Optimal. Leaf size=38 \[ \frac{2}{a \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]
[Out]
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Rubi [A] time = 0.0357687, antiderivative size = 38, 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 \[ \frac{2}{a \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 5.04108, size = 32, normalized size = 0.84 \[ \frac{2}{a \sqrt{a + b x}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0331102, size = 38, normalized size = 1. \[ \frac{2}{a \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*x)^(3/2)),x]
[Out]
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Maple [A] time = 0.012, size = 31, normalized size = 0.8 \[ -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{a\sqrt{bx+a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x+a)^(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(1/((b*x + a)^(3/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230624, size = 1, normalized size = 0.03 \[ \left [\frac{\sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \, \sqrt{a}}{\sqrt{b x + a} a^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) + \sqrt{-a}\right )}}{\sqrt{b x + a} \sqrt{-a} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.84359, size = 146, normalized size = 3.84 \[ \frac{2 a^{3} \sqrt{1 + \frac{b x}{a}}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{3} \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{2} b x \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{2} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.201086, size = 50, normalized size = 1.32 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{2}{\sqrt{b x + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*x),x, algorithm="giac")
[Out]