Optimal. Leaf size=64 \[ 2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{a+b x}}{\sqrt{x}} \]
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Rubi [A] time = 0.0485427, antiderivative size = 64, 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 \[ 2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{a+b x}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(3/2)/x^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 8.03064, size = 60, normalized size = 0.94 \[ 2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} - \frac{2 b \sqrt{a + b x}}{\sqrt{x}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)/x**(5/2),x)
[Out]
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Mathematica [A] time = 0.0482045, size = 56, normalized size = 0.88 \[ 2 b^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x} (a+4 b x)}{3 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(3/2)/x^(5/2),x]
[Out]
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Maple [A] time = 0.029, size = 67, normalized size = 1.1 \[ -{\frac{8\,bx+2\,a}{3}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}}+{1{b}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)/x^(5/2),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^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220613, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b^{\frac{3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (4 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}, \frac{2 \,{\left (3 \, \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) -{\left (4 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)/x^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 27.8618, size = 71, normalized size = 1.11 \[ - \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)/x**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)/x^(5/2),x, algorithm="giac")
[Out]