Optimal. Leaf size=77 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a+b x}}{4 b^2}+\frac{x^{3/2} \sqrt{a+b x}}{2 b} \]
[Out]
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Rubi [A] time = 0.0540794, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{x} \sqrt{a+b x}}{4 b^2}+\frac{x^{3/2} \sqrt{a+b x}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 8.02062, size = 70, normalized size = 0.91 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} - \frac{3 a \sqrt{x} \sqrt{a + b x}}{4 b^{2}} + \frac{x^{\frac{3}{2}} \sqrt{a + b x}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.041313, size = 67, normalized size = 0.87 \[ \frac{3 a^2 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} (2 b x-3 a)}{4 b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/Sqrt[a + b*x],x]
[Out]
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Maple [A] time = 0.007, size = 84, normalized size = 1.1 \[{\frac{1}{2\,b}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}-{\frac{3\,a}{4\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{3\,{a}^{2}}{8}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(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(x^(3/2)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226435, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, b x - 3 \, a\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{5}{2}}}, \frac{3 \, a^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (2 \, b x - 3 \, a\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.3327, size = 100, normalized size = 1.3 \[ - \frac{3 a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 12.7125, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x + a),x, algorithm="giac")
[Out]