3.1664 \(\int \frac{x^{3/2}}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}+\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a} \]

[Out]

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

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Rubi [A]  time = 0.0766494, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}+\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b/x),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8625, size = 65, normalized size = 0.96 \[ \frac{2 x^{\frac{5}{2}}}{5 a} - \frac{2 b x^{\frac{3}{2}}}{3 a^{2}} + \frac{2 b^{2} \sqrt{x}}{a^{3}} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0456197, size = 61, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (3 a^2 x^2-5 a b x+15 b^2\right )}{15 a^3}-\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \[{\frac{2}{5\,a}{x}^{{\frac{5}{2}}}}-{\frac{2\,b}{3\,{a}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{{b}^{2}\sqrt{x}}{{a}^{3}}}-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(a+b/x),x)

[Out]

2/5/a*x^(5/2)-2/3*b*x^(3/2)/a^2+2*b^2*x^(1/2)/a^3-2*b^3/a^3/(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(x^(3/2)/(a + b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239522, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt{x}}{15 \, a^{3}}, -\frac{2 \,{\left (15 \, b^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) -{\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt{x}\right )}}{15 \, a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 23.197, size = 121, normalized size = 1.78 \[ \begin{cases} \frac{2 x^{\frac{5}{2}}}{5 a} - \frac{2 b x^{\frac{3}{2}}}{3 a^{2}} + \frac{2 b^{2} \sqrt{x}}{a^{3}} + \frac{i b^{\frac{5}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{4} \sqrt{\frac{1}{a}}} - \frac{i b^{\frac{5}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{4} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{7}{2}}}{7 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(a+b/x),x)

[Out]

Piecewise((2*x**(5/2)/(5*a) - 2*b*x**(3/2)/(3*a**2) + 2*b**2*sqrt(x)/a**3 + I*b*
*(5/2)*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(a**4*sqrt(1/a)) - I*b**(5/2)*log(I*s
qrt(b)*sqrt(1/a) + sqrt(x))/(a**4*sqrt(1/a)), Ne(a, 0)), (2*x**(7/2)/(7*b), True
))

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GIAC/XCAS [A]  time = 0.217959, size = 80, normalized size = 1.18 \[ -\frac{2 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{2 \,{\left (3 \, a^{4} x^{\frac{5}{2}} - 5 \, a^{3} b x^{\frac{3}{2}} + 15 \, a^{2} b^{2} \sqrt{x}\right )}}{15 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) + 2/15*(3*a^4*x^(5/2) - 5*a^3
*b*x^(3/2) + 15*a^2*b^2*sqrt(x))/a^5