Optimal. Leaf size=79 \[ -\frac{2 (b x+2)^{5/2}}{\sqrt{x}}+\frac{5}{2} b \sqrt{x} (b x+2)^{3/2}+\frac{15}{2} b \sqrt{x} \sqrt{b x+2}+15 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
[Out]
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Rubi [A] time = 0.0555846, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 (b x+2)^{5/2}}{\sqrt{x}}+\frac{5}{2} b \sqrt{x} (b x+2)^{3/2}+\frac{15}{2} b \sqrt{x} \sqrt{b x+2}+15 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(2 + b*x)^(5/2)/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.25842, size = 76, normalized size = 0.96 \[ 15 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{5 b \sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{2} + \frac{15 b \sqrt{x} \sqrt{b x + 2}}{2} - \frac{2 \left (b x + 2\right )^{\frac{5}{2}}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+2)**(5/2)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.049166, size = 56, normalized size = 0.71 \[ \frac{\sqrt{b x+2} \left (b^2 x^2+9 b x-16\right )}{2 \sqrt{x}}+15 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(2 + b*x)^(5/2)/x^(3/2),x]
[Out]
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Maple [A] time = 0.024, size = 81, normalized size = 1. \[{\frac{{b}^{3}{x}^{3}+11\,{b}^{2}{x}^{2}+2\,bx-32}{2}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}}+{\frac{15}{2}\sqrt{b}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+2)^(5/2)/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 + 2)^(5/2)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253828, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) +{\left (b^{2} x^{2} + 9 \, b x - 16\right )} \sqrt{b x + 2} \sqrt{x}}{2 \, x}, \frac{30 \, \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (b^{2} x^{2} + 9 \, b x - 16\right )} \sqrt{b x + 2} \sqrt{x}}{2 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(5/2)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 76.2539, size = 94, normalized size = 1.19 \[ 15 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{b x + 2}} + \frac{11 b^{2} x^{\frac{3}{2}}}{2 \sqrt{b x + 2}} + \frac{b \sqrt{x}}{\sqrt{b x + 2}} - \frac{16}{\sqrt{x} \sqrt{b x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+2)**(5/2)/x**(3/2),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 + 2)^(5/2)/x^(3/2),x, algorithm="giac")
[Out]