3.1677 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{7/2}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{1}{b \sqrt{x} (a x+b)}-\frac{3}{b^2 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.064875, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{1}{b \sqrt{x} (a x+b)}-\frac{3}{b^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^(7/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0875, size = 51, normalized size = 0.91 \[ - \frac{3 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{5}{2}}} + \frac{1}{b \sqrt{x} \left (a x + b\right )} - \frac{3}{b^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0623813, size = 54, normalized size = 0.96 \[ \frac{-3 a x-2 b}{b^2 \sqrt{x} (a x+b)}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^(7/2)),x]

[Out]

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

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Maple [A]  time = 0.018, size = 48, normalized size = 0.9 \[ -2\,{\frac{1}{{b}^{2}\sqrt{x}}}-{\frac{a}{{b}^{2} \left ( ax+b \right ) }\sqrt{x}}-3\,{\frac{a}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247833, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a x + b\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 6 \, a x - 4 \, b}{2 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{x}}, \frac{3 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) - 3 \, a x - 2 \, b}{{\left (a b^{2} x + b^{3}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^(7/2)),x, algorithm="fricas")

[Out]

[1/2*(3*(a*x + b)*sqrt(x)*sqrt(-a/b)*log((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x
+ b)) - 6*a*x - 4*b)/((a*b^2*x + b^3)*sqrt(x)), (3*(a*x + b)*sqrt(x)*sqrt(a/b)*
arctan(b*sqrt(a/b)/(a*sqrt(x))) - 3*a*x - 2*b)/((a*b^2*x + b^3)*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**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222794, size = 66, normalized size = 1.18 \[ -\frac{3 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} - \frac{3 \, a x + 2 \, b}{{\left (a x^{\frac{3}{2}} + b \sqrt{x}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^(7/2)),x, algorithm="giac")

[Out]

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