3.1680 \(\int \frac{x^{3/2}}{\left (a+\frac{b}{x}\right )^3} \, dx\)

Optimal. Leaf size=110 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{11/2}}+\frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{9 x^{7/2}}{4 a^2 (a x+b)}-\frac{x^{9/2}}{2 a (a x+b)^2} \]

[Out]

(63*b^2*Sqrt[x])/(4*a^5) - (21*b*x^(3/2))/(4*a^4) + (63*x^(5/2))/(20*a^3) - x^(9
/2)/(2*a*(b + a*x)^2) - (9*x^(7/2))/(4*a^2*(b + a*x)) - (63*b^(5/2)*ArcTan[(Sqrt
[a]*Sqrt[x])/Sqrt[b]])/(4*a^(11/2))

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Rubi [A]  time = 0.121253, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{11/2}}+\frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{9 x^{7/2}}{4 a^2 (a x+b)}-\frac{x^{9/2}}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b/x)^3,x]

[Out]

(63*b^2*Sqrt[x])/(4*a^5) - (21*b*x^(3/2))/(4*a^4) + (63*x^(5/2))/(20*a^3) - x^(9
/2)/(2*a*(b + a*x)^2) - (9*x^(7/2))/(4*a^2*(b + a*x)) - (63*b^(5/2)*ArcTan[(Sqrt
[a]*Sqrt[x])/Sqrt[b]])/(4*a^(11/2))

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Rubi in Sympy [A]  time = 22.2779, size = 102, normalized size = 0.93 \[ - \frac{x^{\frac{9}{2}}}{2 a \left (a x + b\right )^{2}} - \frac{9 x^{\frac{7}{2}}}{4 a^{2} \left (a x + b\right )} + \frac{63 x^{\frac{5}{2}}}{20 a^{3}} - \frac{21 b x^{\frac{3}{2}}}{4 a^{4}} + \frac{63 b^{2} \sqrt{x}}{4 a^{5}} - \frac{63 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(9/2)/(2*a*(a*x + b)**2) - 9*x**(7/2)/(4*a**2*(a*x + b)) + 63*x**(5/2)/(20*a
**3) - 21*b*x**(3/2)/(4*a**4) + 63*b**2*sqrt(x)/(4*a**5) - 63*b**(5/2)*atan(sqrt
(a)*sqrt(x)/sqrt(b))/(4*a**(11/2))

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Mathematica [A]  time = 0.0854287, size = 92, normalized size = 0.84 \[ \frac{\sqrt{x} \left (8 a^4 x^4-24 a^3 b x^3+168 a^2 b^2 x^2+525 a b^3 x+315 b^4\right )}{20 a^5 (a x+b)^2}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(315*b^4 + 525*a*b^3*x + 168*a^2*b^2*x^2 - 24*a^3*b*x^3 + 8*a^4*x^4))/(
20*a^5*(b + a*x)^2) - (63*b^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*a^(11/2)
)

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Maple [A]  time = 0.019, size = 90, normalized size = 0.8 \[{\frac{2}{5\,{a}^{3}}{x}^{{\frac{5}{2}}}}-2\,{\frac{b{x}^{3/2}}{{a}^{4}}}+12\,{\frac{{b}^{2}\sqrt{x}}{{a}^{5}}}+{\frac{17\,{b}^{3}}{4\,{a}^{4} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{15\,{b}^{4}}{4\,{a}^{5} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{63\,{b}^{3}}{4\,{a}^{5}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(a+b/x)^3,x)

[Out]

2/5*x^(5/2)/a^3-2*b*x^(3/2)/a^4+12*b^2*x^(1/2)/a^5+17/4/a^4*b^3/(a*x+b)^2*x^(3/2
)+15/4/a^5*b^4/(a*x+b)^2*x^(1/2)-63/4/a^5*b^3/(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(x^(3/2)/(a + b/x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239578, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (8 \, a^{4} x^{4} - 24 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} + 525 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt{x}}{40 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}, -\frac{315 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) -{\left (8 \, a^{4} x^{4} - 24 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} + 525 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt{x}}{20 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/40*(315*(a^2*b^2*x^2 + 2*a*b^3*x + b^4)*sqrt(-b/a)*log((a*x - 2*a*sqrt(x)*sqr
t(-b/a) - b)/(a*x + b)) + 2*(8*a^4*x^4 - 24*a^3*b*x^3 + 168*a^2*b^2*x^2 + 525*a*
b^3*x + 315*b^4)*sqrt(x))/(a^7*x^2 + 2*a^6*b*x + a^5*b^2), -1/20*(315*(a^2*b^2*x
^2 + 2*a*b^3*x + b^4)*sqrt(b/a)*arctan(sqrt(x)/sqrt(b/a)) - (8*a^4*x^4 - 24*a^3*
b*x^3 + 168*a^2*b^2*x^2 + 525*a*b^3*x + 315*b^4)*sqrt(x))/(a^7*x^2 + 2*a^6*b*x +
a^5*b^2)]

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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(x**(3/2)/(a+b/x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229399, size = 119, normalized size = 1.08 \[ -\frac{63 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{5}} + \frac{17 \, a b^{3} x^{\frac{3}{2}} + 15 \, b^{4} \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a^{5}} + \frac{2 \,{\left (a^{12} x^{\frac{5}{2}} - 5 \, a^{11} b x^{\frac{3}{2}} + 30 \, a^{10} b^{2} \sqrt{x}\right )}}{5 \, a^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-63/4*b^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/4*(17*a*b^3*x^(3/2) +
15*b^4*sqrt(x))/((a*x + b)^2*a^5) + 2/5*(a^12*x^(5/2) - 5*a^11*b*x^(3/2) + 30*a^
10*b^2*sqrt(x))/a^15