Optimal. Leaf size=51 \[ -\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]
[Out]
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Rubi [A] time = 0.0762548, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*Sqrt[x]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{3} \log{\left (a + b \sqrt{x} \right )}}{b^{4}} - \frac{2 a \int ^{\sqrt{x}} x\, dx}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} + \frac{2 \int ^{\sqrt{x}} a^{2}\, dx}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*x**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0162078, size = 51, normalized size = 1. \[ -\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*Sqrt[x]),x]
[Out]
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Maple [A] time = 0.004, size = 44, normalized size = 0.9 \[ -{\frac{ax}{{b}^{2}}}+{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}+2\,{\frac{{a}^{2}\sqrt{x}}{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*x^(1/2)),x)
[Out]
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Maxima [A] time = 1.44189, size = 82, normalized size = 1.61 \[ -\frac{2 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{4}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*sqrt(x) + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237705, size = 58, normalized size = 1.14 \[ -\frac{3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt{x}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*sqrt(x) + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.6424, size = 49, normalized size = 0.96 \[ - \frac{2 a^{3} \log{\left (1 + \frac{b \sqrt{x}}{a} \right )}}{b^{4}} + \frac{2 a^{2} \sqrt{x}}{b^{3}} - \frac{a x}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*x**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.220052, size = 61, normalized size = 1.2 \[ -\frac{2 \, a^{3}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, b^{2} x^{\frac{3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt{x}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*sqrt(x) + a),x, algorithm="giac")
[Out]