3.1679 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{11/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}-\frac{7 a^2}{b^4 \sqrt{x}}+\frac{7 a}{3 b^3 x^{3/2}}+\frac{1}{b x^{5/2} (a x+b)}-\frac{7}{5 b^2 x^{5/2}} \]

[Out]

-7/(5*b^2*x^(5/2)) + (7*a)/(3*b^3*x^(3/2)) - (7*a^2)/(b^4*Sqrt[x]) + 1/(b*x^(5/2
)*(b + a*x)) - (7*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(9/2)

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Rubi [A]  time = 0.101034, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}}-\frac{7 a^2}{b^4 \sqrt{x}}+\frac{7 a}{3 b^3 x^{3/2}}+\frac{1}{b x^{5/2} (a x+b)}-\frac{7}{5 b^2 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^(11/2)),x]

[Out]

-7/(5*b^2*x^(5/2)) + (7*a)/(3*b^3*x^(3/2)) - (7*a^2)/(b^4*Sqrt[x]) + 1/(b*x^(5/2
)*(b + a*x)) - (7*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(9/2)

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Rubi in Sympy [A]  time = 17.7116, size = 80, normalized size = 0.95 \[ - \frac{7 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{9}{2}}} - \frac{7 a^{2}}{b^{4} \sqrt{x}} + \frac{7 a}{3 b^{3} x^{\frac{3}{2}}} + \frac{1}{b x^{\frac{5}{2}} \left (a x + b\right )} - \frac{7}{5 b^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**(11/2),x)

[Out]

-7*a**(5/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/b**(9/2) - 7*a**2/(b**4*sqrt(x)) + 7*a
/(3*b**3*x**(3/2)) + 1/(b*x**(5/2)*(a*x + b)) - 7/(5*b**2*x**(5/2))

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Mathematica [A]  time = 0.10279, size = 79, normalized size = 0.94 \[ \frac{-105 a^3 x^3-70 a^2 b x^2+14 a b^2 x-6 b^3}{15 b^4 x^{5/2} (a x+b)}-\frac{7 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^(11/2)),x]

[Out]

(-6*b^3 + 14*a*b^2*x - 70*a^2*b*x^2 - 105*a^3*x^3)/(15*b^4*x^(5/2)*(b + a*x)) -
(7*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(9/2)

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Maple [A]  time = 0.022, size = 72, normalized size = 0.9 \[ -{\frac{2}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-6\,{\frac{{a}^{2}}{{b}^{4}\sqrt{x}}}+{\frac{4\,a}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{{a}^{3}}{{b}^{4} \left ( ax+b \right ) }\sqrt{x}}-7\,{\frac{{a}^{3}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^(11/2),x)

[Out]

-2/5/b^2/x^(5/2)-6*a^2/b^4/x^(1/2)+4/3*a/b^3/x^(3/2)-a^3/b^4*x^(1/2)/(a*x+b)-7*a
^3/b^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(1/((a + b/x)^2*x^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240822, size = 1, normalized size = 0.01 \[ \left [-\frac{210 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} - 28 \, a b^{2} x + 12 \, b^{3} - 105 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right )}{30 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{x}}, -\frac{105 \, a^{3} x^{3} + 70 \, a^{2} b x^{2} - 14 \, a b^{2} x + 6 \, b^{3} - 105 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right )}{15 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^(11/2)),x, algorithm="fricas")

[Out]

[-1/30*(210*a^3*x^3 + 140*a^2*b*x^2 - 28*a*b^2*x + 12*b^3 - 105*(a^3*x^3 + a^2*b
*x^2)*sqrt(x)*sqrt(-a/b)*log((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)))/((a*
b^4*x^3 + b^5*x^2)*sqrt(x)), -1/15*(105*a^3*x^3 + 70*a^2*b*x^2 - 14*a*b^2*x + 6*
b^3 - 105*(a^3*x^3 + a^2*b*x^2)*sqrt(x)*sqrt(a/b)*arctan(b*sqrt(a/b)/(a*sqrt(x))
))/((a*b^4*x^3 + b^5*x^2)*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**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221498, size = 95, normalized size = 1.13 \[ -\frac{7 \, a^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} - \frac{a^{3} \sqrt{x}}{{\left (a x + b\right )} b^{4}} - \frac{2 \,{\left (45 \, a^{2} x^{2} - 10 \, a b x + 3 \, b^{2}\right )}}{15 \, b^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^(11/2)),x, algorithm="giac")

[Out]

-7*a^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) - a^3*sqrt(x)/((a*x + b)*b^4)
- 2/15*(45*a^2*x^2 - 10*a*b*x + 3*b^2)/(b^4*x^(5/2))