Optimal. Leaf size=42 \[ \sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}} \]
[Out]
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Rubi [A] time = 0.0547881, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x + c*x^2]/x,x]
[Out]
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Rubi in Sympy [A] time = 5.71714, size = 37, normalized size = 0.88 \[ \frac{b \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c}} + \sqrt{b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.0422125, size = 59, normalized size = 1.4 \[ \sqrt{x (b+c x)} \left (\frac{b \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} \sqrt{x} \sqrt{b+c x}}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x + c*x^2]/x,x]
[Out]
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Maple [A] time = 0.005, size = 43, normalized size = 1. \[ \sqrt{c{x}^{2}+bx}+{\frac{b}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/2)/x,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(c*x^2 + b*x)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24782, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}}{2 \, \sqrt{c}}, \frac{b \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x} \sqrt{-c}}{\sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.220815, size = 65, normalized size = 1.55 \[ -\frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} + \sqrt{c x^{2} + b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x,x, algorithm="giac")
[Out]