3.1663 \(\int \frac{x^{5/2}}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=83 \[ \frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a} \]

[Out]

(-2*b^3*Sqrt[x])/a^4 + (2*b^2*x^(3/2))/(3*a^3) - (2*b*x^(5/2))/(5*a^2) + (2*x^(7
/2))/(7*a) + (2*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(9/2)

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Rubi [A]  time = 0.113574, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b/x),x]

[Out]

(-2*b^3*Sqrt[x])/a^4 + (2*b^2*x^(3/2))/(3*a^3) - (2*b*x^(5/2))/(5*a^2) + (2*x^(7
/2))/(7*a) + (2*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(9/2)

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Rubi in Sympy [A]  time = 17.7212, size = 80, normalized size = 0.96 \[ \frac{2 x^{\frac{7}{2}}}{7 a} - \frac{2 b x^{\frac{5}{2}}}{5 a^{2}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3 a^{3}} - \frac{2 b^{3} \sqrt{x}}{a^{4}} + \frac{2 b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(7/2)/(7*a) - 2*b*x**(5/2)/(5*a**2) + 2*b**2*x**(3/2)/(3*a**3) - 2*b**3*sqr
t(x)/a**4 + 2*b**(7/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/a**(9/2)

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Mathematica [A]  time = 0.056533, size = 72, normalized size = 0.87 \[ \frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}+\frac{2 \sqrt{x} \left (15 a^3 x^3-21 a^2 b x^2+35 a b^2 x-105 b^3\right )}{105 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(a + b/x),x]

[Out]

(2*Sqrt[x]*(-105*b^3 + 35*a*b^2*x - 21*a^2*b*x^2 + 15*a^3*x^3))/(105*a^4) + (2*b
^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(9/2)

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Maple [A]  time = 0.012, size = 65, normalized size = 0.8 \[{\frac{2}{7\,a}{x}^{{\frac{7}{2}}}}-{\frac{2\,b}{5\,{a}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}}{3\,{a}^{3}}{x}^{{\frac{3}{2}}}}-2\,{\frac{{b}^{3}\sqrt{x}}{{a}^{4}}}+2\,{\frac{{b}^{4}}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(a+b/x),x)

[Out]

2/7*x^(7/2)/a-2/5*b*x^(5/2)/a^2+2/3*b^2*x^(3/2)/a^3-2*b^3*x^(1/2)/a^4+2*b^4/a^4/
(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^(5/2)/(a + b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240978, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (15 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} + 35 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt{x}}{105 \, a^{4}}, \frac{2 \,{\left (105 \, b^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) +{\left (15 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} + 35 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt{x}\right )}}{105 \, a^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*b^3*sqrt(-b/a)*log((a*x + 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x + b)) + 2
*(15*a^3*x^3 - 21*a^2*b*x^2 + 35*a*b^2*x - 105*b^3)*sqrt(x))/a^4, 2/105*(105*b^3
*sqrt(b/a)*arctan(sqrt(x)/sqrt(b/a)) + (15*a^3*x^3 - 21*a^2*b*x^2 + 35*a*b^2*x -
105*b^3)*sqrt(x))/a^4]

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Sympy [A]  time = 73.2032, size = 136, normalized size = 1.64 \[ \begin{cases} \frac{2 x^{\frac{7}{2}}}{7 a} - \frac{2 b x^{\frac{5}{2}}}{5 a^{2}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3 a^{3}} - \frac{2 b^{3} \sqrt{x}}{a^{4}} - \frac{i b^{\frac{7}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{5} \sqrt{\frac{1}{a}}} + \frac{i b^{\frac{7}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{5} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{9}{2}}}{9 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(a+b/x),x)

[Out]

Piecewise((2*x**(7/2)/(7*a) - 2*b*x**(5/2)/(5*a**2) + 2*b**2*x**(3/2)/(3*a**3) -
2*b**3*sqrt(x)/a**4 - I*b**(7/2)*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(a**5*sqrt
(1/a)) + I*b**(7/2)*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))/(a**5*sqrt(1/a)), Ne(a, 0
)), (2*x**(9/2)/(9*b), True))

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GIAC/XCAS [A]  time = 0.224548, size = 95, normalized size = 1.14 \[ \frac{2 \, b^{4} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{2 \,{\left (15 \, a^{6} x^{\frac{7}{2}} - 21 \, a^{5} b x^{\frac{5}{2}} + 35 \, a^{4} b^{2} x^{\frac{3}{2}} - 105 \, a^{3} b^{3} \sqrt{x}\right )}}{105 \, a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^4*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + 2/105*(15*a^6*x^(7/2) - 21*a
^5*b*x^(5/2) + 35*a^4*b^2*x^(3/2) - 105*a^3*b^3*sqrt(x))/a^7