Optimal. Leaf size=41 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]
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Rubi [A] time = 0.0367683, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a + b*x]),x]
[Out]
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Rubi in Sympy [A] time = 2.2741, size = 32, normalized size = 0.78 \[ - \frac{\sqrt{a + b x}}{a x} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x+a)**(1/2),x)
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Mathematica [A] time = 0.0230001, size = 41, normalized size = 1. \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{\sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[a + b*x]),x]
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Maple [A] time = 0.005, size = 40, normalized size = 1. \[ 2\,b \left ( -1/2\,{\frac{\sqrt{bx+a}}{axb}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^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(1/(sqrt(b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218636, size = 1, normalized size = 0.02 \[ \left [\frac{b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \, \sqrt{b x + a} \sqrt{a}}{2 \, a^{\frac{3}{2}} x}, -\frac{b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) + \sqrt{b x + a} \sqrt{-a}}{\sqrt{-a} a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*x^2),x, algorithm="fricas")
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Sympy [A] time = 3.45636, size = 44, normalized size = 1.07 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.200883, size = 63, normalized size = 1.54 \[ -\frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x + a} b}{a x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*x^2),x, algorithm="giac")
[Out]