3.1674 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 \sqrt{x}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}+\frac{3 \sqrt{x}}{a^2}-\frac{x^{3/2}}{a (a x+b)} \]

[Out]

(3*Sqrt[x])/a^2 - x^(3/2)/(a*(b + a*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sq
rt[b]])/a^(5/2)

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Rubi [A]  time = 0.0635048, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}+\frac{3 \sqrt{x}}{a^2}-\frac{x^{3/2}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*Sqrt[x]),x]

[Out]

(3*Sqrt[x])/a^2 - x^(3/2)/(a*(b + a*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sq
rt[b]])/a^(5/2)

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Rubi in Sympy [A]  time = 11.1167, size = 49, normalized size = 0.86 \[ - \frac{x^{\frac{3}{2}}}{a \left (a x + b\right )} + \frac{3 \sqrt{x}}{a^{2}} - \frac{3 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(3/2)/(a*(a*x + b)) + 3*sqrt(x)/a**2 - 3*sqrt(b)*atan(sqrt(a)*sqrt(x)/sqrt(b
))/a**(5/2)

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Mathematica [A]  time = 0.0612655, size = 54, normalized size = 0.95 \[ \frac{\sqrt{x} (2 a x+3 b)}{a^2 (a x+b)}-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(3*b + 2*a*x))/(a^2*(b + a*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sq
rt[b]])/a^(5/2)

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Maple [A]  time = 0.016, size = 47, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{{a}^{2}}}+{\frac{b}{{a}^{2} \left ( ax+b \right ) }\sqrt{x}}-3\,{\frac{b}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^(1/2),x)

[Out]

2*x^(1/2)/a^2+b/a^2*x^(1/2)/(a*x+b)-3*b/a^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(1/((a + b/x)^2*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241943, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a x + b\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (2 \, a x + 3 \, b\right )} \sqrt{x}}{2 \,{\left (a^{3} x + a^{2} b\right )}}, -\frac{3 \,{\left (a x + b\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) -{\left (2 \, a x + 3 \, b\right )} \sqrt{x}}{a^{3} x + a^{2} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 32.9017, size = 411, normalized size = 7.21 \[ \begin{cases} \tilde{\infty } x^{\frac{5}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 \sqrt{x}}{a^{2}} & \text{for}\: b = 0 \\\frac{2 x^{\frac{5}{2}}}{5 b^{2}} & \text{for}\: a = 0 \\\frac{4 i a^{2} \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{1}{a}}}{2 i a^{4} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{3} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{6 i a b^{\frac{3}{2}} \sqrt{x} \sqrt{\frac{1}{a}}}{2 i a^{4} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{3} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{3 a b x \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{4} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{3} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{3 a b x \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{4} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{3} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{3 b^{2} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{4} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{3} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{3 b^{2} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{4} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{3} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233561, size = 62, normalized size = 1.09 \[ -\frac{3 \, b \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{2 \, \sqrt{x}}{a^{2}} + \frac{b \sqrt{x}}{{\left (a x + b\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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