Optimal. Leaf size=48 \[ \frac{\sqrt{x} \sqrt{a+b x}}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.0353076, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{x} \sqrt{a+b x}}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 5.1483, size = 41, normalized size = 0.85 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} + \frac{\sqrt{x} \sqrt{a + b x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0303264, size = 51, normalized size = 1.06 \[ \frac{\sqrt{x} \sqrt{a+b x}}{b}-\frac{a \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/Sqrt[a + b*x],x]
[Out]
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Maple [A] time = 0.007, size = 65, normalized size = 1.4 \[{\frac{1}{b}\sqrt{x}\sqrt{bx+a}}-{\frac{a}{2}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(b*x+a)^(1/2),x)
[Out]
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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(sqrt(x)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224826, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x}}{2 \, b^{\frac{3}{2}}}, -\frac{a \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) - \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{\sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.22372, size = 44, normalized size = 0.92 \[ \frac{\sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{b} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/sqrt(b*x + a),x, algorithm="giac")
[Out]