Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2} \]
[Out]
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Rubi [A] time = 0.0574782, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(3/2)/x^3,x]
[Out]
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Rubi in Sympy [A] time = 6.83348, size = 31, normalized size = 0.82 \[ \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{2}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(3/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0378322, size = 36, normalized size = 0.95 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^2 (2 a x-5 b)}{35 b^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(3/2)/x^3,x]
[Out]
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Maple [A] time = 0.007, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-5\,b \right ) }{35\,{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(3/2)/x^3,x)
[Out]
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Maxima [A] time = 1.44632, size = 41, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{5 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225054, size = 66, normalized size = 1.74 \[ \frac{2 \,{\left (2 \, a^{3} x^{3} - a^{2} b x^{2} - 8 \, a b^{2} x - 5 \, b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{35 \, b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.52838, size = 360, normalized size = 9.47 \[ \frac{4 a^{\frac{15}{2}} b^{\frac{3}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} + \frac{2 a^{\frac{13}{2}} b^{\frac{5}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{18 a^{\frac{11}{2}} b^{\frac{7}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{26 a^{\frac{9}{2}} b^{\frac{9}{2}} x \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{10 a^{\frac{7}{2}} b^{\frac{11}{2}} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{4 a^{8} b x^{\frac{9}{2}}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{4 a^{7} b^{2} x^{\frac{7}{2}}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(3/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.261406, size = 239, normalized size = 6.29 \[ \frac{2 \,{\left (35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}}{\rm sign}\left (x\right ) + 105 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b{\rm sign}\left (x\right ) + 140 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{2}{\rm sign}\left (x\right ) + 98 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{3}{\rm sign}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{4}{\rm sign}\left (x\right ) + 5 \, b^{5}{\rm sign}\left (x\right )\right )}}{35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2)/x^3,x, algorithm="giac")
[Out]