3.1685 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{7/2}} \, dx\)

Optimal. Leaf size=70 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{3 \sqrt{x}}{4 b^2 (a x+b)}+\frac{\sqrt{x}}{2 b (a x+b)^2} \]

[Out]

Sqrt[x]/(2*b*(b + a*x)^2) + (3*Sqrt[x])/(4*b^2*(b + a*x)) + (3*ArcTan[(Sqrt[a]*S
qrt[x])/Sqrt[b]])/(4*Sqrt[a]*b^(5/2))

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Rubi [A]  time = 0.0681785, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{3 \sqrt{x}}{4 b^2 (a x+b)}+\frac{\sqrt{x}}{2 b (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^(7/2)),x]

[Out]

Sqrt[x]/(2*b*(b + a*x)^2) + (3*Sqrt[x])/(4*b^2*(b + a*x)) + (3*ArcTan[(Sqrt[a]*S
qrt[x])/Sqrt[b]])/(4*Sqrt[a]*b^(5/2))

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Rubi in Sympy [A]  time = 11.5646, size = 61, normalized size = 0.87 \[ \frac{\sqrt{x}}{2 b \left (a x + b\right )^{2}} + \frac{3 \sqrt{x}}{4 b^{2} \left (a x + b\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**(7/2),x)

[Out]

sqrt(x)/(2*b*(a*x + b)**2) + 3*sqrt(x)/(4*b**2*(a*x + b)) + 3*atan(sqrt(a)*sqrt(
x)/sqrt(b))/(4*sqrt(a)*b**(5/2))

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Mathematica [A]  time = 0.051301, size = 59, normalized size = 0.84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{\sqrt{x} (3 a x+5 b)}{4 b^2 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^(7/2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \[{\frac{1}{2\,b \left ( ax+b \right ) ^{2}}\sqrt{x}}+{\frac{3}{4\,{b}^{2} \left ( ax+b \right ) }\sqrt{x}}+{\frac{3}{4\,{b}^{2}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24224, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b}{\left (3 \, a x + 5 \, b\right )} \sqrt{x} + 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{8 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{-a b}}, \frac{\sqrt{a b}{\left (3 \, a x + 5 \, b\right )} \sqrt{x} - 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(7/2)),x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(-a*b)*(3*a*x + 5*b)*sqrt(x) + 3*(a^2*x^2 + 2*a*b*x + b^2)*log((2*a*
b*sqrt(x) + sqrt(-a*b)*(a*x - b))/(a*x + b)))/((a^2*b^2*x^2 + 2*a*b^3*x + b^4)*s
qrt(-a*b)), 1/4*(sqrt(a*b)*(3*a*x + 5*b)*sqrt(x) - 3*(a^2*x^2 + 2*a*b*x + b^2)*a
rctan(b/(sqrt(a*b)*sqrt(x))))/((a^2*b^2*x^2 + 2*a*b^3*x + b^4)*sqrt(a*b))]

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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)**3/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222764, size = 63, normalized size = 0.9 \[ \frac{3 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{2}} + \frac{3 \, a x^{\frac{3}{2}} + 5 \, b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(7/2)),x, algorithm="giac")

[Out]

3/4*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/4*(3*a*x^(3/2) + 5*b*sqrt(x)
)/((a*x + b)^2*b^2)