3.1670 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^{9/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0814021, 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 \[ -\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}-\frac{2 a^2}{b^3 \sqrt{x}}+\frac{2 a}{3 b^2 x^{3/2}}-\frac{2}{5 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^(9/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.0524, size = 65, normalized size = 0.96 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{7}{2}}} - \frac{2 a^{2}}{b^{3} \sqrt{x}} + \frac{2 a}{3 b^{2} x^{\frac{3}{2}}} - \frac{2}{5 b x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)/x**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.0552518, size = 61, normalized size = 0.9 \[ -\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}-\frac{2 \left (15 a^2 x^2-5 a b x+3 b^2\right )}{15 b^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x^(9/2)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 54, normalized size = 0.8 \[ -{\frac{2}{5\,b}{x}^{-{\frac{5}{2}}}}-2\,{\frac{{a}^{2}}{{b}^{3}\sqrt{x}}}+{\frac{2\,a}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{{a}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^(9/2),x)

[Out]

-2/5/b/x^(5/2)-2*a^2/b^3/x^(1/2)+2/3*a/b^2/x^(3/2)-2*a^3/b^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(1/((a + b/x)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242949, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} x^{\frac{5}{2}} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 30 \, a^{2} x^{2} + 10 \, a b x - 6 \, b^{2}}{15 \, b^{3} x^{\frac{5}{2}}}, \frac{2 \,{\left (15 \, a^{2} x^{\frac{5}{2}} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) - 15 \, a^{2} x^{2} + 5 \, a b x - 3 \, b^{2}\right )}}{15 \, b^{3} x^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^(9/2)),x, algorithm="fricas")

[Out]

[1/15*(15*a^2*x^(5/2)*sqrt(-a/b)*log((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b
)) - 30*a^2*x^2 + 10*a*b*x - 6*b^2)/(b^3*x^(5/2)), 2/15*(15*a^2*x^(5/2)*sqrt(a/b
)*arctan(b*sqrt(a/b)/(a*sqrt(x))) - 15*a^2*x^2 + 5*a*b*x - 3*b^2)/(b^3*x^(5/2))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220148, size = 70, normalized size = 1.03 \[ -\frac{2 \, a^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{2 \,{\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )}}{15 \, b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)*x^(9/2)),x, algorithm="giac")

[Out]

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