Optimal. Leaf size=49 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]
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Rubi [A] time = 0.0701896, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(3/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 8.35686, size = 46, normalized size = 0.94 \[ - \frac{2 A}{a \sqrt{x}} - \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(3/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0517566, size = 49, normalized size = 1. \[ \frac{2 (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{2 A}{a \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(3/2)*(a + b*x)),x]
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Maple [A] time = 0.013, size = 53, normalized size = 1.1 \[ -2\,{\frac{A}{a\sqrt{x}}}-2\,{\frac{Ab}{a\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{B}{\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/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((B*x + A)/((b*x + a)*x^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222826, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (B a - A b\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \, \sqrt{-a b} A}{\sqrt{-a b} a \sqrt{x}}, -\frac{2 \,{\left ({\left (B a - A b\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) + \sqrt{a b} A\right )}}{\sqrt{a b} a \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{3}{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**(3/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.213324, size = 53, normalized size = 1.08 \[ \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{2 \, A}{a \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^(3/2)),x, algorithm="giac")
[Out]