Optimal. Leaf size=56 \[ \frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.0700286, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.46243, size = 49, normalized size = 0.88 \[ \frac{2 \sqrt{x}}{b \sqrt{b x + c x^{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.039948, size = 56, normalized size = 1. \[ \frac{2 \sqrt{x} \left (\sqrt{b}-\sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{b^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.015, size = 51, normalized size = 0.9 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{{b}^{3/2}\sqrt{x} \left ( cx+b \right ) } \left ({\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}-\sqrt{b} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(c*x^2+b*x)^(3/2),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)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235416, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c x^{2} + b x\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{{\left (b c x^{2} + b^{2} x\right )} \sqrt{b}}, -\frac{2 \,{\left ({\left (c x^{2} + b x\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) - \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}\right )}}{{\left (b c x^{2} + b^{2} x\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213929, size = 90, normalized size = 1.61 \[ \frac{2 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{2}{\sqrt{c x + b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]