Optimal. Leaf size=56 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}} \]
[Out]
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Rubi [A] time = 0.106577, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[b*Sqrt[x] + a*x],x]
[Out]
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Rubi in Sympy [A] time = 9.47211, size = 49, normalized size = 0.88 \[ \frac{2 \sqrt{a x + b \sqrt{x}}}{a} - \frac{2 b \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**(1/2)+a*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0471505, size = 62, normalized size = 1.11 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{b \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[b*Sqrt[x] + a*x],x]
[Out]
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Maple [A] time = 0.005, size = 83, normalized size = 1.5 \[ -{1\sqrt{b\sqrt{x}+ax} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b \right ){\frac{1}{\sqrt{a}}}} \right ) -2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^(1/2)+a*x)^(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(1/sqrt(a*x + b*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a*x + b*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x + b \sqrt{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**(1/2)+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.274202, size = 73, normalized size = 1.3 \[ \frac{b{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{a^{\frac{3}{2}}} + \frac{2 \, \sqrt{a x + b \sqrt{x}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a*x + b*sqrt(x)),x, algorithm="giac")
[Out]