Optimal. Leaf size=54 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}-\frac{\sqrt{b x+c x^2}}{x^{3/2}} \]
[Out]
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Rubi [A] time = 0.0692107, antiderivative size = 54, 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 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}-\frac{\sqrt{b x+c x^2}}{x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x + c*x^2]/x^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 7.97257, size = 48, normalized size = 0.89 \[ - \frac{\sqrt{b x + c x^{2}}}{x^{\frac{3}{2}}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/2)/x**(5/2),x)
[Out]
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Mathematica [A] time = 0.0340337, size = 68, normalized size = 1.26 \[ -\frac{\sqrt{x (b+c x)}}{x^{3/2}}-\frac{c \sqrt{x (b+c x)} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{x} \sqrt{b+c x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x + c*x^2]/x^(5/2),x]
[Out]
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Maple [A] time = 0.013, size = 53, normalized size = 1. \[{1 \left ( -{\it Artanh} \left ({1\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) cx-\sqrt{cx+b}\sqrt{b} \right ) \sqrt{x \left ( cx+b \right ) }{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/2)/x^(5/2),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(c*x^2 + b*x)/x^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238076, size = 1, normalized size = 0.02 \[ \left [\frac{c x^{2} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{2 \, \sqrt{b} x^{2}}, -\frac{c x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{\sqrt{-b} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/2)/x**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228687, size = 51, normalized size = 0.94 \[ c{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{c x + b}}{c x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x^(5/2),x, algorithm="giac")
[Out]