3.1687 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{11/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{7}{4 b^2 x^{3/2} (a x+b)}+\frac{1}{2 b x^{3/2} (a x+b)^2}-\frac{35}{12 b^3 x^{3/2}} \]

[Out]

-35/(12*b^3*x^(3/2)) + (35*a)/(4*b^4*Sqrt[x]) + 1/(2*b*x^(3/2)*(b + a*x)^2) + 7/
(4*b^2*x^(3/2)*(b + a*x)) + (35*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^
(9/2))

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Rubi [A]  time = 0.100227, antiderivative size = 95, 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 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}+\frac{35 a}{4 b^4 \sqrt{x}}+\frac{7}{4 b^2 x^{3/2} (a x+b)}+\frac{1}{2 b x^{3/2} (a x+b)^2}-\frac{35}{12 b^3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^(11/2)),x]

[Out]

-35/(12*b^3*x^(3/2)) + (35*a)/(4*b^4*Sqrt[x]) + 1/(2*b*x^(3/2)*(b + a*x)^2) + 7/
(4*b^2*x^(3/2)*(b + a*x)) + (35*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^
(9/2))

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Rubi in Sympy [A]  time = 17.5837, size = 88, normalized size = 0.93 \[ \frac{35 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{9}{2}}} + \frac{35 a}{4 b^{4} \sqrt{x}} + \frac{1}{2 b x^{\frac{3}{2}} \left (a x + b\right )^{2}} + \frac{7}{4 b^{2} x^{\frac{3}{2}} \left (a x + b\right )} - \frac{35}{12 b^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**(11/2),x)

[Out]

35*a**(3/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/(4*b**(9/2)) + 35*a/(4*b**4*sqrt(x)) +
1/(2*b*x**(3/2)*(a*x + b)**2) + 7/(4*b**2*x**(3/2)*(a*x + b)) - 35/(12*b**3*x**
(3/2))

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Mathematica [A]  time = 0.0800841, size = 81, normalized size = 0.85 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}+\frac{105 a^3 x^3+175 a^2 b x^2+56 a b^2 x-8 b^3}{12 b^4 x^{3/2} (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^(11/2)),x]

[Out]

(-8*b^3 + 56*a*b^2*x + 175*a^2*b*x^2 + 105*a^3*x^3)/(12*b^4*x^(3/2)*(b + a*x)^2)
+ (35*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^(9/2))

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Maple [A]  time = 0.023, size = 79, normalized size = 0.8 \[ -{\frac{2}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{a}{{b}^{4}\sqrt{x}}}+{\frac{11\,{a}^{3}}{4\,{b}^{4} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,{a}^{2}}{4\,{b}^{3} \left ( ax+b \right ) ^{2}}\sqrt{x}}+{\frac{35\,{a}^{2}}{4\,{b}^{4}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(11/2),x)

[Out]

-2/3/b^3/x^(3/2)+6*a/b^4/x^(1/2)+11/4/b^4*a^3/(a*x+b)^2*x^(3/2)+13/4/b^3*a^2/(a*
x+b)^2*x^(1/2)+35/4/b^4*a^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^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24596, size = 1, normalized size = 0.01 \[ \left [\frac{210 \, a^{3} x^{3} + 350 \, a^{2} b x^{2} + 112 \, a b^{2} x - 16 \, b^{3} + 105 \,{\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right )}{24 \,{\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )} \sqrt{x}}, \frac{105 \, a^{3} x^{3} + 175 \, a^{2} b x^{2} + 56 \, a b^{2} x - 8 \, b^{3} - 105 \,{\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right )}{12 \,{\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(11/2)),x, algorithm="fricas")

[Out]

[1/24*(210*a^3*x^3 + 350*a^2*b*x^2 + 112*a*b^2*x - 16*b^3 + 105*(a^3*x^3 + 2*a^2
*b*x^2 + a*b^2*x)*sqrt(x)*sqrt(-a/b)*log((a*x + 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x
+ b)))/((a^2*b^4*x^3 + 2*a*b^5*x^2 + b^6*x)*sqrt(x)), 1/12*(105*a^3*x^3 + 175*a
^2*b*x^2 + 56*a*b^2*x - 8*b^3 - 105*(a^3*x^3 + 2*a^2*b*x^2 + a*b^2*x)*sqrt(x)*sq
rt(a/b)*arctan(b*sqrt(a/b)/(a*sqrt(x))))/((a^2*b^4*x^3 + 2*a*b^5*x^2 + b^6*x)*sq
rt(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**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222389, size = 96, normalized size = 1.01 \[ \frac{35 \, a^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{4}} + \frac{2 \,{\left (9 \, a x - b\right )}}{3 \, b^{4} x^{\frac{3}{2}}} + \frac{11 \, a^{3} x^{\frac{3}{2}} + 13 \, a^{2} b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(11/2)),x, algorithm="giac")

[Out]

35/4*a^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/3*(9*a*x - b)/(b^4*x^(3
/2)) + 1/4*(11*a^3*x^(3/2) + 13*a^2*b*sqrt(x))/((a*x + b)^2*b^4)