Optimal. Leaf size=40 \[ -\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.0335752, antiderivative size = 40, 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 \[ -\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(3/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 6.2403, size = 37, normalized size = 0.92 \[ - \frac{2}{a \sqrt{x}} - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(3/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0197599, size = 40, normalized size = 1. \[ -\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(3/2)*(a + b*x)),x]
[Out]
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Maple [A] time = 0.012, size = 32, normalized size = 0.8 \[ -2\,{\frac{1}{a\sqrt{x}}}-2\,{\frac{b}{a\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(3/2)/(b*x+a),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)*x^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219406, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 2}{a \sqrt{x}}, \frac{2 \,{\left (\sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 1\right )}}{a \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*x^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.0093, size = 37, normalized size = 0.92 \[ - \frac{2}{a \sqrt{x}} - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(3/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.202624, size = 42, normalized size = 1.05 \[ -\frac{2 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{2}{a \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*x^(3/2)),x, algorithm="giac")
[Out]