3} \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{1}{a}}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{20 i a^{2} b^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{a}}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{30 i a b^{\frac{5}{2}} \sqrt{x} \sqrt{\frac{1}{a}}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{15 a b^{2} x \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{15 a b^{2} x \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{15 b^{3} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{15 b^{3} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo*x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a**2), Eq(b, 0)),
(2*x**(7/2)/(7*b**2), Eq(a, 0)), (4*I*a**3*sqrt(b)*x**(5/2)*sqrt(1/a)/(6*I*a**5*
sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 20*I*a**2*b**(3/2)*x**(3/2)
*sqrt(1/a)/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 30*I*a
*b**(5/2)*sqrt(x)*sqrt(1/a)/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sq
rt(1/a)) + 15*a*b**2*x*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(6*I*a**5*sqrt(b)*x*s
qrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 15*a*b**2*x*log(I*sqrt(b)*sqrt(1/a) +
sqrt(x))/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) + 15*b**3*
log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**
(3/2)*sqrt(1/a)) - 15*b**3*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))/(6*I*a**5*sqrt(b)*
x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)), True))

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GIAC/XCAS [A]  time = 0.228191, size = 88, normalized size = 1.26 \[ \frac{5 \, b^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{b^{2} \sqrt{x}}{{\left (a x + b\right )} a^{3}} + \frac{2 \,{\left (a^{4} x^{\frac{3}{2}} - 6 \, a^{3} b \sqrt{x}\right )}}{3 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) - b^2*sqrt(x)/((a*x + b)*a^3)
+ 2/3*(a^4*x^(3/2) - 6*a^3*b*sqrt(x))/a^6