Optimal. Leaf size=100 \[ \frac{32 b^3}{5 a^4 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{16 b^2 \sqrt{x}}{5 a^3 \sqrt{a+\frac{b}{x}}}-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}} \]
[Out]
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Rubi [A] time = 0.116165, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{32 b^3}{5 a^4 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{16 b^2 \sqrt{x}}{5 a^3 \sqrt{a+\frac{b}{x}}}-\frac{4 b x^{3/2}}{5 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{5/2}}{5 a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(a + b/x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.78856, size = 87, normalized size = 0.87 \[ \frac{2 x^{\frac{5}{2}}}{5 a \sqrt{a + \frac{b}{x}}} - \frac{4 b x^{\frac{3}{2}}}{5 a^{2} \sqrt{a + \frac{b}{x}}} + \frac{16 b^{2} \sqrt{x}}{5 a^{3} \sqrt{a + \frac{b}{x}}} + \frac{32 b^{3}}{5 a^{4} \sqrt{x} \sqrt{a + \frac{b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(a+b/x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0523591, size = 59, normalized size = 0.59 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (a^3 x^3-2 a^2 b x^2+8 a b^2 x+16 b^3\right )}{5 a^4 (a x+b)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(a + b/x)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 54, normalized size = 0.5 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{3}{x}^{3}-2\,{a}^{2}b{x}^{2}+8\,a{b}^{2}x+16\,{b}^{3} \right ) }{5\,{a}^{4}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(a+b/x)^(3/2),x)
[Out]
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Maxima [A] time = 1.43729, size = 97, normalized size = 0.97 \[ \frac{2 \, b^{3}}{\sqrt{a + \frac{b}{x}} a^{4} \sqrt{x}} + \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}} - 5 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b x^{\frac{3}{2}} + 15 \, \sqrt{a + \frac{b}{x}} b^{2} \sqrt{x}\right )}}{5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(a + b/x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240053, size = 65, normalized size = 0.65 \[ \frac{2 \,{\left (a^{3} x^{3} - 2 \, a^{2} b x^{2} + 8 \, a b^{2} x + 16 \, b^{3}\right )}}{5 \, a^{4} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(a + b/x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 67.3706, size = 320, normalized size = 3.2 \[ \frac{2 a^{5} b^{\frac{19}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{10 a^{3} b^{\frac{23}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{60 a^{2} b^{\frac{25}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{80 a b^{\frac{27}{2}} x \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} + \frac{32 b^{\frac{29}{2}} \sqrt{\frac{a x}{b} + 1}}{5 a^{7} b^{9} x^{3} + 15 a^{6} b^{10} x^{2} + 15 a^{5} b^{11} x + 5 a^{4} b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(a+b/x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227292, size = 76, normalized size = 0.76 \[ -\frac{32 \, b^{\frac{5}{2}}}{5 \, a^{4}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x + b} b^{2} + \frac{5 \, b^{3}}{\sqrt{a x + b}}\right )}}{5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(a + b/x)^(3/2),x, algorithm="giac")
[Out]