3.1686 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{9/2}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2}}+\frac{5}{4 b^2 \sqrt{x} (a x+b)}+\frac{1}{2 b \sqrt{x} (a x+b)^2}-\frac{15}{4 b^3 \sqrt{x}} \]

[Out]

-15/(4*b^3*Sqrt[x]) + 1/(2*b*Sqrt[x]*(b + a*x)^2) + 5/(4*b^2*Sqrt[x]*(b + a*x))
- (15*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^(7/2))

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Rubi [A]  time = 0.0843754, antiderivative size = 82, 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 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2}}+\frac{5}{4 b^2 \sqrt{x} (a x+b)}+\frac{1}{2 b \sqrt{x} (a x+b)^2}-\frac{15}{4 b^3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^(9/2)),x]

[Out]

-15/(4*b^3*Sqrt[x]) + 1/(2*b*Sqrt[x]*(b + a*x)^2) + 5/(4*b^2*Sqrt[x]*(b + a*x))
- (15*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^(7/2))

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Rubi in Sympy [A]  time = 14.3435, size = 75, normalized size = 0.91 \[ - \frac{15 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{7}{2}}} + \frac{1}{2 b \sqrt{x} \left (a x + b\right )^{2}} + \frac{5}{4 b^{2} \sqrt{x} \left (a x + b\right )} - \frac{15}{4 b^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**(9/2),x)

[Out]

-15*sqrt(a)*atan(sqrt(a)*sqrt(x)/sqrt(b))/(4*b**(7/2)) + 1/(2*b*sqrt(x)*(a*x + b
)**2) + 5/(4*b**2*sqrt(x)*(a*x + b)) - 15/(4*b**3*sqrt(x))

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Mathematica [A]  time = 0.0706817, size = 70, normalized size = 0.85 \[ -\frac{15 a^2 x^2+25 a b x+8 b^2}{4 b^3 \sqrt{x} (a x+b)^2}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^(9/2)),x]

[Out]

-(8*b^2 + 25*a*b*x + 15*a^2*x^2)/(4*b^3*Sqrt[x]*(b + a*x)^2) - (15*Sqrt[a]*ArcTa
n[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^(7/2))

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Maple [A]  time = 0.02, size = 66, normalized size = 0.8 \[ -2\,{\frac{1}{{b}^{3}\sqrt{x}}}-{\frac{7\,{a}^{2}}{4\,{b}^{3} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{9\,a}{4\,{b}^{2} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{15\,a}{4\,{b}^{3}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(9/2),x)

[Out]

-2/b^3/x^(1/2)-7/4/b^3*a^2/(a*x+b)^2*x^(3/2)-9/4/b^2*a/(a*x+b)^2*x^(1/2)-15/4/b^
3*a/(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^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246333, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, a^{2} x^{2} + 50 \, a b x - 15 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) + 16 \, b^{2}}{8 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{x}}, -\frac{15 \, a^{2} x^{2} + 25 \, a b x - 15 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) + 8 \, b^{2}}{4 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(9/2)),x, algorithm="fricas")

[Out]

[-1/8*(30*a^2*x^2 + 50*a*b*x - 15*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(x)*sqrt(-a/b)*l
og((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)) + 16*b^2)/((a^2*b^3*x^2 + 2*a*b
^4*x + b^5)*sqrt(x)), -1/4*(15*a^2*x^2 + 25*a*b*x - 15*(a^2*x^2 + 2*a*b*x + b^2)
*sqrt(x)*sqrt(a/b)*arctan(b*sqrt(a/b)/(a*sqrt(x))) + 8*b^2)/((a^2*b^3*x^2 + 2*a*
b^4*x + b^5)*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)**3/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216672, size = 80, normalized size = 0.98 \[ -\frac{15 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3}} - \frac{2}{b^{3} \sqrt{x}} - \frac{7 \, a^{2} x^{\frac{3}{2}} + 9 \, a b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(9/2)),x, algorithm="giac")

[Out]

-15/4*a*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^3) - 2/(b^3*sqrt(x)) - 1/4*(7*a
^2*x^(3/2) + 9*a*b*sqrt(x))/((a*x + b)^2*b^3)