Optimal. Leaf size=96 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^3}-\frac{5 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^2}+\frac{x^3 \sqrt{a+\frac{b}{x}}}{3 a} \]
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Rubi [A] time = 0.129853, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^3}-\frac{5 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^2}+\frac{x^3 \sqrt{a+\frac{b}{x}}}{3 a} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[a + b/x],x]
[Out]
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Rubi in Sympy [A] time = 12.4221, size = 82, normalized size = 0.85 \[ \frac{x^{3} \sqrt{a + \frac{b}{x}}}{3 a} - \frac{5 b x^{2} \sqrt{a + \frac{b}{x}}}{12 a^{2}} + \frac{5 b^{2} x \sqrt{a + \frac{b}{x}}}{8 a^{3}} - \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b/x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.11871, size = 77, normalized size = 0.8 \[ \frac{x \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2-10 a b x+15 b^2\right )}{24 a^3}-\frac{5 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{16 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[a + b/x],x]
[Out]
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Maple [B] time = 0.014, size = 168, normalized size = 1.8 \[{\frac{x}{48}\sqrt{{\frac{ax+b}{x}}} \left ( 16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{9/2}-36\,b\sqrt{a{x}^{2}+bx}x{a}^{9/2}-18\,\sqrt{a{x}^{2}+bx}{b}^{2}{a}^{7/2}+48\,{b}^{2}\sqrt{x \left ( ax+b \right ) }{a}^{7/2}+9\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}-24\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b/x)^(1/2),x)
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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^2/sqrt(a + b/x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236446, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{2} x^{3} - 10 \, a b x^{2} + 15 \, b^{2} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{48 \, a^{\frac{7}{2}}}, \frac{15 \, b^{3} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (8 \, a^{2} x^{3} - 10 \, a b x^{2} + 15 \, b^{2} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-a} a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a + b/x),x, algorithm="fricas")
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Sympy [A] time = 20.7485, size = 128, normalized size = 1.33 \[ \frac{x^{\frac{7}{2}}}{3 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{5}{2}}}{12 a \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{5}{2}} \sqrt{x}}{8 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b/x)**(1/2),x)
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GIAC/XCAS [A] time = 0.253, size = 169, normalized size = 1.76 \[ \frac{1}{24} \, b{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{33 \, a^{2} b^{2} \sqrt{\frac{a x + b}{x}} - \frac{40 \,{\left (a x + b\right )} a b^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{15 \,{\left (a x + b\right )}^{2} b^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{{\left (a - \frac{a x + b}{x}\right )}^{3} a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a + b/x),x, algorithm="giac")
[Out]