3.1678 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{9/2}} \, dx\)

Optimal. Leaf size=69 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (a x+b)}-\frac{5}{3 b^2 x^{3/2}} \]

[Out]

-5/(3*b^2*x^(3/2)) + (5*a)/(b^3*Sqrt[x]) + 1/(b*x^(3/2)*(b + a*x)) + (5*a^(3/2)*
ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(7/2)

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Rubi [A]  time = 0.0798524, antiderivative size = 69, 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 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (a x+b)}-\frac{5}{3 b^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^(9/2)),x]

[Out]

-5/(3*b^2*x^(3/2)) + (5*a)/(b^3*Sqrt[x]) + 1/(b*x^(3/2)*(b + a*x)) + (5*a^(3/2)*
ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(7/2)

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Rubi in Sympy [A]  time = 14.0516, size = 65, normalized size = 0.94 \[ \frac{5 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{7}{2}}} + \frac{5 a}{b^{3} \sqrt{x}} + \frac{1}{b x^{\frac{3}{2}} \left (a x + b\right )} - \frac{5}{3 b^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(3/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/b**(7/2) + 5*a/(b**3*sqrt(x)) + 1/(b*x*
*(3/2)*(a*x + b)) - 5/(3*b**2*x**(3/2))

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Mathematica [A]  time = 0.0753566, size = 68, normalized size = 0.99 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{15 a^2 x^2+10 a b x-2 b^2}{3 b^3 x^{3/2} (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 60, normalized size = 0.9 \[ -{\frac{2}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{a}{{b}^{3}\sqrt{x}}}+{\frac{{a}^{2}}{{b}^{3} \left ( ax+b \right ) }\sqrt{x}}+5\,{\frac{{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^(9/2),x)

[Out]

-2/3/b^2/x^(3/2)+4*a/b^3/x^(1/2)+a^2/b^3*x^(1/2)/(a*x+b)+5*a^2/b^3/(a*b)^(1/2)*a
rctan(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*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239867, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, a^{2} x^{2} + 20 \, a b x + 15 \,{\left (a^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 4 \, b^{2}}{6 \,{\left (a b^{3} x^{2} + b^{4} x\right )} \sqrt{x}}, \frac{15 \, a^{2} x^{2} + 10 \, a b x - 15 \,{\left (a^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) - 2 \, b^{2}}{3 \,{\left (a b^{3} x^{2} + b^{4} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(30*a^2*x^2 + 20*a*b*x + 15*(a^2*x^2 + a*b*x)*sqrt(x)*sqrt(-a/b)*log((a*x +
2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)) - 4*b^2)/((a*b^3*x^2 + b^4*x)*sqrt(x)),
1/3*(15*a^2*x^2 + 10*a*b*x - 15*(a^2*x^2 + a*b*x)*sqrt(x)*sqrt(a/b)*arctan(b*sqr
t(a/b)/(a*sqrt(x))) - 2*b^2)/((a*b^3*x^2 + b^4*x)*sqrt(x))]

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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)**2/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219286, size = 78, normalized size = 1.13 \[ \frac{5 \, a^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{a^{2} \sqrt{x}}{{\left (a x + b\right )} b^{3}} + \frac{2 \,{\left (6 \, a x - b\right )}}{3 \, b^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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