Optimal. Leaf size=63 \[ -\frac{2 (a+b x)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{a+b x}+3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]
[Out]
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Rubi [A] time = 0.0503474, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 (a+b x)^{3/2}}{\sqrt{x}}+3 b \sqrt{x} \sqrt{a+b x}+3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(3/2)/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.92637, size = 60, normalized size = 0.95 \[ 3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )} + 3 b \sqrt{x} \sqrt{a + b x} - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0476551, size = 55, normalized size = 0.87 \[ \frac{\sqrt{a+b x} (b x-2 a)}{\sqrt{x}}+3 a \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(3/2)/x^(3/2),x]
[Out]
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Maple [A] time = 0.045, size = 71, normalized size = 1.1 \[ -{(-bx+2\,a)\sqrt{bx+a}{\frac{1}{\sqrt{x}}}}+{\frac{3\,a}{2}\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)/x^(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((b*x + a)^(3/2)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220659, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \, \sqrt{b x + a}{\left (b x - 2 \, a\right )} \sqrt{x}}{2 \, x}, \frac{3 \, a \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) + \sqrt{b x + a}{\left (b x - 2 \, a\right )} \sqrt{x}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.6099, size = 92, normalized size = 1.46 \[ - \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 + \frac{b x}{a}}} + 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 14.3209, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)/x^(3/2),x, algorithm="giac")
[Out]