Optimal. Leaf size=53 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]
[Out]
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Rubi [A] time = 0.0452411, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(a + b*x),x]
[Out]
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Rubi in Sympy [A] time = 8.7056, size = 49, normalized size = 0.92 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{2 a \sqrt{x}}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0306921, size = 49, normalized size = 0.92 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (b x-3 a)}{3 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(a + b*x),x]
[Out]
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Maple [A] time = 0.009, size = 43, normalized size = 0.8 \[{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{a\sqrt{x}}{{b}^{2}}}+2\,{\frac{{a}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(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(x^(3/2)/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221336, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (b x - 3 \, a\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (b x - 3 \, a\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.45422, size = 49, normalized size = 0.92 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{2 a \sqrt{x}}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.205621, size = 61, normalized size = 1.15 \[ \frac{2 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (b^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(b*x + a),x, algorithm="giac")
[Out]